How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all positive \(y\). The French language has the Académie française to decide its arcane details. There is no equivalent for mathematics, so there is no one deciding once and for all what \(0^0\) equals, or if it even equals anything at all. But that doesn’t matter. While some definitions are so well-established (e.g., “polynomial”, “circle”, “prime number”, etc.) that altering them only causes confusion, in many situations we can define terms as we please, as long as we are clear and consistent.
Don’t get me wrong; the notion of mathematics as proceeding in a never-ending sequence of “definition-theorem-proof” is essential to our understanding of it, and to its rigorous foundations. My mathematical experience has trained me to ask, “What are the definitions?” before answering questions in (and sometimes out of) mathematics. Yet, while we tell students that the definition needs to come before the proof of the theorem, what students apparently hear is that the definition needs to come before the idea, as opposed to the definition coming from the idea.
Why definitions?
What is a definition anyway? Or rather, what gets defined? We could make a special name for the function that maps \(x\) to \(5x^{17} – 29x^2 + 42\), but we don’t. On the other hand, we give the name “sine function” to \(\sin(x)\), the ratio of the length of the side opposite an angle with measure x to the length of the hypotenuse of a right triangle. We give a name to the sine function, even though it takes much longer to describe than \(5x^{17} – 29x^2 + 42\); in fact, we give it a name in part precisely because it takes longer to describe. If we need to refer to \(5x^{17} – 29x^2 + 42\), it’s not that hard, but we do not want to have to write down that definition of sine every time we use it in a statement or problem. We give definitions to ideas for two related reasons:
Brevity: It’s clearly easier to write “\(\sin(x)\)” instead of the huge sentence above. Further, packing this idea into a single word helps make it easier to chunk ideas in an even longer statement, such as a trigonometric identity.
Repetition: If we have to use the same idea more than once, then giving it a compact name increases the efficiency described above that much more. Sometimes an idea repeats just locally, within a single argument or discussion, and then we might temporarily give it a name; for instance when finding the maximum value \(x e^{-x}\), we would write \(f(x)=x e^{-x}\), so we could then write \(0 =f'(x)\), but we are only using \(f\) this way in this one problem. On the other hand, the ideas that show up over and over again, in many different contexts, such as \(\sin(x)\) or “vector space”, get names that stick.
This begs the question, “Why do certain ideas, or combinations of conditions, repeat?” Consider “vector space”. The idea of \(R^n\) is clear enough, but of all its properties, why focus on the simple rules satisfied by vector addition and scalar multiplication?
First, because several additional examples have been found that satisfy these rules, such as the vector space of continuous functions, the vector space of polynomials, and the vector space of polynomials of degree at most 5. Second, because once the key properties that make up the definition are identified, we may find that the proofs only depend on those key properties: The Fundamental Theorem of Linear Algebra, for instance, is true for arbitrary finite-dimensional vector spaces, so we don’t need a separate proof for \(R^n\), for polynomials of degree at most 5, etc. (Purists may argue that all finite-dimensional vector spaces of the same dimension are isomorphic, but this isomorphism is defined in terms of vector addition and scalar multiplication, just reinforcing the significance of those operations.)
Choices
But there are often still choices to be made. Must a vector space include the zero vector, or could it be empty? (Is the empty set a vector space)? For that matter, since vectors are often described as being determined by “a direction and a magnitude” and the zero vector has no direction, is the zero vector even a vector? The answers to these questions are no and yes, respectively, but why? The zero vector is a vector, because it is so helpful for a vector space to be a group under addition, which requires an identity element. (I know — this only takes us back to why are groups defined the way they are. Let’s just take this as a piece of evidence for why groups are an important definition.)
As for the empty vector space, there’s nothing inherently wrong with it, except perhaps for the need for a zero vector as discussed above. (This also takes us back to why groups are not allowed to be empty. Let’s stick to vector spaces for now.) But how would we define the dimension of an empty vector space? How would we define the sum of the empty vector space with another vector space? And then, even if we do make those definitions, how do we reconcile them with this identity?:
\[
\dim (A+B) =\ \dim A\ +\ \dim B\ -\ \dim (A \cap B)
\]
This example shows that, even though we cannot write the proof of a theorem until all the relevant definitions are stated, we do often look ahead at the theorem before settling on the fine points of the definition. At research-level mathematics, we might even modify our definitions substantially to make our theorems stronger, or to deal with potential counterexamples. (For more details on this, read Imre Lakatos’ classic Proofs and Refutations [1].) I will stick to smaller cases where we adjust definitions mostly just to make the theorems easier to state.
More examples
Why is 1 considered to be neither prime nor composite? When you first learn this, it may seem silly. The definition of prime is so simple and elegant — an integer \(n\) is prime if its only factors are 1 and \(n\) — and 1 seems to fit that definition just fine. Why make an exception? The answer lies in the Fundamental Theorem of Arithmetic, that every integer has a unique factorization. Well, except of course that we could change the order of the factors around; for instance, it makes sense to consider \(17 \times 23\) to be the same factorization as \(23 \times 17\). And also we need to leave out any factors of 1, otherwise we might consider \(17 \times 23, 1 \times 17 \times 23, 1 \times 1 \times 17 \times 23\), … to all be different factorizations. If we take a little extra effort at the definition, and rule out 1 as a prime number, then the theorem becomes more elegant to state.
Is a square also a rectangle? In other words, should we define rectangle to include the possibility that the rectangle is a square, or exclude that possibility? When children first learn about shapes, it’s easier to simply categorize shapes, so a shape could be either a rectangle or a square, but not both. But when writing a careful definition of rectangle, it takes more work to exclude the case of a square than to simply allow it. Similarly, theorems about rectangles are easier to state if we don’t have to exclude the special cases where the rectangle happens to be a square: “Two different diameters of a circle are the diagonals of a rectangle” is more elegant than “Two different diameters of a circle are the diagonals of a rectangle, unless the diameters are perpendicular, in which case they are the diagonals of a square.”
Is 0 is a natural number? It doesn’t really matter; just pick an answer, be consistent, and move on. It’s even better if we can use non-ambiguous language instead, such as “positive integers” or “non-negative integers.” To be sure, mathematics is picky, but let’s not be picky about the wrong things.
Finally, what about \(0^0\)? If you just look at limits, you’d be ready to declare that this expression is undefined (the limit of \(x^y\) as \(x\) and \(y\) approach 0 is not defined, even just considering \(x \geq 0\) and \(y \geq 0\)). And that’s fine. But in combinatorics, where I work, setting \(0^0 =1\) makes the binomial theorem (\((x+y)^n = \sum \binom{n}{k} x^k y^{n-k}\)) work in more cases (for instance when \(y=0\)). And so we simply declare \(0^0=1\), at least in combinatorics, even though it might remain undefined in other settings.
(See here for a list of other “ambiguities” in mathematics definitions.)
In each of these examples, there is a human choice about how to exactly state the definition. This is a great freedom. But, to alter a popular phrase, with great freedom comes great responsibility. If you declare \(0^0\) is a value other than 1, now you are limiting, not expanding, the applicability of the binomial theorem. And if you want to declare that \(\frac{1}{0}\) has any numerical value, you will have to sacrifice at least some of the field axioms in your new number system.
In the classroom
The issues that arise with developing precise mathematical definitions is well-known to mathematicians, but we generally don’t share it with our students enough. If we stop hiding this story from our students, then they will see that mathematics is a human endeavor, and that mathematical subjects are not handed down to us from on high. This can be one factor in convincing students that mathematics, even advanced mathematics, is something they can do, that it is not just reserved for other people. And even students who already “get it” will not be turned off — we should not abandon definition-theorem-proof, we can just pay more attention to sharing why each of our definitions is written the way it is. If students know where a definition comes from, what motivated it, and why we made the choices we did, they may have a better chance of making sense of the idea instead of memorizing the string of words or symbols. (See also my earlier blog post, A Call for More Context.)
