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:

- Which blend of these pedagogical approaches have you found congenial for specific audiences?
- Are there other approaches that are essentially different that I did not include in this list?
- What steps do you find helpful when transitioning students who are used to one pedagogical approach to another?
- There is a tendency for students from high poverty schools to be exposed primarily to Traditional Math approaches. The Common Core State Standards represent an effort to slightly improve on this prevailing norm, by pushing teachers in the direction of Conceptual Math instruction. What do you think people who love math and teaching should do to improve access to high quality math education for these students?

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:

- Traditional Math instruction is efficient — the very simplest methods for solving the most common problems are taught directly to the students, whose only task is to remember what they have been told. The arrangement of topics leads students in an orderly way through the topics so that they have precisely the skills they need at each later point in the curriculum.

- Traditional Math instruction is controlled — the instructor does not need to take student responses into consideration when planning the content and order of the course, though he or she may respond to students who need help by adjusting the pace of the course slightly. There is no chance that the instructor or the students will contemplate a question that cannot immediately be answered by recourse to the authority of the text or the algorithm.

- Traditional Math instruction is straight-forward — problems do not have more than one possible answer or interpretation, there is always one that is designated as correct. Students never work on problems that take weeks or months to resolve.

- Traditional Math instruction is egalitarian — every aspect of every lesson is expected to be within the grasp of every student. All students are expected to gain the same knowledge from each lesson.

- Students memorize facts and algorithms without developing conceptual understanding. In K-12 education, the development of student strategies is actively discouraged in favor of memorization by instituting timed tests and by banishing manipulative aids such as the use of fingers. Cute sayings and songs are used to memorize facts.
- Students are eased into difficult topics by careful scaffolding, often divided into specific types of problems. Conceptually messy or difficult ideas are papered over by giving students a definition or rule for dealing with those situations.
- Providing practice so that students readily recall facts and perform procedures is more important than developing the ability of students to solve realistic, practical problems or to think logically and critically.
- Students work word problems which represent unrealistic, overly simplified situations. The problem statements include no extraneous information, provide all needed information, and usually involve only one math concept. Students are actively discouraged from reading and interpreting the text of these problems. Students are often instead encouraged to look in the section of the book they are in to find an example problem which is just like the one they are asked to solve. Students are told to focus only on certain key words which are supposed to tell them which operation to apply without thinking. They are discouraged from drawing accurate pictures representing the problem (even young students are told to draw only circles or a bar or tally marks to show the quantities efficiently). Older students are praised for not needing to draw any diagrams or pictures to aid their thinking. Students are not expected to write substantively to explain their thinking. Because the original context is not actually that important, students are not expected to re-contextualize their answers to consider whether they make sense in the real world.
- Students are trained that being good at math means being quick and not needing to think. Needing to struggle is a sign of a lack of intelligence, practice, or attentiveness.

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:

