How can we help students learn to write about their mathematics?

Question

As a guiding example, imagine an undergraduate Calculus II course where students have to complete a guided "research project" and write a "paper" about their work. This can be a shockingly new experience for them, even if the creativity component is minimized by heavy-handed guidance through the steps. Most of them just aren't used to having to present their thoughts/work in a way beyond "here's a bunch of equations in a row and here's the final answer in a box".

Are there any effective ways we can help them with the writing and presentation component of such a project, or something like it? In general, are there ways (short of meeting with every student individually and reading through several successive drafts of their work) that we can foster good, cogent presentation of a student's own mathematical thoughts and efforts?

I am interested in both personal experiences (good and bad) and research about this.

However, I don't intend this question to be about getting students to write in "math speak" as if for a journal article, nor about doing creative research. I just want them to learn how to share their ideas.

Answer 1

I would like to suggest that one can incorporate writing in a calculus course as a matter of routine mathematical activity, rather than using it only as a part of a big "research project". And integrating mathematical writing into the everyday parts of a course might help to make the point that writing well is a fundamental part of mathematical activity.

I have long held the view that one of the most important skills that students might learn in an undergraduate math class, such as calculus, is the ability to explain a technical idea with clarity and precision. Most of our students will not actually be computing integrals or testing series for convergence in their future careers (though some will), but most of them will probably have to explain something technical to their colleagues, boss or clients, and a mathematics class is a great place to learn and practice this extremely important skill.

In order to do so, I insist that students submit their mathematical work on homework, quizzes and exams in the form of a mathematical explanations, written essay-style with properly punctuated complete English sentences, and with embedded mathematical computations or diagrams to support whatever explanation is being made. The textbook, of course, is full of examples of the kind of writing I mean, with the worked-out examples and theorems, and this is of course the standard way that mathematicians communicate ideas. Weaker students are sometimes mystified by my instructions, and they are sometimes stuck in the habit of writing a series of mathematical expressions separated by equal signs (whether they are equal or not) and a box around the "answer" at the end. So I try to unstick them from that habit. Occasionally students misunderstand my instructions as a request to justify every picky little step, and so I have to correct this, telling them that the more valuable explanations concentrate on the big idea of the solution method, emphasizing especially what might be different in this solution than in another similar one. In time, they seem to get it. (But to be honest, my expectations student writing in calculus are not actually very high, although they are much higher in more advanced courses.)

Answer 2

For several years I taught a crypto course, and an error-correcting codes course, that both emphasized coherent writing. These were for math-major juniors, seniors, and grad students in engineering departments. The homework and exams were mostly about execution of algorithms rather than "proofs", but/and explanation, at least some sort of narrative was required. Coping with students' incredulity ("the symbols speak for themselves!" ?!?!) was time-consuming and frustrating, but seemed worth the effort.

I did distribute "approved solutions" as models for what kind of writing-out was desired.

Also, each term, there was a "writing project" required. We did not do revisions... since I usually had 100 students in a class... but/and by the end of the term most of the kids were catching on to writing in English rather than math-eze. In fact, it was gratifying that the issue shifted a bit from "coherent writing" to "what is plagiarism, versus using sources...".

(This was quite a lot more work than for "standard" math courses, and the other people now teaching these courses have reverted to more typical structure...)

It is indeed hard to communicate the idea that mathematics should be imbedded in natural language, to populations who've been conditioned by typical k-12 that the goal is usually a number-in-a-box, and even "showing your work" is a crazy luxury of sorts.

The point of having people write about their execution of algorithms was intended originally to have people write about more down-to-earth activities (as opposed to the stylized "proof" notion). I discovered to my dismay that many or most had at least initial difficulty getting the verbal part and the computational part of their brains to cooperate... it seemed. That is, in some amazing cases, people were unable (or maybe merely balked...) to say in words what they were doing in some computation, and/or why they chose one thing over another, and so on. Surprisingly subverbal or non-verbal... An example was to explain how someone knew that $15$ factors into primes as $3\cdot 5$. One kid I recall could say "it's obvious", and could not say what they'd do if the number were large enough that a factorization was not "obvious" to them. (This in the days after discussion of trial division and so on... certainly not in a vacuum...)

