In my experience, many students in K12 and postsecondary mathematics courses believe that:
As long as students believe in this mythology, it is hard to motivate them to develop quality mathematical practices. In an effort to undercut these misunderstandings and unproductive beliefs about the nature of mathematics, over the past several years I’ve experimented with assignments and activities that purposefully range across the intellectual, behavioral, and emotional psychological domains. In this article, I provide a toolbox of activities for faculty interested in incorporating these or similar interventions in their courses.
Psychological Domains
A useful oversimplification frames the human psyche as a threestranded model:
The intellectual, or cognitive, domain regards knowledge and understanding of concepts. The behavioral, or enactive, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or affective, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning. In postsecondary mathematics courses, our classroom activities and assessments often focus primarily on intellectual knowledge and understanding, with emotional and behavioral aspects of learning addressed either implicitly or not at all. A partial antidote to this is found in the many active learning techniques being used in postsecondary mathematics courses, such as thinkpairshare, “clicker” systems, oneminute papers, inquirybased learning, and service learning, among others. A strength of active learning methods is that they challenge students’ unhelpful beliefs and practices through public dialogue and activities. What active learning techniques might not explicitly do is frame these discussions and activities within a broader context involving the nature of intelligence and the process of successful learning.
A goal for my courses is to incorporate direct interventions that provide students with three things:
The ways in which these interventions are realized in my classes will change over time, and I am willing to follow current educational trends if they are effective tools for my students. Many of the interventions I have used are based on research in psychology regarding mindsets, a topic that I’ve written about previously on this blog. While the literature on mindset research contains contradictory empirical findings, this is not a problem for me since my main goal is to use the language and motivation that this research provides as a tool for engaging students across psychological domains. Mindset research is only one among many possible sources of motivation for meeting the goals above; what is critical is to make sure that my mathematics courses include activities that explicitly promote student development across all three of these psychological domains.
A Toolbox of Interventions
What follows are student assignments and activities that I’ve used in classes ranging from 20student upperlevel courses for math majors to 150student Calculus courses for STEM majors. They have a common purpose of promoting student development in one or both of the emotional or behavioral domains, complementing other work that my students do to develop intellectually in mathematics. An important disclaimer: none of these activities are original with me; rather, these are all adaptations of the work of others, to whom I will always be indebted.
Introductions. On the first day of class each semester, I begin with students introducing themselves to each other. In a small class with less than 3050 students, there is time for everyone to take turns sharing with the entire class their name and the reason they are taking the course. In a largelecture course, I tell students to do the same thing with 46 people sitting next to each other. I teach at the University of Kentucky, and many of our STEM majors are primarily enrolled in large lecture courses during their first year. By beginning every course with a 5minute activity that recognizes the students and promotes discussion, a collaborative tone is set for the remainder of the course, and some of the isolation that students feel (especially as one among many in a large lecture) can be countered.
Day 1, small classes: reading and autobiography assignment. During the first week of class, I assign an article regarding mindset research by Carol Dweck along with a onepage autobiographical essay. I have used Dweck’s articles “The Secret to Raising Smart Kids” and “Is Math a Gift? Beliefs that put females at risk” for this with success. I assign a grade to the essay based on completion only, completely ignoring the quality of the writing, editing, or ideas. The goal is to get students to reflect and be honest, not necessarily to train them to write well. If students respond to the prompt in a relevant manner, they get full credit.
Day 1, large classes: video and small group discussions. In large classes with 150 or more students, especially in courses that are coordinated across sections, the autobiography assignment is harder to implement. Another way to introduce students to the language of mindsets (or other tools) is to have students students watch a 10minute video about mindset research during class on the first day. Following the video, have students spend 23 minutes free response writing about the video. Following the writing, have students spend 23 minutes discussing their response with a neighbor in the class.
Course policy on supportive language. I have a course policy on supportive language that I use in all of my classes: Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. I give students a variety of examples of “banned” phrases and suggested replacements that can be found here. The important aspect of this policy is that it must be enforced — if I hear students making negative comments, I say “course policy” and have them create a neutral rephrasing of their negative selfcomment. This is tougher to implement in large lectures, but even in this context the policy sets a positive tone for the first month of class. In large lectures with accompanying recitations, it is important that graduate student teaching assistants are aware of this policy and enforce it during their recitation sections. It is also important that students know that the policy applies to faculty and teaching assistants as well. I had a student in a large Calculus II lecture call me out for violating this policy last semester when I was frustrated at making errors during an example, and it was an excellent moment for the class.
Video regarding effectiveness of science videos. During class, I have students watch a video about research regarding the effectiveness of science videos. As with the video on the first day of class, students complete a twominute free writing followed by a twominute discussion with their neighbors regarding their response to the video. For many students, a common behavioral practice is that if they are stuck on a math problem, they immediately search the internet for videos that explain how to do this type of problem. This is typically an unproductive behavior, and dedicating some class time to confront it directly sets the stage for further discussions regarding the processes students use for completing homework and solving problems.
Assign an unsolved problem as homework. As I’ve written before, assigning an unsolved math problem as homework can serve as a gateway to discussions about the nature of highlevel mathematical problem solving and the processes, practices, and attitudes that students bring to authentic mathematical challenges. When I assign an unsolved problem, e.g. those given in the article linked to above, I provide students with the following prompt.
This is a famous unsolved problem in mathematics. Work on it for a while — the goal isn’t for you to solve this, but rather to get a feel for the problem. Create an essay by recording your thoughts and attempts as you work. Focus on responding to the following questions: What did you try to do? Why did you try this? What did you discover as a result? Why is this problem challenging? (Seriously, write down everything you’re thinking and every idea you try, even if it doesn’t go anywhere.)
It’s good to grade this problem generously regarding mathematical content, keeping in mind that the goal is for students to be rewarded for demonstrating persistence and good mathematical processes.
Reflective essay about homework. In most of my upperlevel courses, especially those in which I assign an unsolved problem as homework, I have students write a 23page essay explaining what they found most and least challenging in the homework so far, and what their most and least favorite homework problems have been. The prompt can ask them to directly link to mindset or another external topic, or can be left relatively openended to see what connections students make on their own. This can be either graded with a rubric for writing or graded based on completion. The majority of my students have discussed at length their experience working on the unsolved problem, both what they did and how they felt about their work.
Createyourown homework assignment. A recent assignment that I’ve used is to have students write their own homework assignment toward the end of the semester. The specific prompt I used was this:
Create your own homework assignment containing three problems. The homework assignment should be typed. There should be a mix of easy and hard problems that represent a broad spectrum of ideas from the entire course. For each of these problems, type a paragraph explaining why you chose that problem, whether you think it is easy, medium, or hard in difficulty, and what area of the course the problem represents. Once you have created the homework assignment, you should include complete solutions to each of the problems. Your solutions to the problems may be either typed or handwritten, but they should be complete and correct.
It was fascinating to see what the students came up with for their homework. What I found particularly noteworthy was the large number of students who included as one problem a critical analysis essay or short reflective essay similar to what I had assigned in the course to complement mathematical content work. I had honestly expected their assignments to contain a range of standard problems focused on mathematical content, and was pleasantly surprised to see the students incorporating into their homework tasks that addressed behavioral and emotional aspects of doing mathematics.
Endofcourse reflective essay. In my smaller classes, I assign as the final homework assignment the following short essay prompt. The grade is based only on completion, because I want students to write honestly without fear of being penalized for their opinions.
What were six of the most important discoveries or realizations you made in this class? In other words, what are you taking away from this class that you think might stick with you over time and/or influence you in the future? What have you experienced that might have a longterm effect on you intellectually or personally? These can include things you had not realized about mathematics or society, specific homework problems or theorems from the readings, etc. These can be things that made sense to you, or topics where you were confused, points that you agreed/disagreed with in the readings or class discussions, issues that arose while working on your course project, etc. Explain why these six discoveries or realizations are important to you.
I have found that reading through these essays is a fascinating exercise, because of the wide range of messages that the students perceived as being central to the course. Using this assignment consistently over time has helped me improve my ability to create focused courses with clearly defined and communicated learning outcomes.
Final Thought
If you experiment with any of these activities in your own courses, I would love to hear about your experiences!