- Conceptual Math instruction is less efficient — it takes longer to get to a fluent procedure for solving a given kind of problem. However, the arrangement of topics in a course is still carefully chosen so that students are guided to an understanding of the course topics in an orderly way.
- Conceptual Math instruction is somewhat risky — the instructor must respond to the ideas of the students. The instructor must help them to see whether their ideas make sense and how they can be expressed using formal mathematical notation. There is a chance that a student will bring up an approach that is unknown to the instructor. However, the domain of inquiry is usually restrictive enough that most questions and interpretations can be anticipated.
- Conceptual Math instruction is fairly straight-forward — students sometimes tackle problems that have more than one possible answer or interpretation, but the domain of possibilities is usually somewhat restricted. Concepts do often take days, weeks, or even years to be developed, but individual problems usually are resolved within one day.
- Conceptual Math instruction is egalitarian — every aspect of every lesson is expected to be within the grasp of every student. While students may use different methods to arrive at answers, they all cover the same content.
- When learning a new concept, students are first introduced to concrete models and situations that illustrate the operation and they use manipulatives and drawings to solve problems. They then devise and discuss their own strategies based on place value, properties of operations, or an understanding of other math concepts. Finally, students learn algorithms and are expected to be able to justify why standard approaches work verbally and in writing.
- Students are given word problems which represent unrealistically simplified situations, usually involving unrealistic numbers and measurements, and usually expressed with a minimal number of sentences using simplistic vocabulary. The problem statements often do include extraneous information (often these problems are featured in a special section in the curriculum). Students are encouraged to read and interpret the text of the problems and are trained not to rely on keywords too much by the practice of inserting unknowns in various parts of the story rather than always at the end. However, students are usually trained to use generic circles, a bar, or tables to show quantities rather than freely representing them in whatever way makes sense to them. Students are expected to write substantively to explain their thinking. Students are also expected to re-contextualize their answers enough to decide whether they make sense.
- Instructors strive to respond to student questions by helping the students to explore them. This can be very challenging. When attempting to support the development of student strategies, instructors are often tempted to tell the students how they should be thinking and how they should record their thinking rather than allowing these processes to originate with the students. It is difficult to avoid teaching all of the strategies as if they were algorithms.
- Students are trained that being good at math means being willing and able to think carefully and explain their reasoning. Students are trained to believe that struggling is a sign that their brains are growing, but they do not usually struggle with any given concept for very long before they are rescued by a hint from an instructor. They are carefully supported through the process of learning by being given problems that are just within their zone of proximal development.

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:

- Inquiry instruction is rather inefficient — lots of time is necessarily used to brainstorm, discuss, decide, and resolve conflicts. Excellent classroom management and organizational skills on the part of the instructor are necessary to avoid complete chaos or students spending large amounts of time doing trivial tasks that support no real learning.
- Inquiry instruction is very risky — it is likely that students will come up with questions the instructor does not know how to address, or that they will pose questions that don’t make any sense, or that the class will fail to cover the objectives of the course by side-stepping the approach the instructor had in mind. Students have authority to invent definitions and procedures, but they may end up producing results which are mathematically incorrect or which lack the appropriate level of rigor. Coming up with good topics for a given group of students and ways to provide structure for students as they work is very challenging.
- Inquiry instruction is complicated — everything depends on the individual choices of the students and it can be difficult to manage a classroom full of divergent thinkers and personalities. Materials can be expensive and material management can be challenging. Problems and questions take a long time to resolve and careful planning and management is needed to avoid getting bogged down. Students can easily get stuck and give up in frustration.
- Inquiry instruction is not egalitarian — different students will learn different things as a result of their choices. Social and academic hierarchies will be more obvious in the classroom due to frequent group work.
- Students are provided with much less scaffolding than is typical in either Traditional Math or Conceptual Math approaches. Students must learn to devise methods of making complicated problems tractable. Students wrestle with conceptually messy and difficult ideas through small and large group discussions and experiments of their own devising.
- Students are trained that being good at math means being willing to stick with a complicated problem, struggling to find a solution, and being able to communicate their thinking to others. \item The process of developing, refining, investigating, solving, and presenting questions and their solutions is more important than the specific math topics studied. The history of mathematics as a discipline and cultural aspects of doing mathematics are not emphasized.

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:

- Math Circle instruction is deliberately inefficient — this approach is usually not concerned with mastery of specific math topics. However, when using this approach to teach externally specified content, a Math Circle instructor will seek a mathematically interesting question that will likely lead students past the content to be addressed. The question will not usually be directly about the content to be taught (although sometimes it is). It often takes weeks or months for students to resolve the many questions they pose along the way.
- Math Circle instruction is a high wire act — the instructor never knows where the students will go next. The instructor usually must do quite a bit of research into possible directions the discussion might go to scope out the landscape for possible scenic overlooks and pitfalls. However, the students might take off in another direction which the instructor has not prepared. No matter how much mathematics the instructor knows it is impossible to prepare for everything that might happen.
- Math Circle instruction is complicated — it is very challenging to orchestrate an intellectually satisfying conversation that allows all students to participate. The instructor must think hard to choose an opening gambit for the topic which is low entry, high ceiling, and compelling to the specific students in the class. It can be difficult to find good questions for a given group of students, especially if there are also specific content goals to incorporate.
- Math Circle instruction is not egalitarian — different students will learn different things. This is because the instructor is usually not careful to make sure that all students have mastered any of the content discussed. Some students will have mastered all aspects of the proofs discussed, and others may only have practiced some basic math skills while having fun looking for patterns. A Math Circle instructor chooses to keep aiming high even though not everyone may be ready to appreciate the nuances yet.
- Math Circle instructors tend not to choose topics that are straightforward applications of standard math concepts. This means that real-world problems that are uninteresting mathematically are not usually included. Real-world applications tend to come in only when the mathematics that emerges from those applications is intrinsically compelling.
- Math Circles drop students into the deep end — often giving them impossible or unsolved problems, often exposing them to concepts well beyond those typical for their grade level. This has nothing to do with the level of preparation of the students. The expectation of critical thinking and problem solving is simply set much higher than usual, and the instructor establishes the expectation that the group will need to struggle and might sometimes fail to answer some questions.
- Students are trained not to worry about being good at math. Mathematics is a huge discipline and it is impossible for anyone to know most of what there is to know. There will always be someone who knows more than you and who can work faster, and who cares anyway? We do mathematics because it is fun, beautiful, and is a means of solving interesting and important problems. We learn to appreciate the beauty of mathematical ideas and the joy of feeling our minds at work

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.

- We could assign low grades to work that did not meet our expectations; this would have the advantage of sending students a clear message about whether their work meets the standards of the course, but it might demotivate students or limit the potential of an otherwise competent student to earn a good grade in the course.
- We could assign moderate-to-high grades to such work; this would lower the stakes of failure for students, but it would also require us to endorse work that does not meet a high standard.

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:

- To pass the course, each student must complete a specified number of writing assignments successfully; an assignment is not successful until it meets a set of predetermined standards.
- Whether a student’s attempt at an assignment is successful or not – but especially when it is not – provide specific feedback, aligned with the stated standards for the assignment, that provides a clear direction for improvement so that the student can revise and resubmit their assignment. One of the key mathematical practices that we want to instill in our students is the fact that their mathematical thinking can and should be revised, and that this revision process is an important part of the process of intellectual growth.

**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.

**Assign work and specify expectations.**When we first hand out the writing assignments, we remind students to refer to a list of general expectations that we have set for their work. These expectations cover aspects such as professionalism, completeness, and mathematical thinking. We try to be as explicit with these as possible, with examples. Here’s one example: “Every mathematical statement you make should be justified. For example, if you say, ‘the sum of two even numbers is even,’ you should explain briefly why that statement is true. This is doubly important for statements upon which you later rely to explain larger or more complicated concepts.” For students who have little experience writing in mathematics or even with justifying their thinking, this level of detail can be helpful, since they are often unsure of the level of mathematical depth we expect from their work.**Initial assessment.**Once students turn in their writing assignments by a set deadline, we read them over and provide feedback based on the principles outlined above. If a student’s work is clear, well-reasoned, and meets the stated expectations for the assignment (with minimal spelling and grammatical errors), we will assign them credit for having completed the assignment. Otherwise, we ask them to revise their work.**Assessing the revision.**The most important aspect of the revised writing assignment is whether students have addressed all of the comments that were made on their first draft. We stress that these assignments should be considered as cohesive essays, and even though only one part of the assignment may have a comment by it, addressing that comment can affect their whole essay. For example, one of us (Priya) has observed that preservice elementary students often struggle with the implications of the definition of a rhombus, which can have pervasive effects on their writing assignment about quadrilaterals. In such a case, we may only comment once on a paper that a student needs to check her understanding of the definition of a rhombus; but in the revision, we expect that she will revise the entire paper with this comment in mind. Revisions that meet expectations are assigned credit at this stage. If a revised version of a paper substantially improves upon the previous version but still does not earn credit, we may ask the student to submit another revision or to speak with us in person so that we can address any lingering issues in greater detail.

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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As editor for this blog for the coming year, I would like to invite you to continue its lively and meaningful conversation, of the quality that has been established by my predecessors. I am most grateful to Ben Braun for setting up this useful tool for communication, and hope to continue and expand the dialogue it has afforded us.