An anecdote that Keith Devlin tells, near the end of a blog post about mathematical thinking, illustrates the power of crafting the right definition. To summarize much too briefly, his task was to “look at ways that reasoning and decision making are influenced by the context in which the data arises” in a national security setting. His first step was to “write down as precise a mathematical definition as possible of what a context is.” When he presented his work to government bigwigs, they never got past his first slide, with that definition, because the entire room spent the whole time discussing that one definition; later he was told “That one slide justified having you on the project.”
We might not have the luxury of spending an entire hour discussing a single definition, but we can still let students in on the secret that the definitions are up to us, and that writing them well can make all the difference.
References
[1] Lakatos, Imre. Proofs and refutations. The logic of mathematical discovery. Edited by John Worrall and Elie Zahar. Cambridge University Press, Cambridge-New York-Melbourne, 1976.
]]>This past spring, I received an email from a graduate student who was concerned about applying for jobs in industry. The student wrote: “I’m having a difficult time trying to market my teaching experience. I’ve been teaching for three years now and I want to leverage that in my applications. I’m just not sure what to say beyond ‘improving communication skills’.”
Whether their interests are in academic positions or not, many graduate teaching assistants (GTAs) are concerned about the jobs they will find and whether they are prepared for those jobs. I have led the graduate teaching assistant training in the Department of Mathematics at Oregon State University since 2013. In that time, I have come to realize it is critical to help GTAs understand the professional skills they develop during their graduate careers, particularly as they learn to teach. My goal in this note is to unpack and describe some of the processes of teaching to help the GTAs appreciate the skills they learn through teaching, and see that these skills can be applied to a variety of jobs beyond academia.
I searched the internet for recent articles that describe the skills employers are looking for, now and in the future. In the list that follows, I highlight some of skills that were common across these articles and discuss how GTAs develop these skills through their teaching. This list is not meant to be exhaustive.
In providing this list, I want GTAs to see that teaching is much more than writing mathematics on a board, and that there is much to be learned through the processes of teaching. Illuminating the skills learned through the processes of teaching will help our GTAs reflect on their practices, help them to reflect on what they are doing as teachers, and inspire further exploration. This reflection in turn helps GTAs better describe their relevant experience in cover letters, on CVs, in their teaching statements, and in conversations about their work as teachers. I believe that explicit attention to these skills can contribute greatly to the professional development of GTAs.
Communication is a critical skill recognized on multiple websites, and teaching is all about communication. As teachers, we learn to communicate complex ideas in multiple ways, and we communicate much more than the mathematical concepts we write on the board. We use many modes of communication (speaking, writing, body language, facial expressions, written assignments, handouts, and online materials). We communicate many messages (encouragement, positivity, enthusiasm), and of course we communicate mathematical content (mathematical ideas, problem solving strategies, multiple representations of mathematical concepts).
Most of what we communicate to our students comes through public speaking in classrooms – standing in front of small or large groups of people, who look at us all at the same time, waiting for us to speak to them and to get the class going. Through the processes of teaching, we learn to build skills of responsiveness and to adapt our instruction to the different ways students solve problems. Teaching naturally provides GTAs with opportunities to cultivate the ability to speak publicly and give presentations. Beyond this, teaching helps GTAs learn to convey abstract ideas effectively, in ways that people with varying backgrounds and learning styles can understand. Effective public speaking is a skill that applies in many different situations: job interviews, conference presentations, or presentations in the work place.
What other communication skills might we learn and hone through the processes of teaching?
Several websites noted that facilitation is an important skill for people interested in leadership positions. In fact, one author [1] referred to facilitation as the “key to the future of work.” What work is involved in facilitating student learning? Leading a group of 10, 20, 30, or more students in productive group learning and problem-solving activities is rich, complex work. Facilitating students’ group work on mathematical tasks and conversations about their work requires giving clear instructions and setting expectations. Effective facilitation requires attention to inclusivity and equity to ensure that all students’ voices are heard and supported, and that every student’s work is recognized as valuable and contributing to the course. Facilitation also means actively listening, responding in ways that lead to productive conversations, helping students learn to how to communicate their mathematical thinking, and getting them to work as a team and support each other.
Developing interpersonal skills is also essential for any kind of work with people. Some websites noted that people with strong interpersonal skills are more successful in both their personal and professional lives ([2], [3]). Another website noted that interpersonal skills are considered “employability skills” [4] because hiring managers do not want to hire people without them. A few websites noted that applicants should highlight interpersonal skills in cover letters and/or resumes. So, what are interpersonal skills? They are also called ‘people skills’ – the behaviors and characteristics we use when we communicate and collaborate with others, such as active listening, empathy, collaboration, problem-solving, adaptability, and leadership.
Interpersonal skills can also be learned outside the classroom, during office hours and in tutoring sessions when teachers work with individual or small groups of students. Through these interactions with students, GTAs can learn what mathematical explanations and representations might be best for helping students understand a mathematical idea. In addition, GTAs can reflect on their interactions to learn how their communication with students can be positive and encouraging. Office hours can also include tough situations and conversations, which may require honest or critical feedback that is also supportive and encouraging. GTAs can learn a great deal from their work with individual students and translate that learning into deeper interpersonal skills that can be applied to any number of future work places.
Much of what we do as teachers is lesson planning. Planning for and teaching a course is a form of project management. Before I became a teacher, I was a project manager at an educational software company. Multiple websites offer anywhere from five to ten steps for successful project management ([5], [6]). These steps include determining the objectives of the project (learning outcomes), initiating the project (writing the syllabus, planning the term, finding course materials), executing the project (doing the work of teaching and working with students), managing the project (monitoring progress, re-calibrating, staying connected to project outcomes), and completing the project (getting to the end of the term having covered the material of the course). Indeed, the work of planning a course, sequencing concepts, problems and tasks, connecting past ideas to current ideas, foreshadowing what students will encounter later in the class or in future classes helps our TAs learn about project management.
Once a course is mapped out for the term, there is the day-to-day implementation of the project through specific lessons. GTAs can learn to plan specific features of the lesson, such as lecturing, group work, and student presentations, when and how those features will occur, and how they might create alternatives should a lesson not go as planned. Planning for specific lessons gives GTAs the opportunity to think deeply about what mathematical concepts they will present to students and how they might have students engage in mathematical activities around those concepts. Some questions GTAs might ask when planning for a lesson include: What mathematical concepts do we want to communicate? What representations might we use? At what points in a lesson might we pause and let students do some mathematical work – and why would we pause for those particular tasks? These questions can be applied to multiple work places in terms of how work happens and why, the sequencing of tasks, and revisiting and revising work to improve outcomes. How might homework sets reinforce classroom learning and prepare students for the next class, to keep the mathematical momentum going?
An important skill listed on several websites is analysis of data. Often, the work of grading is simply described as marking students’ work as correct, incorrect, or somewhere in between. I would argue that ‘grading’ does not adequately describe the various ways that teachers assess students’ learning through homework, quizzes, exams, formative and summative assessment. Assessing students’ work is much more complex and requires much more thought. Teachers regularly analyze data as they administer assessments, analyze student work, and make conclusions about how they might modify their teaching to improve student outcomes. Assessing student learning requires the teacher to meaningfully interpret student work and try to understand what students were thinking when they solved a problem. By assessing student learning, GTAs will learn how to analyze data, understand it, and respond to it. They will learn how to keep electronic records, compute statistics, and make decisions on how to proceed.
GTAs learn to use software programs during their graduate course work (e.g., Matlab, Python, SAS, SPSS, and so on), and they also learn about various learning technologies used in the classes and recitations they lead. These technologies include the use of Clickers, Canvas, Blackboard, MyMathLab, Geogebra, Desmos, Learning Catalytics, and open resource materials, among many others. Not only do GTAs learn those programs, but they also learn how to help students to use those technologies. In these ways, GTAs are developing many skills that can be applied to multiple work places.