]]>One common instructional approach during the first two years of undergraduate mathematics in courses such as calculus or differential equations is to teach primarily analytic techniques (procedures) to solve problems and find solutions. In differential equations, for example, this is true whether the course is strictly analytical or focuses on both analytic techniques and qualitative methods for analysis of solutions.
While these analytic techniques play a major part of the early undergraduate mathematics curriculum, there is significant discussion and research about the importance of learning the concepts of mathematics. Many researchers in mathematics education encourage teaching mathematics where students learn the concepts before the procedures and are guided through the process of reinventing traditional procedures themselves (e.g., Heibert, 2013). Additionally, educators who have developed mathematical learning theories often set up a dichotomy between the two kinds of learning (e.g., Skemp, 1975; Haapasalo & Kadijevich, 2000). At the collegiate level, we as professors may agree that these educational ideas hold merit, but also firmly believe that students have a significant amount of content to learn and may not always be able to spend the time necessary to allow students to participate fully in the development of conceptual understanding and the reinvention of the mathematics (including procedures).
However, some researchers, including ourselves, provide evidence that “teaching the procedures to solve problems and find solutions” and “providing ways for teaching concepts first so students will truly understand” can be integrated, and that the notion of learning procedures does not need to be shallow and merely a memorized list (Star, 2005; Hassenbrank & Hodgson, 2007). Our framework to merge these two ways of teaching is titled the Framework for Relational Understanding of Procedures. It was developed as part of Rasmussen and colleagues’ work in differential equations teaching and learning (Rasmussen et. al., 2006). Skemp coined the original definition; she defines relational understanding as “knowing both what to do and why” and contrasts it to instrumental understanding as “rules without reason” (1976, p. 21).
Following, we describe the six components of the Framework for Relational Understanding of Procedures. The idea is that each category can be used to consider and enhance students’ learning as they study a procedure. For each one, we provide a brief explanation, questions about student thinking, and an example of an exam question related to each component taken from our work in differential equations. Likely, each instructor could add other algorithms in differential equations as well as other courses.
Components of Relational Understanding of Procedures
Student can anticipate the outcome of carrying out the procedure without actually having to do so and they can anticipate the relationship of the expected outcome to outcomes from other procedures.
This component suggests that a student understands what kind of solution would be expected before solving. A student might need to consider the following: Is the solution going to be a number, or a function? When is the solution one or two functions? Are there different forms to show the answer? How do the answers compare to other answers from similar procedures?
Example:
Suppose that a differential equation can be solved with either separation of variables or with a general technique for solving first order linear differential equations. Let \(y_{sep}(t)\) be the solution for an initial value problem using separation of variables, and let \(y_{lin}(t)\) be the solution for the same initial value problem using the technique for linear differential equations. Which of the following statements correctly states the relationship between \(y_{sep}(t)\) and \(y_{lin}(t)\)?
Student can identify when it is appropriate to use a specific procedure.
Students often can do the procedure when they know that is what is needed. However, they often are unable to decide before they start which procedure is needed. Ultimately, one reason that this is an issue is because of the structure of typical textbooks (e.g., the homework always matches the section). How many of you have had students say, “I could do all the problems in the homework, but then I didn’t know what to do for the exam”?
Example:
Circle all that apply. A differential equation can be solved with the technique for first order linear ODEs if:
Student can correctly carry out the entire procedure or a selected step in the procedure.
This is what we typically think of as doing a problem, or performing the framework. Can the student do the steps necessary to complete a problem correctly? Can the student analyze where they are in the procedure and know what to do next?
Example:
A student is solving a first order linear differential equation and at some point in her solution process she correctly gets the expression to \(e^{2y}\left(\frac{dy}{dt}+2y\right)\). This expression is equivalent to which of the following?
a) \((e^{2t}y)’\)
b) \((e^{3t}y)’\)
c) \(e^{3t}y’\)
d) \(e^{2t}y’\)
Student understands the reasons why a procedure works overall. Additionally, student knows the motivation or rationale for key steps in the procedure.
This step fundamentally involves the conceptual idea behind the procedure. As instructors, we make efforts to teach these ideas in our classes on a regular basis. However, are we concerned about how the students grow to understand the “why” of the procedure? Do the reasons for the steps play a part in the students’ solving? Can the students go back and make modifications because they understand what is really happening?
Example:
Which of the following would be a justification for one or more of the steps needed to solve a first order linear differential equations? Circle all that apply.
Student can symbolically or graphically verify the correctness or reasonableness of a purported outcome to a procedure without repeating the procedure.
This component is about thinking through the answer in a way that you can decide if it makes sense. Our experience says that if you ask students to check for the reasonableness, they often just repeat the procedure, and this indicates a need to push for the bigger picture of making sense of a solution beyond just doing. Showing competence in this component might involve either checking in terms of seeing if the solution works, or using a graphical or numerical technique to see if the two solutions are compatible. Can the student find a way to check for correctness? Can the student decide if answers are reasonable?
Example:
Joey is solving an autonomous differential equation of the form \(\frac{dy}{dt}=f(x)\), using separation of variables to find the general solution. At one point in his solution process he correctly gets \(e^x=t^2+c \) . His final answer is then \(x=ln(t^2)+c \). We can verify that Joey’s final answer is:
a) Correct because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).
b) Correct because \(x=ln(t^2)+c \) says that graphs of solution are shifts of each other along the x axis (that is, they are vertical shifts of each other).
c) Incorrect because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).
d) Incorrect because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the x axis (that is, they are vertical shifts of each other).
e) Incorrect because \(e^x\) is always positive.
Student can make connections within and across representations involved in the problem and solution: symbolic, graphical, and numerical.
Educational literature suggests that one way to demonstrate deep understanding is to make connections among representations. Traditionally, in upper level mathematics, the representations are often symbolic, but in differential equations, linear algebra, and other freshman and sophomore classes, there are several representations, and students who can be flexible and move among them have better understanding.
Example:
Jung Hee uses a slope field to determine the long term behavior (that is what happens as \(t \to \infty \)) of the solution to the initial value problem \(\frac{dy}{dt}=0.4y(70y) \). Which of the following methods could be used to corroborate the long term behavior she found by using the slope field? Circle all that apply.
Conclusion
The framework described here and the examples from an assessment developed for relational understanding (Keene, Glass, Kim, 2011) may offer some ways to think about teaching procedures that are the foundation of many of the early undergraduate mathematics class. It may not be a matter of trying to teach the procedures or the concepts (as a dichotomy) but of developing a relational understanding of the procedures so that students can not only find answers, but also understand the underpinnings and development of the procedures. We believe that if students have this relational understanding, not only will they perform better in their classes, they will retain the skills and understandings over periods of time. This will result in students doing better in all their mathematics classes.
We would like to acknowledge Dr. Chris Rasmussen for his contributions to the work.
References
Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. Journal für MathematikDidaktik, 21(2), 139157.
Hassenbrank, J. & Hodgson., T. (2007). A framework for developing algebraic understanding & procedural skill: An initial assessment. In Proceedings of Research in Undergraduate Mathematics Annual Conference.
Hiebert, J. (2013). Conceptual and procedural knowledge: The case of mathematics. Routledge.
Keene, K. A., Glass, M. & Kim, J. H. (2011). Identifying and assessing relational understanding in ordinary differential equations. In Proceedings of the 41st Annual Frontiers in Education Conference, Rapid City, SD.
Rasmussen, C., Kwon, O., Allen, K., Marrongelle, K. & Burtch, M. (2006). Capitalizing on advances in K12 mathematics education in undergraduate mathematics: An inquiryoriented approach to differential equations. Asia Pacific Education Review, 7, 8593.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 2026.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404415.
A good educator must facilitate learning for a classroom full of students with different attitudes, personalities, and backgrounds. But how? This question was the starting point for a new Faculty Teaching Seminar in the math and statistics department at Sam Houston State University. In the conversation that transpired, we looked to identify the most important components of creating a class culture that best enables us to achieve learning outcomes. What are our goals? How do we get the ball rolling each semester? How do we get our students on board? Read on to find out…
What is a “classroom culture”? What are you after? Why is it important?
Taylor: The environment in my classroom is a necessary component of a successful semester. The rapport that I build with my students, the tone of the class, and the ways that my students interact with each other are just as much a component of learning as the lectures or textbook.
In my experience, setting up a productive class culture can determine the potential for learning for the entire semester. A productive class culture is one where the students feel supported, protected, and valued.