I am equally grateful to Art Duval, Steven Klee, and Diana White for graciously agreeing to continue on this editorial board, and for Priscilla Bremser, who has retired from the board, for her service to the community. At https://blogs.ams.org/matheducation/about-the-editors/ you will find brief biographies of each of us on the editorial board.

Meanwhile, I would like to look at two aspects of blogging that we can focus on in the coming year.

BRIDGES, NOT WALLS

My intent in taking responsibility for this blog was to further communication in the mathematical community. For me, communication is the most important stimulus for synergy, and lack of communication its most stubborn obstacle.

I have spent all my professional life in three distinct mathematical communities: research mathematics, mathematics education as an academic field, and classroom mathematics education. Their interactions have always been fruitful, but also problematic. The problems are rarely personal. I seem to get along with most of my colleagues. Even when we disagree, even to the extent of having words, things eventually return to a normal, collegial state. The problems arise, I think, from the institutions we live in. Each group is rewarded for different goals and charged with different responsibilities. And different value systems have grown up around these circumstances.

Classroom mathematics, especially on the pre-college level, is mainly the charge of our public schools, which are organized in the US by the smallest and most local units of government. So responsibility tends to be to the community, the family, the individual student. Teachers more and more face the problem of test preparation and accountability. Are the students actually learning good mathematics? Could they be learning in more efficient or more accurate ways? The importance of these questions is—often—eclipsed by the need to demonstrate achievement by standards external to the schools in which teachers work.

Oddly, the accrual of knowledge, the collection of experiences of teachers, is the charge of a different set of institutions: our schools of education. These are academic institutions, and people working in these schools are judged, famously, by publication. But are their research findings having the desired effect in schools and classrooms? Are research questions crafted to respond to the problems of teachers? Is the mathematics being learned precise and pertinent? These are questions that often go unasked by tenure and promotion committees in an academic environment, and sometimes also by journal editors. In its worst cases, the dialogue spins away from the working classroom and the actual mathematics being taught.

The study of mathematics is likewise an academic discipline, and mathematicians are judge by research publications. Mathematicians who get involved in education, who work with schools of education or public schools are sometimes seen as neglecting their duty to their own profession. Why work on curriculum, or outreach, or teacher education, when you could publish two more research articles this year?

These three descriptions, of course, are simply slander against the very people I work with most—those who dare to cross the lines drawn by our institutions around us. And, Dear Reader, you are more than likely to be among these renegades.

I personally would like to hear more about your successes, about how my somewhat cynical descriptions are wrong. Perhaps most important, I would like to hear about how the problems I raise above, of institutional demands thwarting personal efforts, have been dealt with.

We need bridges, not walls. We need doors, not fences. How have you been building them? What help did you get? What obstacles did you face?

THE PLURAL OF ANECDOTE IS DATA

The negation of this subtitle is an old saw, whose veracity I dare to question.

It seems to me that educational research does not pay enough attention to anecdotes. Anecdota (the more traditional plural of the word) offer two important opportunities to academic research. The first is the formation of hypotheses. The scientific method, the usual model for seeking knowledge, does not tell us what questions to ask or what to observe. The wellsprings of hypotheses are unconscious: they lie in our reactions to the thoughts and actions of others, our responses to something that catches our attention in our environment. We are not in control of our unconscious thoughts.

And I think this is a good thing. The unconscious is a source of creativity, of new ideas. So the best we can do is free ourselves, at times, from rational constraint—then later go back and examine our ideas more rationally. But we dare not talk about this process in formal scientific investigations. I think this blog is an excellent venue for just such conversation. What anecdota have you found important in your life? What have you learned from them? Can we use them as springboards for more disciplined investigation?

More formal investigation involves collection of anecdotes, or shaping of experiences into experiments, or refining the nature of the tale. But I would argue that formal investigation begins with informal observation. This is one sense in which data is a plural of anecdote.

Is this true even in the pristine world of mathematics? The creation of the human mind, which may or may not deal with observation of reality? I would argue yes. But in fact I will not argue this. I defer to Pólya, Poincaré, and other mathematicians who have given us glimpses into their mental workshops. And I invite similar glimpses, or analyses of historical work, here in this blog.