On the whole, it seemed beneficial to push students to write coherently, but took much more effort to give useful feedback... and with the ambient curriculum collapsing back to non-verbal mathematics (or stylized math-English), it was somewhat a quixotic battle.

At my university there is nowadays a "writing-intensive" requirement, but it is isolated in the math curriculum and viewed as a tiresome burden. And some of the vehicles for its fulfillment are (in my opinion) somewhat anti-coherent, emphasizing an extreme dialect of math-eze.

Answer 3

Good writing requires style and an intended audience. When we ask our students to explain their mathematics in writing without giving them any guidance, typical results include a string of equations and explanations that display a minimum of effort and reflection. In order to get better writing from our students, we should be more explicit about what we want. For instance, we could have them write for a particular audience:

• Write this as an explanation to an average student who will be taking this same class next semester.
• Write this as a newspaper article that assumes its readers have very little background knowledge.
• Write this in the style of an example in your course textbook.
• Write this in the style of a formal written proposal for grant funding for an important project.

Developing good writing skills also takes a lot of practice (revise, revise, revise!) distributed over a long period of time. If you're really serious about improving the mathematical writing skills of students, you should start by requiring them to explain their answers to homework questions using complete sentences with correct grammar, spelling, and punctuation. Of course, this requires papers to be hand graded and for the grader to enforce this policy (e.g., by having some of the homework points be for mathematical correctness and other points dedicated to exposition). Students who have consistently gotten feedback about their writing on the homework usually have fewer difficulties with longer writing assignments.

Students need explicit guidance on what needs explaining and what does not. Initially, many students will need to be told not to over-explain (e.g., tediously writing a sentence for each algebraic manipulation) and not to under-explain (e.g., assume their reader knows too much). Giving students a long list of rules to follow might unintentionally produce bad results. In general, getting students to make decisions about what to include and exclude boils down to getting them to think about what their reader needs to know in order for things to make sense.

Students need to be told repeatedly that they are expected to explain why they chose to do what they did. The final result of a mathematical computation only has meaning when the problem is put into context, the assumptions are made explicit, and all of the mathematical operations are stated clearly and justified fully.

Creating and grading writing assignments is a daunting and difficult task, as is managing students during the writing process. Good writing assignments take a ton of time to create and grade, and I would encourage you to ask you colleagues in the humanities for tricks they use to make the process go smoothly. To ease your feedback and grading burden, I would encourage you to set aside time in class for students to exchange their work in small groups and give each other feedback on their writing. Another trick is to use examples of good and bad writing (e.g., on a projector screen) in class to start a classroom discussion about the strengths and weaknesses of student work (make sure to get student's permission to show their work beforehand). Using student examples in class also provides the opportunity to openly discuss what feedback you would give and how you would grade such work. This feedback should happen before the final draft is due. You may also want to consider having students turn in both the rough draft (with comments from other students) and the final draft at the same time, and assigning more points to the rough draft to get them to take it seriously. Think very carefully about how to manage your students (e.g., the timeline and incentives to meet deadlines) as this can save you time and anguish when you eventually grade their work.

Finally, take advantage of technology. For instance, would it be appropriate for your students to write their papers on https://stackedit.io (which uses MathJax and LaTeX syntax for typesetting mathematics) and share them with you via Google Drive? They can also make a PDF copy and print it, if you prefer hard copy.

Answer 4

I taught a course in the US which was designated as "Writing in the Major". Once I got used to the idea, I found it invigorating: both to prepare and to teach.

I've managed to find a copy of the handout that I gave to the students. Unfortunately, I've lost the source file but I'll see what ps2ascii can do (yes, it was in the days when PDF was too "cutting edge" to use).

I think that the key parts that made it work were the peer review and the detailed rubric.

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Project on Fourier Series

1 Overview

The aim of this assignment is to give you practice writing mathematical prose that is both readable and precise.