I believe the list above is a good first step in helping GTAs to appreciate the skills they learn through teaching. However, it is essential that we go beyond informing GTAs that they are learning these skills because, eventually, they will have to present themselves as people with these skills. It is critically important that GTAs become conversant in these skills, to develop their own voice and an understanding of how they authentically present themselves as someone with these skills. Consequently, I ask the GTAs to reflect on their experiences in classrooms with these skills in mind. They write about how they see themselves learning these skills through different teaching situations. Some of these situations might include how they communicate a difficult idea to students, how they sequence a lesson and why they thought that sequence was effective (or not), or how and why they might improve on their assessments of student learning. In writing about these situations, GTAs reflect on how they are growing as teachers and what skills they have learned through the processes of teaching, which helps them see the way their teaching experience can be applied across multiple professions.
Through these types of activities, I think we can help GTAs appreciate what they learn from their teaching experiences and help them to translate that learning into a concrete, explicit set of skills they can apply to multiple professions. And, for students like the one cited above, we can provide them with ways to present themselves as learned professionals.
[1] Klein, B. (2017). What “facilitation” really means and why it’s the key to the future of work. Retrieved August 28, 2018 from https://www.fastcompany.com/40467377/what-facilitation-really-means-and-why-its-key-to-the-future-of-work.
[2] Terrell, S. (2018). What are interpersonal skills and why are they so important? Retrieved August 28, 2018 from https://blog.mindvalley.com/what-are-interpersonal-skills/?utm_source=google_blog
[3] The Skills You Need. (n.d.). Interpersonal skills. Retrieved August 28, 2018 from https://www.skillsyouneed.com/interpersonal-skills.html
[4] Doyle, A. (2018). Interpersonal skills list and examples. Retrieved August 28, 2018 from https://www.thebalancecareers.com/interpersonal-skills-list-2063724
[5] Sundwall, H. (1996). Seven steps to success for project managers. PM Network, 10(4), 31–32. Retrieved August 30, 2018 from https://www.pmi.org/learning/library/seven-steps-success-project-managers-3313
[6] Lucidchart Content Team. (2017). 5 essential project management steps. Retrieved August 30, 2018 from https://www.lucidchart.com/blog/5-essential-project-management-steps.
]]>Real Analysis is a rite of passage for undergraduate math majors. It is one of my favorite courses to teach, but I recognize that the course is challenging for students, and, for many, downright intimidating. In Fall 2017 I was scheduled to teach Real Analysis for the third time in my career. Prior to the semester starting, I knew that I wanted to alter the grading scheme of the course to de-emphasize exams in favor of effort. Ultimately, I wanted to promote a growth mindset and to help students identify their strengths and weaknesses independent of exam performance. During our annual summer visit, my good friend and graduate school classmate Matthew Pons described to me his new project with Allison Henrich, Emille Lawrence, and David Taylor called The Struggle is Real: Stories of Struggle and Resilience on the Path to Becoming a Mathematician. (For more information on their project, check out https://math.roanoke.edu/tsir/.) I loved their idea of gathering and sharing personalized stories around this topic and immediately thought of adapting the exercise for my students. Since I was teaching Real Analysis, I decided to include reflective homework problems and activities under the label #thestruggleisREAL. I was worried that the hashtag was too gimmicky, but decided that with the right sales pitch students would embrace the pun. In this post I describe how this well-trodden hashtag injected a great deal of reflection, and a bit of levity, into my students’ experience in Real Analysis.
I’ve spent a great deal of time privately considering my own struggles in mathematics. I’ve exchanged stories with classmates from graduate school, with my spouse and other confidantes, and one-on-one with students during office hours. But as a student I never had these conversations as a part of a class, and as a teacher I have not offered a place for sharing such stories beyond one-off conversations. Thus I decided, without figuring out any further details, to incorporate into the class writing prompts where students could share their own stories of struggle and resilience. The two things I did decide upon from the beginning were that the writing assignments would be regular throughout the semester and graded (but low stakes). The pertinent language on the syllabus read:
#thestruggleisREAL: Throughout the semester there will be homework problems and activities under this hashtag. At the end of the semester, students may elect to complete a reflective project elaborating on this work. For students who complete this option, the project will count towards 10% of the final grade, and the exams will each count 15%. [As opposed to the higher of two exam scores counting 25% and the lower exam score counting 15%].
I previously used different grading tracks in Calculus classes when including a Community Engagement component in the course, and I felt that this would be another appropriate time to give students agency in how their work was evaluated. It was important to me that they participate in reflection throughout the semester and that everyone was rewarded equally for this effort. At the end of the semester, students could then choose what type of work was the best measure of their performance in their class. As is to be expected there were questions regarding the mechanics of the grade distributions, but there were no complaints about separate grading tracks. Indeed, several students with high exam averages completed the final reflective project, and several with low exam scores opted out.
The Assignments
As in previous semesters, I assigned weekly homework in the class consisting mainly of proofs. This time around, I added to each assignment one problem labeled #thestruggleisREAL. In addition to the reflection that these problems prompted, I think that by putting narrative writing side-by-side with formal proof writing I was able to strengthen the case that mathematical writing is “writing”. In most cases, students received full credit for completing the problem. I relied on the fact that all Real Analysis students have taken a writing intensive first-year seminar. This means that the quality of exposition was generally acceptable, and besides that I used the formal proofs in the homework to hold them accountable for writing mechanics.
Below is a selection of the writing prompts that I used during the semester.
First assignment: I teach at a small liberal arts college where course reputations carry a great deal of weight. With this in mind, the first reflection asked students to write down their preconceived notions of the class. Students did not turn in their assignments, but shared them in pairs during our second class meeting. I wrote my own reflection and shared it on the class webpage.
Graded assignments: The writing prompts asked students to reflect upon a particular homework problem or proof completed in class. I generally asked for a paragraph response and it counted as a regular homework problem. Some examples are:
Optional (fun) assignments:
Final homework assignment: Should Real Analysis be a requirement for the math major? Why or why not?
Final project: Finally, as stated in the syllabus, students were given the option to complete a more formal writing piece reflecting on their semester in Real Analysis. I directed students who were interested in the project to first read an article from 2010 in Math Horizons, The View from Here: Confronting Analysis, by Tina Rapke. (If I had found the article sooner, I would have assigned it as reading in Week 1!) The written assignment required students to choose 3-5 of their own proofs from the semester as representative examples, and then to write 1-2 pages narrating their experience in Real Analysis using these proofs to illustrate their experiences. Roughly half of the class turned in a narrative and most exceeded the requested two pages in length. I was extremely pleased with the quality of reflection they demonstrated in the final projects. Yes, most students who did the final reflection did so in order to boost their course grade, but many of their writings achieved what I had hoped for at the beginning of the semester: they catalogued moments of growth, success, and failure while taking a look back on their experience in Real Analysis.
Overall Impressions
The most telling evaluation of #thestruggleisREAL is that word got back to me about the hashtag from colleagues in my department. Students were talking about the assignments outside of class, which gives me hope that they were sharing stories of their struggles with each other in addition to completing the writing. The class had a great sense of community throughout the semester and whether by chance or due to the invitation to voice their stories of struggle, they were very supportive of each other.
I use a well-known text when teaching Real Analysis – Understanding Analysis by Stephen Abbott – so the Internet is always an issue when it comes to homework problems. This semester I noticed a decline in blatant misuse of online sources when grading homework. I would like to think that by being asked to document their struggle, and being incentivized to acknowledge the difficulty of the work, students were less inclined to simply copy an answer from an outside source.
I will certainly incorporate a regular reflective component into future Real Analysis classes, using #thestruggleisREAL as long as it has some relevance to students. The components that worked the best were the regularity of the reflective writing prompts, the periodic inclusion of “light” activities (such as the meme contest) to allow for creativity and humor, and the optional nature of the final project. In the future, I have three concrete ideas for improving upon #thestruggleisREAL:
I have recently heard several mathematicians claim that the educational philosophies of Math Circles and the Inquiry Learning Community are essentially the same. I disagree. I will contrast the differences between these two approaches, along with two other common educational philosophies in the United States. All four approaches to math education differ significantly both in terms of the overall instructional goals and in terms of the primary methods used to achieve these goals.