Another sense in which anecdote is important is in the reification of formally achieved results. It happens that, even when an hypothesis is the result of anecdotal observation, the process of formal investigation skews the meaning. The need for rigor of thought, for comparison of data, can constrain the very data we are comparing. This is the deeper meaning of the old joke about psychology, the one whose punchline is “What does it tell us about rats?”

Is this true in public policy? After all, when we make rules for a mass of people, we must ‘act statistically,’ do the greatest good for the greatest number. Do anecdotes have a place in this arena? Well, yes. Let’s get real. And question another old saw.

“Facts are stubborn things.” This quote has been variously attributed (https://quoteinvestigator.com/2010/06/18/facts-stubborn/, accessed 4/2018), most famously to Samuel Adams. And I’m not sure it’s true. In public discussion, facts can be pliable, ductile, malleable. Even when research methods are unquestionably rigorous, the questions of which facts to adduce and how they relate to the decision being taken are themselves not data-driven. They are matters of judgment.

I find that opinions are much more stubborn than facts. And opinions are often based on anecdota, on cases that are personally known to the holder of the opinion, or stories—anecdotes—that ring true on an individual level. So even in the area of public policy, if we don’t pay attention to anecdotes, to their meaning to individuals, we will not be able to act effectively.

Anecdotes about how research is used, how it plays out in the field, what effect research has on practice, can offer valuable feedback to the researcher. I invite readers to use this blog as a place to tell stories of direct experience, of the sort deemed ‘anecdotal’ in more formal academic research.

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Mathematics is the result of human curiosity and our desire to explain, predict, and explore observed and imagined phenomena. Our shared curiosity and sense of wonder is the wellspring of our mathematical culture. Yet a common sadness is felt by those who love mathematics, as we witness people’s wonder and curiosity stilled by strong cultural and social forces. As Paul Lockhart writes in *A Mathematician’s Lament*:

If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done… Everyone knows that something is wrong.

Many mathematicians, mathematics teachers, and mathematics fans and ambassadors share these feelings. It is natural that for many of us, our primary responses to this arise through our teaching, in an effort to help students rediscover their natural sense of mathematical joy and curiosity. However, my belief is that this situation is actually a symptom of an issue that extends beyond teaching and learning, and beyond the confines of mathematics. I believe that at its core, this issue involves our cultural responses to three questions:

- How do we build relationships with those around us?
- What accomplishments do we reward and recognize?
- What stories do we tell, especially about mathematics and mathematicians?

As powerful as classroom cultures and environments can and should be, I believe we must have an even grander vision for ourselves and our community. We need to find ways to change some core qualities of the culture of mathematics itself, qualities related to the three questions above. A central challenge is that these changes are generally orthogonal to cultural norms of society at large. In this article, I share some reflections on these questions, along with ideas for how we can work together to meet the challenge of improving the culture of mathematics both within and beyond the classroom.

**How Do We Build Relationships With Those Around Us?**

The practice of doing mathematics is one infused with emotional complexity. In *Loving and Hating Mathematics: Challenging the Myths of Mathematical Life*, Reuben Hersh and Vera John-Steiner write:

Mathematics is an artifact created by thinking creatures of flesh and blood… Mathematicians, like all people, think socially and emotionally in the categories of their time and place and culture. In any great endeavor, such as the structuring of mathematical knowledge, we bring all of our humanity to the work… It’s a challenge for everyone to achieve balance in one’s emotional life. It’s a particularly severe challenge for those working in mathematics, where the pursuit of certainty, without a clearly identified path, can sometimes lead to despair. The mathematicians’ absorption in their special, separate world of thought is central to their accomplishments and their joy in doing mathematics. Yet all creative work requires support.