The subject of the project is Fourier series. The central part of the project is to prove Fejer's Theorem. This theorem deals with the question of uniform convergence of the Fourier series of a function. The statement of the theorem can be found below, together with some details of the mathematics involved.

There are several rounds of revision for this assignment. The first is to give a copy of your assignment to another student in the class. Part of your grade will be the quality of feedback you give to your partner in this phase. The papers will be then turned in a first time to me, at which point comments and suggestions for improvement will be made. After the papers are returned, you will be required to resubmit edited or rewritten versions. Editing is an integral part of the writing process.

Important dates:

• Tuesday 11th of May. First draft given to peer in class for review.
• Thursday 13th of May. Comments returned to peer.
• Tuesday 18th of May. Complete first draft due to me.
• Tuesday 25th of May. Assignments returned with comments.
• Tuesday 1st of June. Final rewritten versions due.

Note: When submitting the final version, you must also submit your first draft, your peer review, your partner's review of your paper, and the tutor's comments on your first draft. Therefore, it is vital that you keep copies of everything.

2 Parts of the paper

Your paper should consist of the following sections:

1. An Introduction: an ideal introduction gives a clear, concise, and accurate overview of the paper, presenting enough information to interest a reader in the rest of the work and persuade them of its importance without presenting so much that the reader is overwhelmed.

2. The Proof: an ideal section containing the proof of a major theorem proves the theorem in a correct, logical way; it provides any information that the reader is unlikely to already know (or should be reminded of); and is laid out in a manner such that the reader always knows which step of the proof is being done, how it fits into the whole proof, and its relative importance in the proof.

3. An Application: an ideal section about an application explains the application concisely including any context that may be useful for the reader to know; it demonstrates clearly the relevance of the application to the main topic; and does all this with just enough detail to convince the reader without so much that the reader is confused.

3 Guidelines for Writing

Your aim should be to make your paper understandable. You should imagine that another student in the class has asked for your help on this topic.

Use full sentences. The only sentence fragments that are acceptable are headings such as "Theorem" and "Proof". You should not use symbols such as [symbols for therefore, implies, because, and so forth] nor should you use abbreviations such as "s.t." or "iff". These detract from the clarity. It should be possible for someone with basic mathematical knowledge to read your paper aloud without hesitating over symbols or poorly constructed sentences.

Different parts of the paper have different purposes and these lead to different styles. In the first part, the primary aim is to interest a person in your paper, to give them an overview, but not to burden them with minutiae. In the second part, the aim is to present the theorem and its proof. In this part, accuracy is paramount but it is not possible to be accurate without being clear. In the third part - the application - clarity is again foremost.

When proving the theorem, although the proof is in the book you should not regard this as a copying exercise. The style adopted in the book is not the best style for this type of paper.

[Section 4 was some mathematical background on Fejer's theorem.]

5 Rubric

The paper will be graded according to the scheme below. There are five categories: introduction, proof, application, overall presentation, and review and revision. In each section, there are four marks available, making a total of twenty. The specifics for each mark are laid out below. This scale is incremental. That means that to get, say, two marks on the introduction you have to not only satisfy the criteria laid out for two marks but also the criteria for one mark.

Mark Criteria

• Introduction

  1. Introduction is present.
  2. Aims of the paper are stated. Broad area of mathematics to which material belongs is stated.
  3. Aims of the paper are clearly stated. Concise background information is explained. Relevance of subject matter is stated.
  4. A clear, concise, accurate introduction is present.

• Proof of Theorem

  1. Methodological steps have been taken toward proving the result.
  2. Theorem is correctly proven.
  3. Relevant definitions are stated. Theorem is proved in a clear way with correct differentiation to the steps.
  4. A complete, precise, concise and detailed proof is given. Where relevant, especially elegant or informative steps are highlighted.

• Application

  1. An application of the subject matter is given.
  2. The application is set in its proper context.
  3. The relevance of the subject matter to the application has been explained.
  4. The importance of this work to the application is evaluated.

• Overall Presentation

  1. Paper is legible with adequate spelling, punctuation, and grammar.
  2. Paper is neatly and clearly presented with good spelling, punctuation, and grammar.
  3. Secondary sources are cited correctly.
  4. Segué between sections is smooth and logical.