In my experience, even the originators and staunchest advocates for specific philosophies incorporate the other approaches when putting their favored one into practice. Instructors should think carefully about the goals they have for a given set of students, and then choose a combination of approaches they believe most likely to meet those goals.
I would like to invite you to comment on any thoughts that you have about these lists. A few questions that I have for readers include:
I look forward to hearing your ideas!
The primary goal of the Traditional Math approach is to teach students to solve problems of a specified type as easily and efficiently as possible. This approach arose out of a need to broaden the pool of people able to accurately perform specific computations.
Because of these goals, Traditional Math instruction has the following characteristics:
The primary goal of the Conceptual Math approach is to guide students to a deep enough understanding of common math topics that they can devise multiple approaches to solve those kinds of problems, and make sense out of their answers. In our current economy, employees need to know when to apply common computational approaches more than they need to know how to fluently perform multi-digit computations. Most employees are asked to devise ways to solve a range of problems rather than simply following a procedure laid out by someone else.
On the other hand, the Conceptual Math approach still aims to be accessible to all students and to all teachers. This means that course and lesson designs must be simple enough that teachers with many students and busy schedules can implement them easily. As a consequence, the Conceptual Math approach puts more emphasis on deep understanding of traditional math topics rather than developing the ability of students to research and tackle realistic practical problems or to create and tackle their own mathematical questions.
Because of these goals, Conceptual Math instruction has the following characteristics:
The primary goal of the Inquiry approach is to teach students to create and investigate their own questions. This approach to instruction originated with those interested in preparing students to be scientists, engineers, programmers, or entrepreneurs.
The instructor often guides student inquiry by posing the initial question, which usually does not provide all of the needed information, and is deliberately badly defined. Problems often involve messy, realistic numbers. Students pose sub-questions and have substantial control over the direction their investigation will go. Students not only re-contextualize their results, but often present their results to outside audiences in a variety of written and verbal formats (including videos and web pages). During concluding discussions, the group creates anchor charts to codify strategies and facts they have discovered.
Communication and collaboration are explicit goals of the Inquiry approach. Students share their thinking verbally and in writing and give one another meaningful feedback. There is significant emphasis on teaching students about ways they can contribute positively to a team effort.
Because of these goals, the Inquiry approach to instruction has the following characteristics:
The primary goal of the Math Circle approach to instruction is to teach students learn how to work creatively in the discipline of mathematics. They create new mathematical playgrounds, brainstorm new questions for existing mathematical playgrounds, make original approaches to questions posed, generate data for given approaches, design ways to organize information obtained, propose conjectures about patterns they see, seek proofs of conjectures, find ways to define terms that make it easier to explain results, and express their results using diagrams, mathematical notation, and terms the way a mathematician would. Students learn to seek connections between seemingly different situations.
One of the goals of a Math Circle is to enculturate students as mathematicians. Students cannot develop this culture on their own working in small groups, so a Math Circle instructor frequently models the norms of mathematical discourse. Most of the ideas for solving problems come from the students (though the instructor may ask leading questions when needed). However, the instructor frequently intrudes while students are presenting their ideas to impose the cultural norms of math as a discipline.
Students learn about mathematics as a discipline. They learn to value (and collect) failed attempts as an aid to eventually solving a problem. They practice common proof techniques, and learn to use terms and notation so that other mathematicians will understand what they say and write. Students are exposed to the history of the mathematical ideas they encounter. They also learn what makes a question mathematically interesting, and how to deal with being stuck (emotionally and mathematically). Students learn to interact appropriately with fellow researchers, including being able to communicate effectively in verbal and written form, balancing personal emotional needs against those of a group, building a collegial atmosphere capable of producing interesting mathematical insights, and enjoying the process of mathematical discovery.
Because of these goals, the Math Circle approach to instruction has the following characteristics:
Later, the teacher showed the kids a mathematical tug-of-war game. Each pair of children would have a single die, a small plastic bear, and a number line laid out like this.
The bear starts on the 10 and children take turns rolling the die, one child moving the bear that many steps toward 20 and the other child moving the bear toward 0. Each child also each had a sheet to record the bear’s moves, one sheet with addition templates the other with subtraction Using this format, the children were to record where the bear had started when their turn began, the size of their move, and where the bear landed.
They all understood the mechanics—roll the die and move the bear that many spaces toward their side. I was surprised that several didn’t seem to understand that they were playing one game, together, rather than taking turns re-starting the bear at 10 and rolling their die to see how far it went this time. It was no surprise, though, that only a few recorded their jumps. Frankly, that made sense. The recording step may (or may not!) serve learning but, to the children, it was simply an arbitrary rule with no logical role in the game. Nothing about the game was enhanced by recording it.
We played, cleaned up, and then it was snack time.
During snack time, Alli asked me “how do I write positive three?” I thought, of course, of her early morning announcement about negative numbers. Her question was so clear and specific that I didn’t think (as I always should) to say (as I often do) “I’m not sure I understand. Tell me more.” I too quickly assumed that I knew what she meant.
“Well, we usually just write three, just the way you always write it.”
“But I mean positive three.”
I should have realized right then that I’d mistaken what she had in mind, but I plowed on.
“Just 3—we could put a plus sign in front, but we don’t usually.”
“No but I was on 17 and I rolled 6. How do I write positive 3?”
“Well, Alli, what is seventeen plus six?”
“Twenty-three. But how do I write positive three?”
Now I understood.
Communication with kindergarteners can feel like a string of non-sequiturs when we don’t see the connective tissue, the theory in their mind that they assume we know and that they therefore don’t bother communicating.
It turns out that what Alli meant tells us a lot about the theory she had constructed when her father told her about negative numbers. Prior to hearing about them, Alli had never heard of positive numbers, either. There were just numbers. Now she knew there are kinds of numbers. I don’t know what her father did or didn’t say, but it’s easy to believe that he, like I, would have assumed that nothing further needed to be said about positive numbers; after all, Alli was already quite adept with them. But for Alli, it wasn’t yet clear that the familiar numbers were just getting a new name, positive. For all she knew, the designation positive might well be reserved only for some special use.
And that does explain her question. She learned that going below zero called for negative numbers, and that they contrasted with positive numbers somehow. Perhaps she first thought that positive numbers were all the numbers she had already known (or, less likely, that 0 was yet a third category), but in the context of the number line tug of war game, she built a competing theory. The line contained the numbers from 0 to 20—just plain numbers. She knows that there are other numbers, not shown. Now she knows that below 0 were negative numbers. Perhaps the designation positive also refers to numbers not shown, but above 20. In other words, the categories she created were not “above and below zero,” but “above and below the range we’re attending to.” With astonishing ease for a kindergarten child, she mentally computed 17 + 6 = 23, but now she assumed that “positive three” was the way to express that excess above 20 and she wanted to know how to write it.
The point of relating this story is not to show how impressively smart kindergarteners can be. And it’s certainly not to note a “misconception.” It’s to illustrate what I think is a subtle aspect of teaching mathematics. As teachers, we can’t fully control what ideas our students build, even if we believe we are being are quite clear and precise. What people (children and adults) put in their minds is what they construct, not what someone else says or even shows, and it combines what they already know with their interpretation of what they are currently seeing and hearing. Because that construction combines current experience with past, our “clear and precise” communication will reach different people differently: each makes something of it, but not necessarily what someone else would make, and not necessarily what we expected would be made. We say/write what’s in our mind; what gets in the mind of the listener/reader isn’t conveyed there but built there. Communication is not high-fidelity.