Ken: I seek a “learning community” of studentscholars, people who are curious about mathematics and serious about learning. I want calculus students who are proud to be taking calculus. I want an upper level mathematics class where the students see themselves as professionals. I want a graduate class where students focus on exploration of mathematics and its mysteries, and where curiosity is the driving reason for study.
I don’t distribute the class syllabus as a hard copy. I collect work every class period and “speedgrade” it to return it the next day. I work with a department secretary to force late registration students to meet with me before adding my classes. I never have office hours before class. When challenged by colleagues about some of these unusual practices, I realized that I desire a certain type of classroom environment. I push, coach, and manipulate my students to achieve that environment.
What do you do on Day 1 to create a classroom culture?
Ken: My class culture begins with my syllabus, which lays out some “professional” expectations of my students. But I also begin, from Day 1, to set the stage for class expectations. Since much of the material I provide will be online (either via Blackboard or Google Drive), the syllabus is also available there and I do not hand out a hard copy. There will not be handouts during the semester; let’s get the students used to this on the first day!
In classes with a prerequisite, I give a quiz the first day. The intended message is, “We are serious about learning and are on the move!” Early in the semester I keep the class at a fairly brisk pace (emphasizing a steady regime of study) and I make sure to model this on day 1. Since many firstyear students view office hours before class as an invitation to procrastinate, my office hours are not before class, but afterwards!
I never dismiss class early, not even on the first day.
Taylor: The answer to this question depends on what level the class is and what method of teaching I am using in the class, but there are some common themes in all of my classes on Day 1:
Does classroom culture vary by class level?
Taylor: Yes, absolutely. I usually focus on one or two aspects of a successful class culture and hone in on developing those aspects. In a Calculus class, for example, I most want the students to learn to justify their thought processes. To achieve this, I will ask them to buddy up every day – literally push their desk next to someone else’s. I tell them, “Turn to you partner and ask, `Why is it true that…?’” I’ll then solicit feedback in a way that supports their collaboration by asking a student, “What justification did you and your partner come up with?”
In an Inquiry Based Learning class, I most want students to value productive failure as an integral part of the learning process.
I will then carefully praise mistakes and encourage participation from students who know they are wrong. In the photo, you see my IBL Algebra students writing proofs on the board; I have them visit each other’s work and circle anything they don’t agree with. Since we have a safe space where it’s ok to be wrong, my students are professional but thorough when it comes to correcting mathematical errors.
Ken: Yes, certainly this varies by level. At the lower level my expectations are typically overly optimistic. I don’t abandon them, but I recognize that students have been trained to focus on grades and testing. At the graduate level a classroom culture can be relatively easy to create, particularly if the students are already in a cohort and beginning to form a community.
The emphasis on a classroom environment is even important at the grade school level – see this article by Yackel and Cobb on creating a productive classroom environment in second and third grade!
What about students who don’t buy in? How do you create/enforce “buy in” of your culture?
Ken: Some students, in first or secondyear classes, don’t buy in to the steady stream of new material, and the necessary consistent study discipline. I routinely remind everyone of the expectations, and I attempt to motivate these expectations, in the same way that the coach of an athletic team might create team pride. For those students clearly not keeping up, I eventually chat with them briefly about the fact that this class is probably not for them. I encourage these students to either catch up quickly or find a more constructive use of their time. (There is an art to this. I often write an email to a poorlyperforming student in which I express concerns about the progress and suggest some constructive alternatives that include starting fresh in the course next semester. I write these emails with a view to Mom reading over the student’s shoulder!)
At every level there is a fair amount of coaching. “Here is where we are going! Here is what we are trying to achieve! Look how far you’ve come!” I’ve coached competitive youth soccer teams and the speeches are similar. “You are working hard to reach this level! Keep it up! Here is our game plan for today…”
Taylor: I want all my students to take charge of their own education, so I will let a challenging student make his or her own decisions on how to participate in class, as long as the behavior isn’t disruptive. I may gently remind that student that I would prefer her or him to be fully engaged. In general, though, I think that if your class culture is based on a genuine desire to facilitate learning, students recognize and value the effort.
What are pitfalls, mistakes, disasters?
Taylor: A few semesters ago, I had a mutinous Calculus class. Somehow, I encouraged so much communication and collaboration among my students outside of class that a vocal minority opposition sprung up from within the class. I later discovered that there were students campaigning for the class to give me bad course evaluations (which happened). My feelings were hurt for a bit, but I learned valuable lessons that semester. I had been uncompromising in my desire for them to ask themselves “Why?” and this group wasn’t academically ready to do that. I now pay more attention to differentiating instruction, for example when a student asks a question in class.
Ken: My goal is a community of students all going in the same direction. If just one or two students are not swimming with the rest, the general flow of students will often pull them in to the current. But if a significant minority resist the direction of the class then things can go bad quickly. I must keep up with class morale and make sure that the program is flowing (somewhat).
Long ago, in an abstract algebra class where students were supposed to do small projects without discussing their work with others, I uncovered a collaborative ring that included a majority of the class. The students had ignored my published restrictions on collaboration. Rather than punish over half the class for this “plagiarism”, I backed up and restarted the process, admitting that I had not been sufficiently aware of the stress my problems generated. (The memory of that class is still a bit painful.)
In summary, effective learning occurs in a class environment in which curiosity, exploration and even mistakes are part of the norm. We seek to create that culture even before the first class day!
What do other teachers do to facilitate this? We would like to know!
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While one important component of successful teaching and learning is what happens inside the classroom, an equally important component involves decisions made at the administrative level that impact our classroom environments. A challenge that mathematics departments face is to make successful arguments for resources that support highquality programs and courses for our students. Such arguments are often bolstered when the activities of a department are placed within the context of recommendations from professional societies.
In this article we survey a selection of recent reports and recommendations related to courses in the first two years of college study, with the goal of providing an overview of these reports for faculty and department leaders. It is worth noting that most of these were created with grant support from the National Science Foundation (NSF). There are at least seventeen professional societies involved in mathematics education efforts, of which six are represented in these reports: American Mathematical Society (AMS), Mathematical Association of America (MAA), American Statistical Association (ASA), Society for Industrial and Applied Mathematics (SIAM), American Mathematical Association of TwoYear Colleges (AMATYC), and National Council of Teachers of Mathematics (NCTM).
A Common Vision for Undergraduate Mathematical Science Programs in 2025 (AMS, MAA, ASA, SIAM, AMATYC)
The MAA Common Vision project brought together leaders from the AMS, MAA, ASA, SIAM, and AMATYC to collectively reconsider undergraduate curricula and ways to improve education in the mathematical sciences. This was the first time that these five professional societies had engaged in a joint project regarding postsecondary mathematics education, reflecting the current emphasis in the mathematical community on developing coherent responses to the challenges we face across all types of institutions. Project participants represented not only these mathematical sciences associations, but also partner STEM disciplines, higher education advocacy organizations, and industry. The resulting report includes an indepth examination of seven curricular guides published by these five associations, with the primary goal of identifying common themes in the guides. The report reflects a synthesis of these themes with other research and input from project participants and other thought leaders in the mathematical sciences community.
Some of the prominent common themes from these seven curricular guides identified by the report are:
From twoyear colleges to researchfocused universities, from the context of teaching STEMfocused students to engaging students struggling with basic quantitative literacy, it is important to have in mind the common challenges of teaching and learning mathematics. While there are certainly unique challenges for different teaching environments and student populations, the Common Vision report helps identify the ways in which we can provide a coherent response to all these challenges, given a solid level of support and adequate resources. The full 2015 Common Vision report is available at http://www.maa.org/programs/facultyanddepartments/commonvision.
Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra (MAA)
College Algebra is one of the courses that plays a role across the mathematical sciences. Between 1999 and 2011, the MAA Committee for the Undergraduate Program in Mathematics conducted a series of NSFfunded activities as part of their CRAFTY project, i.e., Curriculum Renewal across the First Two Years. In the first phase of this project, a series of workshops involving mathematicians and faculty from partner disciplines were organized to identify desirable student learning outcomes for mathematics courses. In the second phase of this project, the focus narrowed to developing guidelines for College Algebra based on these workshops. Of particular note is the coherence between these guidelines and the main themes from the Common Vision report, especially with regard to:
The reports from CRAFTY are of interest to faculty involved in revising their college algebra courses, as well as to faculty who are searching for a starting point for discussions with faculty in partner disciplines. Further, the CRAFTY reports contain chapters detailing the experiences of various institutions through modelingbased revisions to their college algebra courses, including successes and failures. These reports, including guidelines for College Algebra courses, can be found at the CRAFTY website.