Unfortunately, our mathematical culture does not encourage us to discuss and share the emotional ups and downs of our mathematical lives. While this is also true of our society at large, at least our broader culture acknowledges the role that therapy and counseling can play to improve our lives. In mathematics, our tacit prohibition on discussing the emotional aspects of mathematics has serious consequences for our community, ranging from mental health issues, especially among graduate students, to people unnecessarily working in isolation (e.g. Dusa McDuff’s reflections here), to differences in the sense of belonging and efficacy that people feel in mathematics. As Hersh and John-Steiner point out, these consequences are particularly detrimental in mathematics, due to the nature of our work.

Whether or not individuals choose to actively share their stories with colleagues, teachers, or peers is not the point here — what matters is whether or not individuals in our community feel that they are surrounded by people who are *supportive and willing to listen without judgement*. Creating a culture of authentic inquiry in our interpersonal relationships can provide the social and emotional support that we all require to pursue the creative work of mathematics. Because most mathematicians work in a community framed within professional organizations (colleges, universities, businesses, government entities, etc.), a natural source for inspiration and guidance in this professional context is the literature on organizational culture and leadership. One accessible and relevant resource in this area is Edgar Schein’s book *Humble Inquiry: The Gentle Art of Asking Instead of Telling.* Schein describes Humble Inquiry as the type of inquiry that

maximizes my curiosity and interest in the other person and minimizes bias and preconceptions about the other person. I want to

access my ignoranceand ask for information in the least biased and threatening way. I do not want to lead the other person or put him or her into a position of having to give a socially acceptable response. I want to inquire in the way that will best discover what is really on the other person’s mind. I want others to feel that I accept them, am interested in them, and am genuinely curious about what is on their minds regarding the particular situation we find ourselves in.

Schein is careful to distinguish this form of inquiry from others, such as diagnostic inquiry, confrontational inquiry, and process-oriented inquiry. He emphasizes that “the world is becoming more technologically complex, interdependent, and culturally diverse, which makes the building of relationships more and more necessary to get things accomplished and, at the same time, more difficult.” In other words, the challenge of building authentic interpersonal relationships is not only one for mathematical culture, but for society at large. Schein also emphasizes the importance of individuals in leadership positions learning to use and model authentic inquiry.

To give a concrete idea of how this approach might be used in practice, I share below some questions that we might not ordinarily consider when we are speaking with a student, colleague, employee, or peer. However, questions such as these can powerfully change our perspective toward those around us. I am not suggesting that we routinely ask these questions in regular conversation, but rather that we have these questions in our conscious mind, that we are open to the possibility that the people we interact with have complicated and difficult lives, especially when we are having challenging conversations.

- Does the person I am talking to have access to sufficient food and housing to meet their needs and the needs of their family?
- Does the colleague or student I am talking to have personal challenges or crises I don’t know about, such as a relative, spouse, or child with a serious health issue?
- Has the person I am speaking to been a victim of abuse or assault?
- How many hours each week does this student have to work to pay for their housing and basic expenses?
- Is this student or colleague suffering from PTSD due to military or other service?
- Who is this student or colleague responsible for supporting financially?

Unfortunately, these questions reflect realities that impact many more of our students and colleagues than we might guess. Knowing the answers to these questions would not necessarily change my expectations for student learning in a course, or for job responsibilities for an employee, etc., though it might inspire me to handle situations differently or with more humility. By honestly recognizing and affirming that we are usually ignorant of important aspects in the lives of those around us, we can be more empathetic, flexible, and ethical in our treatment of and relationships with others.

**What Accomplishments Do We Reward and Recognize?**

Complicating our relationships with other people is that, at least in the United States, a dominant social and cultural force is the drive to prize individual achievement over the building of relationships. This force extends throughout our society, not only in mathematics. In *Humble Inquiry*, Schein writes:

When we compare some of the artifacts and behaviors that we observe with some of the [social] values that we are told about, we find inconsistencies, which tell us that there is a deeper level to culture, one that includes what we can think of as tacit

assumptions…The most common example of this in the United States is that we claim to value teamwork and talk about it all the time, but the artifacts — our promotional systems and rewards systems — are entirely individualistic. We espouse equality of opportunity and freedom, but the artifacts — poorer education, little opportunity, and various forms of discrimination… — suggest that there are other assumptions having to do with pragmatism and “rugged individualism” that operate all the time and really determine our behavior.