Alli was working out a piece of mathematics. That’s where her dad was no doubt focused when he mentioned negative numbers and that’s where I focused as I tried (and at first failed) to answer Alli’s question. But Alli was also working out a piece of English, a definition. In many contexts, we do report how far some value is above or below a range. Although she’s unlikely to have examples like blood-pressure or cholesterol levels, any kindergartener does already know that some categories name whole ranges of numbers above and below another range of numbers. For example, with no particular precision about which numbers demarcate the categories, they know that babies are below a certain age and adults are above a certain other age and in between are children. Alli has no information yet from which to conclude that this isn’t how the words negative and positive are used when referring to numbers. But it could be, whence Alli’s interest in knowing how to (or whether we should) treat 23 as “positive three.”
In this story, the uncertainty about the meaning of a word is of no real consequence. Though someone might wonder why knowing about “negative” was insufficient to clarify for her what “positive” meant, there’s no risk that Alli’s confusion would lead anyone to conclude that she’s “bad at math.” And, aside from her own interest, there’s no rush for her to know: she is, after all, still in kindergarten and will surely sort this all out in time.
But there are times when the vagaries of communication cause mischief. In US elementary schools, it’s common (probably close to universal) practice for teachers to instruct children to pronounce numbers like 3.12 as “three and twelve hundredths,” not as “three point one two,” what I call a spelling pronunciation. (In my opinion, the insistence on a fraction pronunciation in school is not helpful—for one thing, just think how you’d be expected to pronounce 3.14159—but I’ll save my many reasons for a later blog post.) In one fourth grade classroom that I was supporting, the teacher asked the students to read 3.12, and then wanted to check their understanding of the place value names, so she asked “how many ones?”
The class chorused “Three!”
“How many tenths?”
“One!”
“And how many hundredths?”
Dead silence.
Then a timid “two?” and a more timid “twelve?”
The context “how many ones, how many tenths” seemed to call for the answer two, which is what we know the teacher wanted to hear, despite the loose wording of her question. But children don’t yet have a way to be sure. They’d just read the number as “three and twelve hundredths,” so twelve was a sensible answer. Nobody, of course, answered “three hundred twelve,” which would have been a delightful response showing deep understanding, just as nobody answered the earlier questions with “3.12 ones” and “31.2 tenths.” All of these answers are mathematically correct but they’re “wise guy” answers because they violate norms for communication. They are correct, but clearly not what the teacher meant by the question. In the case of “how many hundredths,” however, students might well be unsure which the teacher meant.
Because the teacher didn’t recognize the source of the confusion—just as I had not at first understood the source of Alli’s confusion—she heard the hesitation and mixed answers as evidence that the class didn’t really understand the mathematics. I had the luxury of being the observer, hearing and following up individual children’s queries rather than having the full responsibility of the teacher addressing and trying to manage the entire class. What I heard and saw made it clear that virtually all of the children did understand the mathematics; the confusion was only about which of two very reasonable interpretations of the teacher’s question was the one she intended.
Unlike the story of Alli, this miscommunication did have consequences. One consequence was a review that was unnecessary, and therefore a turn-off, and that still didn’t clarify the question (the English) and so left several children feeling like they “don’t get it,” despite being able to respond correctly to unambiguous questions on the same content. The worst consequence, in my opinion, is that the lesson some children are getting is not about decimals but that they “just don’t get math.”
So what can we do to reduce negative consequences of missed communications?
At times, I read laments about teachers’ imprecision in language; these are decent examples and I’ll say a bit more about the issue, but later.
In my view (and in all kinds of circumstances), we give students a valuable message when we try to figure out what is sensible about their responses and explicitly state it: “Ah, you were thinking about the twelve hundredths we had just read, and [to the other student] you were thinking about just the number shown in that hundredths place.” In a case like this, it’s valuable even to acknowledge that can now see why they hesitated to answer and that we didn’t at first understand: “Oops, I wasn’t clear about which of those I meant.” Such responses from us teach several things. Possibly the most important is that students know that their thinking is valued even if it takes us a while to catch on. Another is that students see that our focus is on the logic, the sense they were trying to make even if it did not match our intent, and that we are assuming that’s their focus, too. That sets logic, not an answer to a particular question, at the center of the mathematical game. It values clarity, and it shows that we, too, struggle to communicate clearly. It detoxifies errors without fanfare and without “celebrating mistakes,” which students recognize as school propaganda. (Nobody ever says “Woohoo! I made a mistake!”) It models asking questions when we get lost in communicating an idea. (After all, if the teacher does that, it must be a useful and respectable tool.) And it acknowledges that trying to express mathematical ideas in words is clumsy and difficult—the problem is often not the thinking, but the communication—and that’s why mathematics has special vocabulary, notation and conventions. It’s not because mathematicians like fancy words and symbols.
And when we can’t understand students’ logic, we can admit that, legitimizing “I don’t understand what you mean” by showing that that happens to us, too. Kids’ explanations, even when they are totally correct, are often elliptical or garbled, so there’s plenty of opportunity for us to say, “Wait, I don’t get it. Could you explain again?,” giving you a chance to understand and giving them a chance to clarify and perhaps even rethink.
Finally, what about that issue of teachers’ imprecision in language? Being routinely more precise takes a lot of thought, a lot of knowledge, and a kind of self-consciousness and control that is hard to achieve, but building good “mathematical hygiene” (I attribute that lovely term to Roger Howe) with appropriate use of mathematical vocabulary and correct use of notation is a certainly a thing for teachers to think about. On the other hand we must also recognize that there will remain times when conveying a rough idea of what we mean is the best we can do, times when communication, especially with a child, can’t achieve understandability and precision at the same time. Teaching must walk a fine line.
Mathematics is so much easier than English.
(Just as I was finishing writing this blog post, I saw a brief article “Linguistic Ambiguity” by Ben Hookes in issue 103 of the Primary and Early Years Magazine on the NCTEM website, https://www.ncetm.org.uk/resources/52245, which gives other examples in which kids’ sensible interpretations of language leads to answers we might, but shouldn’t, consider wrong.)
]]>My work and that of my colleagues at Education Development Center has always put mathematical thinking—the habits of mind that are indigenous to our discipline—at the core of our work with teachers. What we’ve learned from expert teachers has led me to think more carefully about what it means to “work like a mathematician.” The attached essay details some of the things I’ve learned.
]]>We frequently use writing assignments to encourage students to examine topics in greater depth than what we cover in class, and we emphasize to our students that writing assignments constitute one of the most important dimensions on which students’ thinking will be assessed. Yet in our early implementation of these assignments, we frequently received work that did not reflect students’ full potential for understanding the topic explored in the assignment. In these cases, because we were using a roughly linear scale to assign each submission a numerical grade, which would then become part of a student’s overall grade in the course, we faced a difficult decision.
Neither option, however, seemed to address our greatest concern: that some of our students had not explored and communicated about the topic of the assignment with the depth desired. Moreover, numerical grades allowed many students to decide that they had gained enough, grade-wise, out of the assignment, and did not need to take advantage of opportunities to revise their work. In this article, we’ll talk about our journey toward crafting and implementing a grading scheme for writing assignments that provides greater opportunity for student learning and growth. While we use writing assignments specifically in the context of content courses for preservice teachers, we believe much of our advice is adaptable to other mathematics courses.
One of the major breakthroughs that helped us support students in submitting higher quality work was to develop clear expectations for these assignments and share them with students, an idea consistent with Braun’s (2014) essay on mathematical writing in PRIMUS and the recent MAA Instructional Practices Guide (2018). (See also two articles by Ben Braun on the blog.) However, it was not enough to be explicit about our expectations. In order to ensure that each student turned in work that met high quality standards, we adopted two principles:
Feedback and opportunities for growth
In order to help students learn to produce higher-quality writing assignments, we had to improve the quality of the feedback we gave. In our own efforts to learn more about assessment, we learned about a study by Butler (1987) in which fifth- and sixth-graders were given a sequence of divergent thinking tasks, and periodically given either numerical grades or individual comments related to their performance on the tasks. Butler found that of these two groups, the students who received comments were more likely to maintain interest in the tasks, and more likely to attribute their success to their potential to grow through sustained effort. On the other hand, students who received grades were more likely to attribute success to innate ability, and tended to maintain interest in the tasks only as long as they received positive messages about their ability relative to other students’. This agreed with our own experience as college mathematics teachers: we knew that given both grades and comments, our students often glanced at the grades and discarded or ignored the feedback. Thus we concluded that the first step we needed to take was to reduce our dependency (and with it, our students’ dependency) on numerical grades. In Specifications Grading, Nilson (2015) discussed the potential of minimal, non-graded feedback on writing assignments, which provided the seed from which our grading systems grew.