Characteristics of Successful Programs in College Calculus (MAA) and the MAA/NCTM Joint Position Statement on Calculus
While College Algebra is important across all partner disciplines, in STEM disciplines it is Calculus courses that play a central role. Beginning in 2009, with support from the NSF, the MAA has undertaken studies regarding the state of college Calculus. The first phase of this project consisted of a national survey of Calculus students and instructors followed by site visits to programs identified as effective based on the survey results. The second phase of this project began in 2015 and will broaden its scope to include Precalculus through Calculus II. The primary result of the first phase of the Calculus study was the identification of the following seven characteristics of successful calculus programs:
The motivation for an ongoing study of Calculus in the United States is that despite its centrality in the postsecondary mathematics curriculum, Calculus instruction is in a state of crisis. In 2012 the MAA and NCTM released a joint position statement on Calculus, including a background document motivating the position statement in which the authors conclude:
The United States has fallen into a seriously dysfunctional system for preparing students for careers in science and engineering, guaranteeing that all but the very best [students] rush through essential parts of the mathematics curriculum [in high school] and then are forced to sit and spin their wheels while they try to compensate for what was missed.
University Calculus courses are taught in a complicated broader context involving high school Calculus courses, the AP exam system, the rapid increase of dual and concurrent enrollment programs, and other factors, significantly complicating postsecondary Calculus instruction. For departments that are interested in rethinking their Calculus courses, these resources can help clarify our conversations and provide refined focal points for improvement.
Articles and reports from the MAA Calculus Study can be found at http://www.maa.org/programs/facultyanddepartments/curriculumdevelopmentresources/nationalstudiescollegecalculus. The MAA/NCTM joint position statement on Calculus, including a background document with more information, can be found at http://www.nctm.org/StandardsandPositions/PositionStatements/Calculus/.
Other Resources
There are several other reports that are worthy of attention from faculty and department leadership, a few of which are briefly discussed here. The NRC Mathematical Sciences in 2025 report is a comprehensive review of the mathematical sciences, including a vision for the future over the next decade. The NRC report was completed at the request of the NSF, and includes information about a wide range of topics in the mathematical sciences, including education and diversity. The MAA CUPM Curriculum Guides from 2004 and 2015 are discussed in the Common Vision report, and together form a substantial set of recommendations for courses, departments, and programs. The other curricular guidelines discussed in the Common Vision report are each worth serious consideration beyond the summaries given in Common Vision. For mathematics departments that teach large numbers of preservice K12 teachers, the CBMS Mathematical Education of Teachers II from 2012 and the NCTM Principles to Actions report are important and informative.
Regardless of the specific focus of an individual department or institution, framing our activities in a broader context and making use of resources produced by the professional societies can significantly strengthen our arguments in favor of increased support and resources for our mission of teaching and learning. These resources also serve to inform and inspire us as we revise our courses and programs.
]]>Many college and university students do volunteer work in local communities, and can learn valuable lessons in the process. The term “service learning” refers more specifically to service activities that are integral parts of academic courses. It can sometimes be difficult for mathematicians to envision how such projects could be included in their courses, especially courses focused on “pure” topics; for example, I have difficulty imagining how one would include such activities in Abstract Algebra. I have found myself, however, teaching courses in which service learning made sense, and I’ve implemented some servicelearning projects with varying outcomes. Below I share some lessons I’ve learned in the process.
First, though, I offer some context. Here at Middlebury College, every entering student takes a writingintensive firstyear seminar (FYS), and every department contributes to the FYS program. For my most recent seminars, I’ve taught “Mathematics for All,” which explores questions of equity in K12 mathematics education. Students develop their writing and reasoning skills by, for example, comparing contemporary critiques of mathematics education, and examining what is meant by “highstakes” testing. Each version of the course has focused on a different age group, and the students did projects at local schools. These seminars are capped at fifteen students, and the instructor is the students’ academic advisors until they choose their majors, which might not happen until sophomore year.
The other relevant course is “Mathematics for Teachers,” a math content course for aspiring educators. I offer it jointly with the Education Studies department; the aims are for students to strengthen their understanding of fundamental mathematics concepts, grow as mathematical thinkers, and gain appreciation for the complexities of teaching math to children. That last aim was the motivation for getting students into classrooms. This course has had a more varied audience, including firstyears through seniors, only some of whom are firmly committed to teaching after graduation, and from 15 through 30 students.
Here are my notes to self about service learning:
Include a servicelearning project only if it supports the learning goals for your course. This may seem obvious, but wellmeaning administrators with the worthy goal of community engagement might conflate “life lessons” with the intellectual development for which you are responsible. Be sure that your own goals for your students are primary.
In the first of my FYS projects, students learned about the statewide school assessment program then in place, and wrote a brochure about it for parents of elementaryschool children. A more recent cohort learned about the nature and importance of math acquisition for preschool children, and then designed and played math games with children in the local Head Start classroom. Next, they wrote a report about their activities for a college committee that was reviewing community engagement efforts. In both cases, I felt that the writingtolearn objective of the FYS program had been met, along with the content goals concerning testing and earlychildhood learning, respectively.
Get help from people on campus. Those enthusiastic administrators in your Office of Community Engagement (or Campus Compact liaison) may have lists of potential community partners, and may even have some funds to cover expenses, from van rentals for site visits to cards for thankyou notes to partners. They can also connect you with colleagues in other departments who have tried projects.
Make sure you have clear communication with community partners ahead of time about what you and they expect. Remember the brochure my students assembled? We ceremoniously delivered a couple of boxes to the school principal at a nice dinner on campus, which also included the teachers whose classes we’d visited. Only later did I learn that the principal never distributed the brochures. What we thought would be helpful – an accurate Q. and A. list, in plain language, about the testing program – didn’t serve the principal’s needs. In the future, I will make sure that at a minimum, expectations and needs for both my students and our community partners are laid out in writing. Bringing the students into that conversation is particularly helpful.
Invite your community partner to visit your class ahead of the project. Before the visit, have your students prepare some questions. Having learned from my first experience, I had the principal of the second school come to my “Math for Teachers” class. She emphasized, in a way that I couldn’t, the importance of maintaining confidentiality, especially given that some of her students were children of college faculty and staff. She also was explicit about the kinds of conversations she wanted my students to have with hers: “Ask them to explain their strategies.” A view into a fourthgrader’s mathematical thinking was exactly what I hoped my students would get.
When the Head Start teachers came to my latest FYS class, they too offered valuable background. One had been a Head Start parent before she went back to school in early childhood education, a life quite different from most of my students’. They emphasized the importance of play in their room, and exuded a love for their charges that was contagious. Most notable to my students was the advice to get down on the kids’ level rather than stand around, and I saw that happen as soon as we entered their classroom. The children engaged with my students immediately, to everyone’s delight.
Be prepared to invest a lot of time in logistics. Scheduling site visits and making sure students get to them can be a challenge. It didn’t make sense for me to send more than three students at a time to a classroom, and schools have frequent alterations to their schedules, for example, so my plans went through many drafts. Consultation with my colleagues in Education Studies was invaluable here, but even so, there were a lot of moving parts to monitor.
Invite your community partner to come to class once the project is complete. This doesn’t have to last for a full class period, but it can be valuable. For example, one group of my students had engaged the preschoolers in a game in the gym, and were a bit discouraged with how it had gone. The teachers, however, said, “You got them to stand on that line – that’s an accomplishment!” My students got a little more insight into the world of three to fiveyearolds, and a bonus lesson in setting realistic expectations.
Have your students reflect on their experiences. Allow time for the class to debrief, and, where appropriate, assign writing that requires them to integrate what they’ve learned in the field with the rest of the course. Cognitive science has identified the importance of metacognition to the learning process (see, for example, How People Learn), and it’s also useful to read students’ reflections to inform your next project. This was my notsosecret agenda in having students write about the Head Start project for the committee; their report included a brief description of the readings and discussions that preceded the project, and explained how working with the children had built on those readings. It also made it clear to me that should I do the project again, I will need to help my students finetune their games, given the developmental range between an immature threeyearold and an almostsixyearold.