How does this manifest itself in the mathematical community? As one example, publications are generally used as the currency of our realm, and it is typical that single-author publications are viewed as more valuable than publications resulting from a team of collaborators. Yet it is reasonable to ask what benefits mathematics more: having a person write a paper on their own, or having researchers build relationships and collaborative teams that are able to pool ideas and resources effectively?

Similarly, faculty are frequently given a higher evaluation for single-PI research grants than for leading a team of co-PIs on an infrastructure or education grant. Yet the NSF Education and Human Resources (EHR) and Undergraduate Education (DUE) divisions have funding available and have been actively seeking proposals in mathematics, as evidenced by Jim Lewis’s 2015 AMS Committee on Education presentation and this blog’s 2016 post by NSF program officers about the type of awards funded by EHR and DUE. Again, is this what we actually want to value in our mathematical culture? Is this what most strongly benefits the community? What is it that we collectively want to achieve, and do these recognition values reflect our common goals?

As a third example, consider two hypothetical students: a “smart” student who solves correctly every problem the instructor provides, or a student who sometimes makes errors yet is engaged, persistent, motivated, and dedicated. Which student is most likely to receive praise and support in a math course? Which is typically considered to be the most successful in mathematics? To have the most mathematical potential? To be the highest achieving? Which of these students do we typically provide with encouragement, awards, and recognition?

How will we make explicit, especially at the local level within departments and colleges, what type of collaborative activity we value, and how it will be rewarded? How will we go about recognizing and rewarding the type of activities that are needed to build supportive communities? The first step is one of the most difficult, in that we have to have clear and articulate discussions about these questions. This will almost certainly lead to serious disagreements among colleagues and peers, as mathematicians have strong beliefs about cultural issues; in many important ways, mathematical culture is quite conservative and deferential to tradition, though in my experience we rarely discuss this. Improving our habits of authentic interpersonal inquiry, which we have already seen is necessary for building better relationships, will be required of everyone involved in these types of discussions.

**What Stories Do We Tell About Mathematics and Mathematicians?**

A noteworthy quality of mathematical culture is that we frequently celebrate mathematical mythology rather than mathematical reality. For example, the stories we tell in mathematics are typically mythological, whether they are stories about “brilliant” mathematicians and their work or descriptions of “typical” career paths for mathematicians. For example, I have often overheard undergraduate and graduate students being told some variation of the story that an ordinary “successful career” in mathematics involves finishing high school, then immediately getting an undergraduate degree, then immediately completing a PhD, then obtaining a postdoc, then getting a tenure-track job, and then staying in that job until death. However, many of the mathematicians I know do not fit this simplified plotline at all. Rather, when we begin asking each other about our real stories, the stories that we usually don’t tell in public because they go against our cultural myths, we find that our realities are often much richer and more interesting than the standard narrative.

A deep commitment to the real instead of the mythological also influences our understanding of the nature and history of our discipline. We must be willing to challenge “traditional” and inherited narratives regarding the origins of mathematics, even when these narratives are strongly embedded in our culture. For example, in *The Crest of the Peacock: Non-European Roots of Mathematics*, George Ghevergese Joseph writes:

Evidence of [the contributions of Egyptian and Mesopotamian mathematicians] is not all hidden away in obscure journals or expressed in languages that tend to be ignored by many Western scholars: much is published in English in “respectable” journals and books… The reason for the neglect [of these contributions] was not that the relevant literature was inaccessible or “unrespectable” but something deeper — a serious flaw in Western attitudes to historical scholarship (one not confined to histories of mathematics or science). An excessive enthusiasm for everything Greek, arising from the belief that much that is desirable and worthy of emulation in Western civilization originated in ancient Greece, has led to a reluctance to allow other ancient civilizations any share in the historical heritage of mathematical discovery. The belief in a “Greek miracle” and the way of attributing any significant mathematical discoveries to Greek influences are part of this syndrome.