In addition to reducing the role of numerical grades, we needed to learn to give useful written feedback efficiently. Black and Wiliam (1998) found that feedback is more effective when it is focused on specific characteristics of tasks rather than simply on whether a learner’s response to a task is correct or incorrect, or on characteristics of the learner. Hattie and Timperley (2007) reinforced these findings, reporting that feedback that concerns a learner’s processing of a task, or their self-regulatory and metacognitive processes, can be more effective than feedback that focuses on characteristics of the learner or of the learner’s performance on a specific task. For example, we will often circle or highlight a paragraph in which a student’s mathematical reasoning is flawed or unclear, and ask a question aimed at prompting them to think more deeply about what they have written or request a clarification. The goal with the feedback is not to provide a clear roadmap of what students need to do to “fix” their work, but instead to prompt further thinking and motivate students to talk to us, a classmate, or a tutor about their work.
Implementation of an assignment
Once we decided to provide process-level feedback rather than simply giving an overall evaluation of students’ work, the next step was to implement a framework for our assignments that would encourage students to interact with our feedback and pursue suggested avenues for further investigation. Over several semesters, through a process of trial and error, we each independently converged toward the following format.
Pass/revise/fail grading can take a little getting used to. Now that we’ve done it for several semesters, our grading time is differently distributed, as well as more purposeful and better aimed at student learning, than it would be if we assigned numerical grades and focused on giving enough feedback to justify those grades. The need for students to resubmit assignments means that the end of the semester can be hectic; however, the quality of work that we get from the students after revisions makes it worth the effort. In particular, we find that grading a revised version of an assignment requires significantly less work than grading the original: the newer version usually contains fewer errors and is written more clearly; and we often remember the issues with the original version well enough to focus our attention on the parts of the paper that have been altered.
As far as students are concerned, although they initially find the requirement of revision and the all-or-nothing grading of these assignments onerous at first, many have expressed the sense that they see how this grading scheme supports our assertion that mathematics requires revision. Over multiple semesters of implementing these assignments as we improved in communicating our expectations, we have seen students’ work get more reflective, more thorough, and more professional. In addition, we have seen that by seizing opportunities to rethink and revise their work, students develop a more reliable command of some of the key ideas in a course; for example, preservice secondary teachers who complete a writing assignment on geometric proof are less likely to make unsupported claims in a geometric proof task on their final exam, and those who complete an assignment on the reasoning behind equation-solving procedures typically give a more mathematically precise explanation for extraneous solutions in an equation-solving task on the final. We believe that these learning gains would not be attained if we allowed students to settle for imprecise thinking on the writing assignments.
Personal notes on implementation
We include here some brief practical observations that have come from our individual implementation of these assignments in courses that we teach. Examples of our writing assignments may be found here.
Priya: My writing assignments for preservice elementary teachers are actually structured reflections/extensions on assigned readings. Students are asked to read about a mathematical topic in an excerpt from van de Walle et al. (2007) or Tobey and Minton (2010), for example, and answer some specific but interrelated questions about it. When I first instituted them, I chose 12 readings addressing key concepts that I wanted to assess with these assignments; trial and error, and a greater understanding of what I felt was truly important in the course has reduced that number to seven. I should also note that the mode of feedback in my class is not comprehensive written feedback, as it is in Cody’s class. Instead, I often provide quite minimal written feedback by simply circling a paragraph that needs to be rethought and allow students to work through the reasoning on their own. If they need further guidance, I encourage them to ask me or their peers questions.
Cody: I use writing assignments in my capstone course for preservice secondary teachers; each assignment asks students to do a thorough conceptual “unpacking” of a problem or procedure that can be found in high school mathematics. My assignments contrast with Priya’s in that each assignment has a specific set of expectations for mathematical reasoning; these expectations are enumerated in the document explaining the assignment. When writing these, I leave myself a bit of room to interpret the expectations flexibly depending on the specific direction a student takes with the assignment. For example, the assignment I have attached to this article asks students to identify two different ways of solving a rational equation – one that appears to lead to one real solution, and one that appears to lead to two (due to a transformation that changes the domain of the expression on each side). In addition to resolving this apparent contradiction and identifying the transformation that alters the solution set, students must explain each method in terms of properties of equality. Thus if a student uses “cross-multiplication” in one of their approaches and does not provide an algebraic explanation of why this strategy is valid, I ask them to revise the response to include such an explanation.
We have now both embraced the transition from grade-based feedback to process-based feedback on writing assignments. We find that this reorientation has allowed us to identify certain “non-negotiable” learning outcomes we want our students to achieve; and our practice of requiring revision and resubmission of papers allows us to provide appropriate assistance to each student, whether that assistance takes the form of a brief comment, a more elaborate marking of a paper, or a one-on-one consultation. One of the greatest benefits of this approach is that the progression in students’ papers provides us with clearer evidence of what students are learning as they work on these assignments. This evidence allows for further fine-tuning of the assignments so that each one provides an appropriate level of challenge for students, and so that success on an assignment clearly signifies attainment of the desired learning outcomes.
References
Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7-74.
Braun, B. (2014). Personal, expository, critical, and creative: Using writing in mathematics courses. PRIMUS, 24(6), 447-464.
Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest, and performance. Journal of Educational Psychology, 79(4), 474.
Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81-112.
Mathematical Association of America. (2018). MAA Instructional Practices Guide.Washington, DC: Mathematical Association of America.
Nilson, L. (2015). Specifications grading: Restoring rigor, motivating students, and saving faculty time. Sterling, VA: Stylus Publishing.
Tobey, C.R. & Minton, L. (2010). Uncovering student thinking in mathematics, grades K-5: 25 formative assessment probes for the elementary classroom. Corwin.
Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M., Wray, J. A., & Brown, E. T. (2007). Elementary and middle school mathematics: Teaching developmentally. Pearson.
]]>When two grandmasters face off in a chess tournament, they are faced with a complicated bit of game theory. If you were in one of their positions, you would prepare for the match by studying your opponent’s games in great depth. You would study board positions they had created, looking for weaknesses in their defenses and blunders their previous opponents (or they themselves) had made. It would be safe for them to assume that you could have a strategy in mind to counter any of their strategies that had previously been successful.
Of course, your opponent would naturally study your body of work in the same way. Therefore, by the time you sat down at the board, there would be a natural expectation that you know that your opponent knows that you know as much as you possibly could about them, and likewise they have the same expectation of you.
As a consequence, the natural strategy for determining who is the better player is to try to avoid these positions in the first place. Don’t allow the board to get to a point where you have been defeated in the past. Don’t allow the board to get to a point where you have been successful in the past because your opponent might know how to turn that position to their advantage. Get away from what has been seen before and create a new position that truly tests the skill of each player. There’s a term for this – chess players call it going “off book.”
To chess enthusiasts, this moment is exciting. It’s the moment in the game when the board reflects a position that has never been recorded in a tournament. It is an opportunity for observers to experience chess history and witness the creation of new knowledge or strategy. Every move is new and the anticipation of what might come next is thrilling.
Why is this relevant to math education?
Employers want students to be prepared to tackle a variety of real world problems. These problems may be vague or imprecise. They may be posed without any sense of what the answer should entail, or perhaps without understanding what the problem truly is. We don’t need to look much further than the annual Mathematical Contest in Modeling to see a wealth of interesting real-world problems that can inspire a wide range of potential solutions.