Read about other mathematicians and statisticians who have done service projects. For much more information and advice, I highly recommend Mathematics in Service to the Community, edited by Charles Hadlock (also the author of the lovely Field Theory and its Classical Problems). There you’ll find a chapter on servicelearning in mathematical modeling courses, with case studies such as “The Baltimore City Fire Department Staffing Problem,” as well as chapters for instructors of statistics and education courses, and a detailed “HowTo” chapter. There is also a special Primus issue on the topic.
As a selfdescribed skeptic, would I include a servicelearning component in a future course? At this point, I would say yes for the firstyear seminar, but no for the Mathematics for Teachers course. In the latter, the semester is already too short to meet my mathematics learning objectives, and students can go into local school classrooms as part of Education Studies “methods” courses. In the seminar, though, if I want my students to hone their analytical skills by asking what an equitable mathematical educational system would look like, the benefits of experience at the children’s eye level are well worth the trouble. The added benefits to my own learning have been a welcome bonus.
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At the 2016 Joint Mathematics Meetings in Seattle this past January, an unusual mix of mathematicians and mathematics educators gathered for an AMS special session on Essential Mathematical Structures and Practices in K12 Mathematics. This was the fourth consecutive special session at JMM organized by Bill McCallum and other folks at Illustrative Mathematics that focused on work in mathematics of mutual concern to mathematicians, mathematics educators, and K12 teachers. The theme this year was inspired by a conversation between Dick Stanley and Kristin Umland about ratios and proportional relationships, and the talks were selected and ordered to highlight the development of mathematical ideas that are both upstream and downstream of this terrain.
Academic mathematicians are able to describe mathematical ideas in an efficient way. Across specialties, they share tools of language and habits of communication that have been shaped in order to facilitate the exchange of abstract knowledge. One purpose of the special session was to apply this cultural skill to selected topics in K12 mathematics. The participants sought to create clearly expressed and easily understood descriptions of topics that are rarely developed clearly in the K12 curriculum, such as measurement, number systems, proportional relationships, and linear and exponential functions. Although many people have been working in this area in recent years, much more needs to be done.
The mathematical community can—and should—contribute to the mathematics of the K12 curriculum by applying to it the same principles of logical clarity that are used in the best expositions at the college level and beyond, so that teachers can have the best possible curricular materials and support. The coming of the Common Core State Standards in Mathematics has created a new opportunity for positive change in all subjects. Still, old habits change very slowly, and the path in this new direction is going to need many more signposts. This means that it is more important than ever for mathematicians to partner with teachers and mathematics educators to lay out a mathematically coherent path that fills in the outline of the standards.
To consider an example for which this is especially true, let’s take proportionality, the subject of two of the talks (Madden and Umland) at the special session. Mathematicians who look in detail at sources on this subject written over the past, say, 30 years will find a disconcerting jumble. Much of what is written is a throwback to procedures that were practiced in the middleages—neither wrong, nor useless, but out of sync with contemporary mathematical practice. Some things are needlessly obscure, ambiguous, confusing. It is easy to identify the difficulty here. Traditional treatments of “ratio and proportion” in school mathematics are rooted in a tradition that goes back to Euclid and entered the school curriculum in the Middle Ages. They are based on the concept of two equivalent ratios relating four fixed quantities.
There is no pathway through the traditional curriculum formulated on the basic idea of the modern understanding of proportionality: One (changing) quantity is proportional to another if their ratio is always the same. To understand why this is so, one must understand the different definitions for a ratio implicitly used by different people. In the traditional curriculum, people talk of a ratio \(a/b\) as a comparison between quantities \(a\) and \(b\), by which they mean that when making a multiplicative comparison between \(a\) and \(b\), the scale factor is \(a/b\) (or \(b/a\) depending on which direction one wishes to compare). So “the same ratio” in this context means “the same scale factor in a multiplicative comparison of two quantities of the same type,” and when people speak of a ratio, they sometimes mean the whole notion of a multiplicative comparison, and they sometimes mean just the scale factor.
In modern times, we have expanded and abstracted the definition of a ratio. We talk of a recipe as a ratio and we are not limited to two ingredients, and ratio equivalence is characterized by multiplying all quantities by the same scale factor rather than taking their pairwise quotients. In addition, we have extended our conception of a ratio to include quantities of different types, like distance and time. Formally, the traditional definition of a ratio \(a \mathbin{:} b\) is limited to quantities of the same type (like length and length) and by definition the ratios \(a \mathbin{:} b\) and \(c \mathbin{:} d\) are equivalent if and only if \(a/b = c/d\). The modern definition does not restrict us to quantities of the same type, and equivalence classes are characterized as \[\{sa \mathbin{:} sb \mid \text{for all}\ s > 0 \in {\mathbb R}\}\] it follows from this definition that ratios of two quantities are equivalent if and only if their related quotients are equal. This extension of the idea of a ratio is what allows us to define proportionality as we do in modern times: two variable quantities (i.e. quantities that take values from a specified subset of the real numbers) \(x\) and \(y\) that are related to one another in such a way that the ratios \(x \mathbin{:} y\) (for all values of \(x\) and \(y\)) are always equivalent, from which it follows that \(y/x\) is constant. This may be expressed in the familiar form \(y = kx\).
Unfortunately, the classical/medieval treatments of proportionality do not lead to this view. In fact, in many traditional treatments, the phrase “proportional to” does not appear at all. (Just as, in many traditional treatments of “ratio”, the idea that equivalent ratios have equal quotients is not mentioned.) What is unfortunate is that many textbooks do not go beyond the classical/medieval paradigm, and many teachers are unaware of the need to do so. An earlier post showed in a striking way that many teachers and others in the mathematics education community are not able to bridge that gap and demonstrate a conceptual understanding of proportional relationships. This is true even though they could readily solve procedural problems involving four quantities that were proportional.
What is surprising (and this was the point of the earlier post) is the “learning curve” that separates the classical/medieval paradigm from the modern formulation. Teaching experience shows that going from one perspective to the other is not a natural transition, but a discontinuous conceptual change. Many people simply do not think in terms of variables. In the meantime, those who have acquired the ability to do so regard it as easy and natural—so much so that they appear to have as much difficulty understanding how anyone could fail to grasp it, as those who do not grasp it have in acquiring it.
There is another approach to proportionality that aims to fix this problem, outlined in the Common Core State Standards. Still, the key idea related to variable quantities is very brief and easy to miss. (Four lines in 7.RP 2c, page 48.) In fact, this is the only place where the phrase “proportional to” occurs in the treatment of proportionality in the standards. So it is easy for people to fall back on the old habit of looking at the subject in a static way in terms of four numbers that form two equivalent ratios.
As a result, the old habits of the traditional ratio and proportion curriculum are still firmly entrenched. It will take a coordinated effort by mathematicians and mathematics educators working together to bring the middle school treatment of proportionality into the modern era. We need not jettison the “Rule of Three” (the old manner of reasoning with two four quantities in two equivalent ratios), but the curriculum needs a clear pathway joining this to the \(y=kx\) paradigm. This work is not so different from something mathematicians do all the time: Find the right definitions that lead to the best explanations and descriptions of mathematical phenomena.
We should make sure that ideas are accompanied by gradelevel appropriate definitions, so students and teachers have something reliable to refer to when they have questions or doubts. We should provide gradelevel appropriate explanations of where formulas and other results come from. We should develop ideas in a way that leads smoothly to subsequent topics and supports them. No one wants to lead students into computational dead ends. In short, this means considering what we often call “grade school mathematics” as a legitimate part of the field of mathematics and treating it as such.
This will not be easy. The culture and practices of school mathematics are not familiar to most mathematicians, but understanding them is essential for success. Moreover, for the most effective work to be done, mathematicians need to get together and talk to each other and to mathematics educators in detail about how we can most fruitfully participate in efforts to improve this situation. Although there has been much work by educational researchers on ratios and proportions (see here and here for instance), we have not yet come across any that works with the modern view of proportional relationships of varying quantities, instead of solving equations about equivalent ratios.