As an example of the mythological twisting of history that occurs in mathematics courses, in the edition of Stewart’s Calculus textbook that my department uses there are no women mathematicians listed in the index except for a reference to the “witch of Maria Agnesi”, which does not discuss Agnesi’s mathematical contributions at all. This perpetuates the mythology that “no women did math” in the past. However, this does not reflect reality, as there were several prominent women mathematicians and mathematical physicists working in the 1700’s and 1800’s, such as Maria Agnesi, Laura Bassi, Emilie du Chatelet, Mary Sommerville, Sophie Germain, and Sofia Kovalevskaya, who have certainly earned a place in our standard textbooks. We need to train ourselves to reflexively identify mythological stories, and to respond to them by actively seeking the real story.

**Conclusion**

These three questions are certainly not the only ones that should be asked about the culture of mathematics, but they are all of central importance. One of the common themes inspiring these questions is that we must insist that the humanity of mathematics and mathematicians be placed on an equal footing as mathematical knowledge and discovery itself, and be recognized as equally valuable. This is certainly not a new idea, but it is one which we must continue to emphasize, speak about, and share. With this observation in mind, I will end with the following passage from Rochelle Gutiérrez’s talk *Rehumanizing Mathematics: A Vision for the Future*:

]]>If we think that mathematics is not political, not cultural, not any of these other things, then how do we remind ourselves that it is a human endeavor?… Why is it useful to me to say “Rehumanizing” instead of saying “equity”?… Rehumanizing for me… is to honor the fact that for centuries, humans… have been doing mathematics in ways that are humane. It’s not that we have to invent something new for people to be doing, we have been doing it. People see themselves as mathematical, everyone is mathematical… The “Re-” part is a way of acknowledging that there are things that have been erased, and yet people persist.

What kinds of mathematical knowledge are necessary for full participation in contemporary democratic society? How well, and how fairly, do our schools educate students in quantitative skills and reasoning? By what measures might we judge success?

To put it another way, what would an equitable mathematics education system look like? In this post, I reflect on some articles published on this blog that support our efforts to move toward fairness.

A good place to start is in our own classrooms. Once we acknowledge the disproportionate distribution of access to mathematics experienced by our own students, we can make use of Six Ways Mathematics Instructors Can Support Diversity and Inclusion, by Natalie LF Hobson. One of the six ways is to “[e]ncourage your students to embrace a growth mindset,” which Cody L. Patterson explores in Theory into Practice: Growth Mindset and Assessment.

My seminar includes a service-learning project. As Ekaterina Yurasovskaya demonstrates in Learning by Teaching: Service-Learning in a Precalculus Classroom, such a project, while challenging on several levels, can benefit both the community being served and the students. If my own experience is any guide, the instructor can also gain some unanticipated lessons about mathematics learning in the early grades.

Attending to equity and inclusion is hard work. When I need to take a step back for an energy recharge, I go straight to contributions from Ben Braun, our founding Editor-in-Chief. His Aspirations and Ideals, Struggles and Realities is rich with inspirational ideas. I’ve assigned The Secret Question (Are We Actually Good at Math?) to my own students. It means a lot to them, and the resulting conversations are deep and illuminating.

Let’s not forget about the struggles our own colleagues may continue to face as they work within the flawed systems that Ben describes so well. A useful reading in this regard is Student Evaluations Ratings of Teaching: What Every Instructor Should Know, by Jacqueline Dewar. The author points out that “‘ratings’ denote data that need interpretation,” and gives useful guidelines for interpretation. While not focusing exclusively on the question of bias, the article does cite sources on that topic, including this study published in 2016.

Moving on to other aspects of our professional lives, Viviane Pons describes An Inclusive Maths Conference: ECCO 2016 . Having been to dozens of conferences, many of them quite worthwhile, I was fascinated by the intentional design details that made this one special, and wish I’d been there to experience it!

A simple Announcement of a Statement from the American Mathematical Society’s Board of Trustees reminds us that we can work toward the greater good within our professional societies.

While I’ve had plenty of my own “secret question” moments in a lifetime of learning mathematics, I recognize the benefits of mathematical habits of mind to me as an individual and as a citizen of the world. Those benefits should be available to everyone. We can all work toward that end, and I hope you’ve found some ideas here on how you might help.

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