Of course, the same principle applies to pure mathematics. Answers to research problems of any sort are exciting because they represent a creation of knowledge. There is a thrill that comes from writing down a formula or idea that no one has written before; from creating mathematics that has not been seen. This happens as a result of getting away from the books.
How can we create a similar experience for our students?
An obvious venue for this is through research experiences for undergraduates, ranging from formal summer programs to community/industry partnerships or projects that are part of a class. However, research experiences are not the only way to get students to think outside the textbook. For me, the most exciting day of any class is the one where a student asks me a question I can’t answer. I love these types of questions, and I try to be as transparent as I can about my thought process. I’ll say, “That’s a really good question! I don’t know the answer right now, and to me that’s really exciting because it means you’re thinking really deeply about the material we’re studying.” Sometimes, I need a few minutes or a night to think about the problem and figure out how to answer it. Sometimes I don’t know the answer because it is completely new.
For example, one time a calculus student observed the periodic cycle of derivatives of trig functions (the derivative of sin(x) is cos(x), whose derivative is –sin(x), whose derivative is –cos(x), whose derivative is sin(x)) and asked “Does that have anything to do with imaginary numbers?” On the way back to my office after class, I realized that his observation was related to Euler’s formula e^(it) = cos(t)+isin(t), which then inspired an exciting homework problem when we reached the chapter on derivatives of exponential functions.
In another instance, I devoted two days at the end of the quarter of a graph theory class to unsolved problems in graph theory. My philosophy was that attempting to cram new material into the last two days of the course, and subsequently testing the students on this new material on the final exam (which was to be given two days after the last day of class) was an unfair assessment of their learning. Perhaps we would be better served pedagogically by exploring applications of what they had learned in a quarter of graph theory.
I came to class with an open problem and asked students to spend 5 minutes in a group brainstorming potential approaches. The students shared their ideas with the class and then split into groups with peers who were thinking about the problem in similar ways. We spent the rest of class working on the problem. My role was to bounce from group to group, hear their ideas, and provide input as best as I could. My most common response was “I don’t know, but that seems interesting.” When I could, I would point to interesting special cases, share my intuition, or point to terminology or references that could be helpful. Some students wrote code. Others drew pictures. Others generated data. Others focused on special families of graphs. But everyone worked productively on something. Everyone generated new ideas. Everyone created new mathematics, or at least, mathematics that was new to us.
Working on unsolved problems allowed the students to showcase the variety of graph-theoretic ideas they had learned over the course of the quarter. It served as a good exercise in preparing for the final exam because they had to reflect broadly on the course material and think about applying those ideas to new problems in a relatively short amount of time. Beyond this, the activity showed the students how much they had learned and let them see that they were capable of applying their knowledge to unsolved problems. Research didn’t require a PhD or an internship in a fancy lab – it just required a blackboard and the willingness to say “what if…?”.
My hypothesis is that the students’ excitement to explore new ideas and ask interesting questions partially stemmed from my openness to hearing their ideas and my willingness to say “I don’t know.” This is not to say that we as professors should not have a deep understanding of mathematics. We absolutely must. We exhibit the breadth of our knowledge through our teaching in the way we present the material, or the different ways we can explain a concept to a student. At the same time, we can be honest when students make keen observations or ask thoughtful questions. We should be excited when our students push us to think deeply about our subject and celebrate their insights.
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I have a secret: For the last year or so, my nine-year-old daughter and I have been trying to develop a meditation practice. This guy, Andy, who leads us daily through meditation sessions facilitated by a phone app, has become a familiar name between my daughter and me. (Even my five-year-old occasionally mentions Andy when going to bed at night; sleepy-time Andy tells us to lie on our backs and close our eyes and start by saying good night to our toes.) One day my daughter posed me a question. We had just completed our ten-minute session for the day. She was not willing to move on yet, it seemed, so I waited. She finally formulated her question and asked, very carefully, “Mom, is Andy perfect?”
This is a profound question even though it has an easy answer: “No”. If Andy is human, he is not perfect. Yet none of his flaws are really my business, because he is effectively teaching us to be better. He is consistently, with kindness, in good humor, and with no sign of condescension, telling us how we can do better. In every session, or let me be honest, in most sessions, we learn from him.
Aren’t there always those we look up to who exemplify ideals we wish to uphold or those who represent the type of character that leaves us in awe? Reading Art Duval’s post on kindness in this very blog, listening to Francis Su’s talk on mathematics for human flourishing, digging into clear critiques of our community fearlessly dispensed by mathematicians such as Piper Harron and Izabella Laba, some of us might wistfully say: but I am not good enough. I am not as capable. I am not as kind. I am not as forgiving. I am not as insightful. I am not as brave.
Now let me rephrase that for you so as to be clear. All of the above are ways of saying the same thing: “I am not ready to be vulnerable.” All these amazing people are amazing partially because they are willing to put themselves out there, trying to live up to their own ideals. (And for some, an alternative may not even exist.) Do they ever falter? Maybe they do. It is not my story to tell. Again, like Andy, any of their possible faltering is none of my business. What is my business is what I learn from them.
Now some might be concerned that I am potentially giving some people free pass to be terrible human beings as long as they try to uphold certain ideals. Slippery slope and such, and where do you stand with respect to Thomas Jefferson and Bertrand Russell and Roman Polanski and Bill Cosby and <pick your favorite fallen idol here>? I could of course share my opinions about those particularly imperfect men here, or I could simply affirm that certain people seem to be allowed more imperfections than others. But that is not my point here.
My point is that those of us who see teaching as our vocation are probably not all perfect, and many of us will never be perfect. But we should allow ourselves this imperfection as we continue to try to teach as well as we possibly can. Ours is a profession in flux: we grow every day we go into the classroom, we have opportunities to learn with every mistake we make, with every new topic we get into, and with every new pedagogical tool we adopt or leave behind. And we are not going to be perfect every day; for many of us, it is actually a rare day that ends without any major snafus. But we are human and we continue to grow and make mistakes and strive to improve till we die. When we can accept this as a fact of life and stop beating ourselves up about our imperfections, we have that breathing room to grow, and perhaps even ironically, to get closer to our own ideals.
As teachers, we owe it to ourselves and to our students to try and be decent human beings. There are simple rules after all: Respect your students, respect the fact that there is a power differential in the classroom and in any teacher-student relationship, and respect the needs and life constraints of your students. Once we are agreed upon these basic rules, however, the test is no longer about perfection. We need to allow ourselves to embrace our humanity and our own “under construction” status.
This, I hope, is a liberating point. As a teacher you are probably not perfect. You are probably not doing everything right. But if you have your heart in the right place (in terms of the three respect-related rules above) and if you are striving to be a better teacher, occasional failures or imperfections are expected and should not stop you from trying and trying again. Francis Su explains this in exquisite language in his talk on the lesson of grace in teaching, and I cannot claim to be able to say it better: “Your accomplishments are NOT what make you a worthy human being.” Here is the friendly amendment summarizing this post: Your imperfections are part of what makes you a worthy human being. Do not reject or hate them. Instead accept them, learn from them, and grow with them.
Next comes the question of transparency. Ok, even if we come to terms with our own humanity and the necessarily associated imperfection, how much of this can we reveal to our students and colleagues? Do we share with students our pedagogical flaws or mathematical troubles? Do we discuss our shortcomings with colleagues?
Francis Su talks about some of this in the context of grace, but my daughter has already noted that Andy does not admit to many flaws in his recordings. In my case, I have noticed that as I got older and became a part of the furniture in my department, my teaching persona has grown more and more comfortable with her faults in front of her students. I have found also that as long as I do not pretend to be perfect, my students are able to show me the compassion I sometimes seem to withhold from myself. Opening myself up in this way and receiving such compassion even if occasionally helped my teaching persona become a much more fluid and connected part of my overall identity.