The AMS Special Session at the Joint Meetings is one example of the work that mathematicians are doing related to school mathematics. Several of the speakers are also involved in curriculum development efforts and teacher professional development, as are mathematicians at many institutions across the US. MSRI sponsors Math Circles, and AIM sponsors Math Teachers’ Circles and both have online resources to help mathematicians who are interested to get started. Jason Zimba’s recent article in the Notices about the Common Core also has many suggestions, including especially the final section, “What Mathematicians Can Do”.
To summarize, there is a lot more work to be done. Will you join us?
]]>As statisticians in mathematics departments, we have both spent many department meetings, departmental reviews, and watercooler conversations discussing the merits of various different curricular decisions with respect to the calculus sequence (“Why not take linear algebra before calculus III??”), upper division electives (“But those classes are needed for graduate school!”), and number and order of courses required for the mathematics major/minor. Recently, more of those discussions have related to critical components of the statistics curriculum, and how courses from mathematics ensure that statistics students have a solid quantitative foundation. These kinds of conversations reinforce the fact that there are strong connections between mathematics and statistics, and these connections can and do affect decisions about undergraduate curricula.
More generally, this is an exciting time to be in a quantitative field. The amount of data available is staggering and there is no end to the need for models that harness the deluge of information presented to us every day. Mathematicians, Statisticians, Data Scientists, and Computer Scientists will all play substantial roles in moving quantitative ideas forward in a new data driven age. To be clear, there are challenges as well as opportunities in what lies ahead, and how we move forward – particularly with respect to training the next generation of mathematical, statistical, and computational scientists – requires deep and careful thought.
The goal of this blog post is to share some of the recent pedagogical ideas in statistics with our mathematician colleagues with whom we – as statisticians – are intimately engaged in building curricula. We hope that the description of the recent developments will open up larger conversations about modernizing both statistics and mathematics curricula. Many of the ideas below on engaging students in and out of the classroom, connecting courses in sequence or in parallel, and assessing new programs are relevant to all of us as we work to better our own classrooms.
We spent 18 months as part of a committee whose purpose was to revise the American Statistical Association’s Undergraduate Curriculum Guidelines (posted here). These new recommendations provide a flexible structure to ensure that students receive the necessary background and critical and problem solving skills to thrive in our increasingly datacentric world. Through our work on the undergraduate statistics guidelines committee, we were excited by many interesting and innovative ideas our colleagues were implementing in and about their own classrooms. This led us to coedit a special issue of The American Statistician on Statistics and the Undergraduate Curriculum (December, 2015, including a guest editorial). In this blog post, we briefly describe several of the articles of this special issue, with the goal of familiarizing the readers with some of the issues and innovations that statisticians have implemented in their undergraduate classrooms. Many mathematicians teach introductory and advanced statistics courses, and we believe they have a vested interest in what statisticians can and should know mathematically. Additionally, they are likely interested in additional reflections on and ideas for their own classes. Our hope is that the special issue of TAS can be a valuable resource for those interested in statistics and the mathematical sciences at the undergraduate level.
The issue brought together a set of articles designed to help undergraduate statistics curricula be forward thinking. We begin with an editorial that includes a list of key papers discussing statistics at the undergraduate level. George Cobb provides a particularly provocative article encouraging all of us to “tear down” current curricula and start over. His piece is accompanied by 19 responses from renowned statisticians and education experts across the world. A link to these responses and George’s spirited rejoinder can be found here.
A number of the articles in the issue work to answer the question: “Do our bachelor’s graduates have the needed skills to compute with data?” Chamandy et al. describe problems they have encountered at Google that required sophisticated understanding of theoretical statistics. They describe excellent case studies for advanced undergraduate statistics students, by demonstrating problem solving in context. In another paper, Nolan and Temple Lang report on their summer program, “Explorations in Statistics Research”, which exposed six summer cohorts of students to the process of posing statistical questions and solving real world industry problems. In his paper, Grimshaw provides a framework for adding real data into a course with metrics arranged on two axes: data source and data management. On each axis the data is considered to be good/better/best for developing student skills of computing with data. Each of these responses provides practical examples of pedagogical innovations that statisticians have developed to help their students become more adept at working with real data in more realistic settings.
On the curricular side, many of the articles provide structure for a specific class designed to modernize the curriculum. For example, Hardin et al. and Baumer discuss different approaches to implementing a data science / computational statistics course in the modern era. Their articles provide templates for structuring the course to integrate data science with statistics while simultaneously allowing students opportunities to practice communicating results to a larger audience. Green and Blankenship provide an updated datafocused approach to the traditional Statistical Theory course. Their paper provides multiple great examples for making both Probability and Statistical Theory courses more modern and more interactive. Blades et al. and Khachatryan describe second courses (after introductory statistics), which are well suited to the goal of building students’ skills at making decisions based on data. Blades et al. discuss using design of experiments as a second course in statistics, which can follow introductory statistics combined with any level of mathematics. Khachatryan brings case studies into a time series course to help students engage with the real world connections.
The issue closes with a pair of articles (Chance & Peck and Moore & Kaplan) that address the curriculum from the assessment perspective, providing a framework for basing programmatic decisions. The set of articles will be of particular interest to readers who seek ideas for program evaluation, new approaches to teaching statistics, and to incorporate some new ideas into their existing courses and programs.
We hope that The American Statistician special issue and the extended bibliography of papers in the guest editorial as well as the online discussion provide useful fodder for further review, assessment, and continuous improvement of the undergraduate statistics curriculum. As mathematicians and statisticians we will work together to ensure that the next generation of students is able to take a leadership role by making decisions using data in the increasingly complex world that they will inhabit.
]]>Those of us who teach mathematics know that students struggle writing the symbolism of mathematics even through they can articulate some of the concepts behind the symbolism. Those of us who interact with children know that they struggle articulating their thoughts even though they can convey their thoughts through gesture. For example, children point to indicate what they want and touch items or use their fingers as they learn to count. It is through such bodily action that children learn to recognize three objects as the quantity three without simultaneously touching and counting one, two, three. Athletes and musicians also apply bodily actions to master their sport or instrument respectively. For example, how many times have you have seen a basketball player shoot an imaginary ball into an imaginary hoop? Consider how a piano teacher places a student’s hand on top of the teacher’s hand as the teacher plays the piano. These are just a few ways in which we use our body to learn, so why not use it purposefully to promote the learning of mathematics?
In layman’s terms the philosophy of embodied cognition argues that learning is a result of interactions with our environment [1]. There are broad interpretations of this philosophy, but like others I interpret it to mean that we learn through bodily movements. Various mathematics education researchers adopt embodied cognition as a theoretical lens because it allows them to use gesture as a source of evidence as students learn linear algebra, differential equations, complex variables, etc. [2,3,4]. I too document research participants’ gestures as they tackle tasks related to my research on the teaching and learning of complex analysis, but embodied cognition also informs my teaching. In this blog, I describe how I highlight students’ gestures to help them articulate their thinking and I illustrate embodied activities designed to elucidate mathematical concepts via bodily movement.
As a first illustration, consider how I expose preservice elementary teachers to Euclidean transformations. I commence by asking students to define a translation, reflection, rotation, and dilation. Similar to children, students tend to gesture when they are unable to articulate their thoughts. For example, after asking students to define a rotation, Sammy (pseudonym) raised his hand and the following dialogue occurred. Although Sammy was unable to provide a definition, he turned his right hand back and forth as though turning a doorknob.
Sammy: Well it’s a, it’s kind of like, well you know, well I don’t know.
Me: Sammy I am going to repeat what you said and regesture your gestures.
I repeated Sammy’s words and emulated his gestures, which quickly prompted him to say, “Oh, it’s like a turn, you are just turning the object.” This exemplifies neuroscientists’ belief that if a person is “not engaged in an intentional action, or watching another person engage in an intentional action” then there is no expectation for neurons in the premotor cortex brain area to activate [5, p. 13]. These neurons are believed to be responsible for helping us interpret other’s actions as well as our own actions. In Sammy’s case, attention to his gestures facilitated creating working definitions based on the gesture characteristics. For example, the students commented that Sammy’s wrist could be perceived as the center of rotation. We have similar conversations when students alternate between turning their palm faceup and facedown as they convey their reflection definition.