Some of my transparency about my mathematical shortcomings has even helped me connect with students. In calculus for example, the first time I shared with my students that epsilon-delta proofs had been awfully confusing to me as I was learning them, I noticed some optimism appear in several students’ strained faces. This revelation is now a routine part of our discussion when we get to that topic. In other contexts, too, I often share with my students that it took me a while to understand some of the connections we are making together. I occasionally share with them several of the treasured but simple-minded tools and mnemonics I use to differentiate between basic ideas or pairs of words (for example a capital “H” has a horizontal line through it and that is how I, a non-native speaker of English, can tell apart the words “horizontal” and “vertical”).
A not-so-trivial question raises its head here: Is the ability to show vulnerability to students or to colleagues a sign of privilege? I do not have proof for this, but my tentative answer has to be yes here. I have been lucky throughout my teaching career to have only rarely faced authority-undermining behavior from students (and most of that happened when I was pregnant). But if you dig into that luck, you will find several layers of privilege. My skin color, my glasses, and my weird accent seem to have protected me for years against student doubt about my mathematical competence. Furthermore my relentlessly growing age has basically immunized me against youth-based stereotypes. In my present context then, it is, I surmise, both personally rewarding and professionally productive for me to not hide my imperfections.
However, many do not feel like they have that kind of freedom. And if you are not tenured or on the tenure track, if you are an adjunct, if you do not have a PhD, if you are not white, if you are not cis-gendered, if you are not able-bodied, if you work at an institution where students consistently challenge instructors’ authority, you might be correct in assuming that your students and even sometimes your colleagues may not always be compassionate about your humanity. People are not always nice and they are not even always good. If people are indifferent, inconsiderate, or just plain deplorable about my imperfections, as occasionally they are bound to be, I try to interpret this as a sign telling me something about them and not about me; it might even be their burnt coffee that morning. But hostility is not the norm in my professional environment today. When there is at least a modicum of mutual respect in a professional context, I’d say that giving people the benefit of the doubt goes a long way.
If on the other hand your professional context is hostile or dehumanizing or if your academic position is vulnerable, then I certainly do not advise displaying imperfections. In fact many people in such situations end up extending the no-defecation-in-your-place-of-employment rule to the level of no-show-of-humanity-in-your-place-of-employment. Such defensive positions are about self-preservation, which comes above all else. I know; I myself have lived in that mode for several years. So I will not suggest that people in such positions do anything that will make them feel more vulnerable. People know their contexts best.
As some of our colleagues find themselves forced to take up defensive positions, the rest of us with various sorts and levels of privilege have the duty to work to make our spaces less dehumanizing. Part of this rehumanization is going to be about accepting our own imperfections and living with them openly. Not giving credence to the genius worship cult, not paying lip service to the mathematical celebrity culture, not acting the part of the perfect professor are some of the follow-up steps. (If you are ambitious enough, let us rehumanize mathematics from its roots!) But showing our humanity and emphasizing to students, to colleagues, and to ourselves that imperfection is part of the human package is a good start.
End Note
For those readers who were disappointed that this post did not turn out to be about contemplation in the mathematics classroom, here are a few leads to pursue: If you are totally new to the topic, I would suggest starting with Tobin Hart’s article on mindfulness in the classroom. Luke Wolcott’s article on contemplation in mathematics brings things much closer to home. On a related note, some might find this article I wrote about metacognition in the mathematics classroom of interest, too. After all, being reflective about our pedagogical practice and encouraging students to be reflective of their own learning go hand in hand and naturally round out a coherent view of contemplative pedagogy.
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I went to New York City Public Schools, in the Bronx. I always enjoyed arithmetic and mastered it easily. I remember not knowing what ‘fractions’ were, but don’t remember learning about them, any more than I remember learning to read. The understanding came to me naturally, and I hardly noticed the process. Even first year algebra didn’t seem like a learning process, more like a set of exercises. So I had mastered a lot of mathematics (well, a lot of algorithms) before I really understood what it was I was learning.
A revelation came in ninth grade, when I was 13. Ms. Blanche Funke, a good math teacher in JHS 135, took some of us during lunch and organized us as a math team, to compete against other local Junior High Schools. Now this is work I have since spent decades doing, and I know now what could have been done. But Ms. Funke didn’t quite. Her idea was to give us advanced training in textbook algebra—not to find ways to make us think differently about that same algebra.
So she gave us the definition of an arithmetic progression, and the standard formulas. And a problem something like: “Insert 3 arithmetic means between 8 and 20.” I loved this work. Plug into one formula, get the common difference, then plug into another formula and get the three required numbers. I could see what I needed to do and took joy in starting the work.
But next to me was my friend David Dolinko, and he was busy drawing something in his notebook—some diagram of a molecule in chemistry. (Professor Dolinko has lately retired from the UCLA School of Law). I poked David. “C’mon. Let’s do this problem. It’s fun!”
David looked at me, as if annoyed at the interruption: “Oh, I did that already. Eight, eleven, fourteen, seventeen, twenty.” And went back to his drawing.
That moment changed my world. Suddenly I realized that these formulas had meaning, were trying to express something. They were expressing that the numbers were ‘equally spaced’. So David could just pick them out—the numbers were small—and didn’t have to bother with the algebra. Algebra has meaning. And if you know its meaning you can use it more effectively. Suddenly, instead of black and white, I saw the world of algebra in color.
I thought about this a long time. The colors attracted me more and more. I wasn’t just good at mathematics. I enjoyed it, and enjoyed being good at it.
Well, the next year I was still sitting next to my friend David, in the last seat, last row of a classroom in the Bronx High School of Science. We were taking geometry, the classic neo-Euclidean syllabus, taught by one Dr. Louis Cohen. He was a somewhat impersonal teacher, or so we thought, but a master of his discipline. And of teaching it. So one day he had covered (I don’t remember how) the theorem that the angles of a triangle sum to 180 degrees. The lesson had gone quickly, so he filled the time with some ‘honors’ problems: the sum of the angles of a quadrilateral, some problems with exterior angles, and so on. And to cap it off, he drew a five-pointed star on the board:
Not a regular figure, but just any one that came to hand, using the usual technique of following the diagonals of an imaginary pentagon. He then asked for the sum of the angles at the points of the star.
My hand shot up, seemingly of its own accord. “180 degrees,” I said, without quite knowing why. And to my horror, Dr. Cohen strode calmly down the aisle to my desk, with a piece of chalk in his hand, handed me the chalk, and asked me to explain to the class how I had figured this out. But I didn’t know how I had figured it out. I just saw it, with intuitive clarity. What was I going to do?
I was lucky that we sat in the back of the room. As I saw him coming towards me, I began to analyze my own thoughts. And as I walked to the front, I figured out what to say. To this day I remember my hand trembling and my voice shaking as I pointed out certain triangles, certain exterior angles, and got the angle measures all to ‘live’ in the same triangle. Dr. Cohen praised me, then gave a slicker version of the proof that must have clarified it for the other students. Of course, there are better ways even than his to prove this statement. If the reader can’t think of a nice proof offhand, take a look (for example) at https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-polygons/v/sum-of-the-exterior-angles-of-convex-polygon (accessed 6/2018). The argument can be adjusted to cover non-convex polygons.
Why is this important? Well, it is important for us to understand that the language of mathematics is a language of thought. And that thought is synonymous with intuitive thought. We sometimes get caught up in the expression of our intuitions, and fail to go back and make clear, even to ourselves, what we are talking about. This phenomenon has deep implications for teaching. How we do this, how we know it has happened, how we integrate it into the teaching of mathematics as a forma language, are all questions we must struggle with. But they are not questions that we can beg. We must somehow be sure that students can eventually understand our results on an intuitive level, whether or not we communicate with them on this level directly. Without that, we are teaching algorithms—even algorithms of proof—and not mathematics.
I invite readers to contribute their ideas to this blog about how to make mathematics accessible on an intuitive level.
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