After creating working definitions, the students collaboratively complete worksheets using manipulatives (see Table 1 for sample tasks). The purpose of these worksheets is for students to determine the image of a figure under a given transformation, to work backwards, and to make discoveries about the properties of Euclidean transformations. After completing the worksheets, the students present their work to the class. After this, we proceed with some embodied activities, which reinforce classroom work.
Table 1. Sample Worksheet Questions 
Suppose the point (x,y) was translated in the direction of (5,2) to obtain the image (4,8). What is the preimage point (x,y)?

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

Consider the figure below.
1. Reflect triangle ABC about line m and label it as triangle A’B’C’. 2. Reflect triangle A’B’C’ about line n and label it as triangle A”B”C”. 3. Construct the circle with center O and radius \(\overline{OA}\). Do the same for radius \(\overline{OB}\) and \(\overline{OC}\). 4. Describe all the points that pass through each circle and explain why this happens. Use mathematical transformation ideas for your explanation. 
As part of the embodied activities, students act out many of the worksheet tasks on a giant grid where the students are the points and rope serves as segments. Figure 1 depicts students as image points after translating in the direction of (1,2). As a result of this activity the students realized that not moving at the “same rate” causes the rope to become loose. With some probing, they connect the notion of the “same rate,” to an equal slope, and to the worksheet discovery that under a given translation, the segments connecting a preimage point and its corresponding image point are parallel and congruent. It is during this activity that the students also use language alluding to rigid motions of the plane. That is, they realize that under a transformation every point on the preimage transforms simultaneously rather than one point at a time as they performed it on the worksheet.
Figure 1. Embodied Translations
Figure 2 illustrates the students determining their image point under the given line of reflection. The combination of the rope and large right triangle helped them make meaning of the fact that a line of reflection is the perpendicular bisector of the segment connecting a preimage point and its image point. While working on the worksheet some students generally forget about the perpendicular aspect or the bisector aspect of the definition, but somehow using the rope facilitated attending to both facets of the definition. This could be because they are able to simply fold the rope over the line of reflection. It is not uncommon to hear comments such as “Oh that’s what perpendicular bisector means.” Similar comments are made with the rotation task. As the students rotate about the center of rotation (another student) while holding the rope they remark, “So a rotation means you are traveling in a circular fashion.”
Figure 2. Embodied Reflections
I now highlight an example where students unconsciously engage in bodily movement. As part of a second semester geometry course for prospective secondary teachers, the students performed stereographic projections using Lénárt spheres (Figure 3).
Figure 3. Stereographic Projection
One group of students was not satisfied with their image sketches because the string should go through the sphere. Thus, they relied on their eyes (another example of embodied cognition) to determine the image of the circles (Figure 4). The group progressed quite rapidly through the tasks until they arrived at the great circle that passes through the North Pole. At this point one of the students, Neil (pseudonym) got up and pointed both of his arms up to denote the North Pole (Figure 5a). While engaged in bodily motion he mentioned that one half of the great circle would be projected down (Figure 5b) and the other half would get projected in the opposite direction (Figure 5c). During this action both he and his teammember remarked, “So it will map to a line and there is a break at the North Pole.” This dynamic engagement did not go unnoticed by the other students and I asked Neil to regesture his discovery. Furthermore, when we started the unit on inversions the students quickly recalled Neil’s bodily action as they hypothesized about the image of a circle that passes through the center (O) of the circle of inversion. They knew the circle would break at point O and map to a line.
Figure 4. Using eyes for stereographic projection
Figure 5a  Figure 5b  Figure 5c 
Figure 5. Stereographic projection of a great circle 
Currently, there is much buzz about active learning of mathematics [6] but the definition of active learning is vague and sometimes it is difficult to determine if students are truly engaged in learning. By paying attention to students’ gestures instructors can hypothesize about students’ mathematical reasoning and ask probing questions that help students convey their mathematical reasoning. It is also an effective technique for assessing mathematical misconceptions, but this can only occur if instructors are attuned to gesture. Furthermore, tasks such as embodied activities help bring to life the mathematics where students are actively learning and have “aha” moments – these aha moments are the best part of teaching.
References:
[1] Anderson, M. L. (2003). Embodied cognition: A field guide. Artificial Intelligence, 149, 91130.
[2] Nemirovky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal for the Learning Sciences, 21(2), 287323.
[3] Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301323.
[4] Tabaghi S. G. & Sinclair, N. (2013). Using dynamic geometry to explore eigenvectors: The emergence of dynamicsyntheticgeometric thinking. Technology, Knowledge and Learning, (18), 149164.
[5] Gallagher, S. (2014). Phenomenology in embodied cognition. In L. Shapiro (Editor), The Routledge Handbook of Embodied Cognition (918). London: Routledge Taylor and Francis Group.
[6] Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., and Wenderoth, M. P. (2015) Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 11(23), 84108415.
]]>I had what seemed the perfect first fulltime teaching position, in that much of the planning for Calculus had already been done when I arrived. The department chair handed me the textbook and the syllabus, essentially a daybyday schedule of book sections and homework assignments. This being the United States Naval Academy at Annapolis, where every student takes Calculus, a lot of wisdom had gone into the schedule. I now look back at that syllabus with a mixture of gratitude for the jump start and recognition that much has changed. What’s in your syllabus? What does your institution require, and what is most important to you? What is decidedly not in your syllabus? Do you hand out a paper copy on the first day, or is it all online? How well does the syllabus reflect what you want your course to be?
For some time after my move to Middlebury College, my syllabi followed that basic first model. Eventually, however, I felt constrained by detailed plans that had made sense in August but didn’t fit the October reality. After too many classes that ended in a rush to cover specific content, I began to offer a general outline of topics for the semester, along with exam dates, reminders about the honor code, and only the first assignment. Putting together assignments a week at a time allowed me to be more responsive to what I was seeing and hearing in the classroom, while honoring my commitment to engage particular concepts. Looking back, I see that shift in the syllabus as an early sign of my disillusionment with a strict lecture format.
Once I started bringing students into InquiryBased Learning territory, a syllabus needed to include a description of what we were doing and why. As I tried to articulate a rationale for more active class sessions, I was compelled to consider what my compositions in the syllabus genre communicated to students, intentionally or not. For example, apart from locking us all into a rigid timetable, devoting most of the typing to a list of homework exercises risks putting the textbook at the center, while I want to put learning at the center. A reproduction of the catalog course description, including terms that novices don’t yet understand, isn’t exactly an invitation into a captivating intellectual experience.
Hence my current syllabus starts with a short description of the course content for the nonexpert. Next comes my contact information, followed by my learning objectives, again with as little technical language as possible. Only then does the reader come to the schedule of topics and important due dates, at the bottom of the first page.
One of the challenges of writing a syllabus is that, as the first document I present to my students, it serves multiple purposes. As much as I would prefer to stick to a conversation about learning, external pressures intrude. In the interest of minimizing disputes, I’ve learned to state clearly my policies on absence and late homework. Middlebury’s honor code necessitates an explicit description of what kind of sharing is acceptable and what is not. More generally, many colleges and universities have explicit requirements for what is included in a syllabus. Also, the audience is not just students; internal and external committees routinely collect syllabi in their reviews of faculty members, departments, and institutions.
Most important to many students, it seems, is information about how I will assess their work. Some tell me bluntly that they allocate their study time in direct proportion to the percentages in my grading policy, which can send me into an internal rant about how The System has driven them to focus on extrinsic rewards rather than the intrinsic satisfactions of intellectual growth.
Actually, though, these seemingly invasive topics are connected to learning, and I try to be explicit about those connections in the syllabus. In an active learning environment, I tell students, it is especially important to arrive to class having put in a good faith effort on the homework due that day. To establish and maintain a learning community, everyone should come to every class on time, barring illness. You must engage your own brain in order to learn, and this is an institution devoted to learning, which is why we have an honor code.
The possibility of a website for each course introduces both flexibility and complexity into syllabus composition. (Did I mention that I started at Middlebury in 1984?) My course sites have included, in addition to daily assignments, a rubric for evaluating proofs (adapted from Keith Devlin’s), a a statement on InquiryBased Learning, and samples of written work. This helps me meet my goal of keeping the actual syllabus to two sides of one sheet of paper, a valuable exercise. My hope is that the shorter length increases the chance that students will actually read it.
Do they read it? Thinking deeply about what is or is not on the syllabus leads quickly into a broader consideration of the nature of communication between me and my students. Should I ensure that they read it by giving a quiz, in the interest of their mathematical progress, or should I allow them to learn from experience that they are responsible for that progress? Given that I’ve decided to do less, and more judicious, telling about the mathematics, could I be more creative about conveying my expectations and teaching philosophy? One thing is clear: the syllabus can only do so much. A technique I learned from a colleague in the humanities is to start the first class not with handing out the syllabus or taking attendance, but rather with the actual material of the course. “Let’s start with some math” seems the best introduction of all.
A careful consideration of the syllabus leads to all kinds of questions about contemporary higher education. In addition to the coddling vs. building character debate, there’s the question of grade inflation. If I value class participation and reward it with a significant portion of the grade, then will the resulting grade distribution be objectionable to some colleagues? Does the need to explicit about late work policies and academic honesty, along with some institutions’ exacting requirements for syllabi, say anything about legalism on campus? I recently took a required online course concerning Title IX regulations on harassment; it ended with “… you can inform students by including information in your syllabus on resources, reporting options, and student rights. Check with your Title IX Coordinator to see if your school has approved language regarding Title IX reporting and resources to include in syllabi.” I’m ready to include such language on the course website, but would including it on the actual syllabus help or hinder my efforts to create a small community centered on learning mathematics?
Evidently the changes in my own syllabi reflect transitions in my attitudes toward my students. As a nervous young lecturer trying to stay on schedule, I prepared my notes on a pad; three pages was usually just about right for a 50minute session. Small classes meant I learned names quickly and thought I knew who was struggling, but I’m afraid I only interacted with those who raised hands in class or came to office hours. My job was to present information clearly; their job was to absorb, and the syllabus contained the map through the textbook.
Now the syllabus lays out what I find most important: the learning goals, which concern both content and practice; a framework for meeting those goals; and the principles, based on what I know about learning, connecting the goals with the framework. The syllabus is only the beginning, and I try to reinforce and elaborate on the goals and principles repeatedly in class and on the website. I now see my job as providing a structure in which students can practice mathematics on the content of the course, and responding to their efforts appropriately along the way.
]]>We want to begin this post with thanks to all of our readers and contributors — we appreciate your feedback and ideas through your writing, social media comments, and inperson conversations at mathematical meetings and events. Inperson conversations have been on the minds of the editors recently because we had our firstever inperson meeting as an editorial board at the 2016 Joint Meetings in Seattle. This was great fun and gave us a chance to seriously reflect on our blog, its role in the mathematical community, and what we want to do over the next year or two. In this post, we give a brief update about a change to the structure of our blog, followed by some highlights of our experiences attending the joint meetings.
Update
As regular readers of our blog know, since we began in June 2014 we have been publishing articles on or around the 1st, 10th, and 20th of each month. Starting with this post, we will be changing to a biweekly publication schedule. Starting on Monday, January 25th, new articles will appear every two weeks. If you want to receive notice when new blog posts appear, please subscribe to our email distribution list on the righthand sidebar of the blog or subscribe to our RSS feed.
JMM Highlights
With so many excellent activities going on at the Joint Meetings, even with five of us we could only attend a small fraction of the offerings. Given that, here are some of the highlights from our experiences.
Diana:
It amazes me how much my focus at conferences has gradually shifted over time as I have progressed through the academic ranks. As a graduate student and postdoctoral fellow, I almost exclusively attended talks and chitchatted with colleagues and friends during breaks. Now, with many projects in progress with folks from around the country, and with several leadership roles, I feel lucky if I can choose one talk per day to attend! I spend most of my time in meetings, but I enjoy this greatly. I find it wonderful to connect in person with colleagues on various projects and committees, work and connect intensely for a bit, and then resume working remotely. How wonderful this combination of modern technology and in person meeting can be when used well!
One of my main foci at the Joint Math Meetings was in my capacity as Director of the National Association of Math Circles (NAMC), one of the main outreach activities of the Mathematical Sciences Research Institute. It’s great to see Math Circles growing and becoming more wellknown across the country. I strongly encourage you to become involved with an existing one, or to start a new one. At the NAMC happy hour, it was particularly wonderful to meet PoShen Loh, coach of the U.S. International Math Olympiad team that won the Gold Medal in 2015. I’m particularly excited about his new Expii website that is posting a wonderful weekly problem set to engage kids of all backgrounds and experiences with mathematics.
Another major focus for my time at the JMM consisted of events surrounding the development of the Mathematical Association of America’s Instructional Practices Guide. I’m honored and excited to be a part of developing this as an accompaniment to the decadal MAA Curriculum Guide. At the JMM, we conducted focus groups, held meetings of the steering committee and advisory board, hosted a panel, and had several leadership team meetings. The 23 year project is off to a great start!
Art:
The entire AMS Special Session on Essential Mathematical Structures and Practices in K–12 Mathematics featured mathematicians and mathematics educators talking with (not at!) each other about treating the mathematical topics that arise in K12 with the rigor that they deserve. Expect a post about this soon, focusing on ratio and proportion as an illustrative example, from some of the speakers. The panelists at the AMS & AWM Committees on Education Panel Discussion Work in Mathematics Education in Departments of Mathematical Sciences showed the different ways that mathematicians can be involved in mathematics education. Yvonne Lai presented research she and her collaborators conducted on the mathematical knowledge for teaching at the high school level, and they will also be contributing a post here in the future. I also learned more about Illustrative Mathematics, a community of educators collecting quality classroom resources that encourage mathematical understanding; their materials are easy to browse at their website.
Priscilla:
The first JMM stop for me was a panel discussion on “Creating a meaningful Calculus I experience for students entering with high school calculus.” David Bressoud (Macalester) suggested that such students “don’t need a course of techniques — they need big ideas,” and presented an outline of his department’s course. He also recommended Michael Oerhtman’s Clear Calculus materials for a laboratory approach. Robin Pemantle (UPenn) described students’ backgrounds as “porous,” so that he and his colleagues offer a course which is ⅓ filling in holes in calculus, algebra, and even arithmetic, ⅓ topics (probability densities, differential equations) suggested by faculty in client departments, and ⅓ multivariable calculus. Uri Treisman encouraged us to accept the fact that “it’s in the culture that you’re supposed to repeat calculus” after taking it in high school. His response is to offer an introduction to university mathematics, rather than simply an introduction to calculus; he wants to “startle students” with the power of the subject and to emphasize proof and precision. “Don’t make high school calculus the enemy,” Treisman insists; instead we should “work collaboratively with the high school teacher community.”
The premiere showing of “Navajo Math Circles” was a treat. I’ve bought a copy for my department so that we can show it on campus. The “extra features” page has some fascinating short videos that go beyond the film content.
Elise:
A highlight for me was a talk by Chris Rasmussen about teaching called “Advances in inquiryoriented instruction at the postsecondary level: Student success and instructor practices.” This was in Alan Schoenfeld’s MAA Invited Paper Session on What Do We Know about University Mathematics Teaching, and How Can It Help Us? (which provided a number of insightful and thoughtprovoking talks). In his talk, Chris highlighted a few papers and projects that demonstrate the progress that is being made at the postsecondary level. It was exciting to see that there are quite a few new advances, and he cited some recent and upcoming papers that speak to this issue. I appreciate that there are opportunities for productive discussions about instruction at the postsecondary level.
Ben:
I had two highlights from the 2016 JMM. First, I greatly enjoyed giving two talks about teaching, one as part of a Project NExT panel on broadening assessment in postsecondary education and another on the topic of growth mindset interventions to support IBL pedagogies in the MAA contributed paper session on inquirybased learning. It was rewarding to have excellent questions and interactions with audience members, both in person and through email, following my talks. Second, I attended Alan Schoenfeld’s talk about improving K16 mathematics, in which he described the Teaching for Robust Understanding (TRU) Math framework. The five dimensions of this framework strongly resonated with me, and I encourage all of our readers to visit the TRU Math website for more information. One of the comments that Schoenfeld made stuck with me: at this point in mathematics education research, we know what to do to teach right, and it is hard work to implement our knowledge. This led me to reflect on my belief that half of our challenge at the postsecondary level is to broaden awareness and understanding about what constitutes powerful mathematics classrooms — it is impossible to implement change for the better if we don’t recognize and value the complexities of our classrooms and our students’ learning.
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