• Review and Revision

  1. First draft, peer review, partner's peer review, and tutor's review are present.
  2. First draft has all sections, peer review comments on all sections of partner's first draft (even if said section is missing).
  3. Final version shows improvement in accordance with partner's review, where relevant. Peer review is conducted according to this rubric. Tutor's recommendations have been implemented.
  4. Peer review gives practical guidelines for improvement in partner's paper.

Answer 5

These are all good answers, however, let me add two general points. These points are more about the general idea of this question than the guiding example of calculus II.

1. Good writing comes from good reading. Generally, we need to encourage our students to make a continual habit of reading technical and nontechnical mathematical treatise. Ideally, some of this is not directly connected to completing a homework or getting a grade in some course. The larger goal, become a scholar in math. From the technical side, we read to learn more about what is out there and in the process, see how we express our mathematical ideas. From the non-technical side, popular books help give you ideas about how we can repackage our ideas to be understood by people who do not have the luxury of our mathematical lexicon.
2. Good writing does not come from online homework and/or emporium style mathematics. If you don't know already, an "emporium" is a context where the students are forced to go to the emporium to watch instructional videos, read e-texts, e-tutorials etc... then complete the online homework while in house. There is little or at least almost no note-taking in this format of instruction. Moreover, there is almost no written homework. In principle, free of this format, you might take the opportunity to force them to write grammatically complete sentences. However, without a major reformat to include a human grading system, it is not possible to grade written work in this format of instruction. In particular, it is antithetical to the popular efficiency of the model to burden it with extra human interaction. So, generically, work to keep human grading wherever possible. The convenience of e-grading is seductive, but, it robs us of instructional opportunities. In short, if the students "do" the assignment they may be able to "do" the math relative to some metric, but, they might not be able to write even a few simple sentences explaining what they are doing.

Answer 6

I'll share my experiences, but definitely want to hear from others, as well.

I've assigned these "research project" papers in Calc II. The actual math is mostly guided, with some open-ended questions and extra credit available. I tell students to meet with me while they're working, to make sure they're on the right track, and they always ask something like, "How am I supposed to write about this?"

Pasted below are the guidelines I put in the project descriptions. I've found them to be somewhat helpful as general guidelines, but they still require the students to think about what they themselves need to say and how to say it. But this is part of the goal of the project, anyway. Mostly, I just keep stressing this for them: "Put yourself in the shoes of a classmate reading your paper. Would you learn from it?"

I've found the results to be generally quite good, actually! I think this is partly because this experience is new for them, so they just have to think about it more. I also think this is because (most of them) they do come to me and ask about it.

Some things that need improvement ...

It's tough to get the students to qualitatively describe things. Often, they write about their work by "translating" the steps into wordy sentences that actually don't complement the accompanying equations. Similarly, they'll include a graph with little description or one plotted on an ill-fitted scale. I think this reflects a general trend that the students (particularly at into calculus college level) still view their work as entirely quantitative in nature. I try to address this in class by describing my own heuristic/qualitative views whenever they're applicable, but it's tough to address this phenomenon head on.

I also think they do very little editing. Once they figure out the right steps and write a description of it, they're done with that part and move on. Without seeing their works in progress, I cannot verify this, and can only guess based on final outputs. Furthermore, I wonder if this just reflects a broader trend of poor writing skills gained from current education. (I actively combat this in my "intro to proofs" course, but that course is much more heavily devoted to developing writing skills, as a whole.)

In the following pages, I will provide a list of projects for you to choose from, for the month of March. These projects will guide you through the solution of an interesting and relevant theoretical problem or an application of the material we see in class (or a combination of the two). You will write your results and discoveries in the form of a "research" paper that describes the steps in solving your particular problem, some reasons as to why it is interesting and related to what we've done in class, and a description of what you learned and what you found useful.

You will be graded not only on the correctness of your work, but also the clarity of your descriptions. The goal is that a fellow classmate can pick up your paper, read and follow along with your work, and feel like they learned something, even if they didn't work on that particular project.

I want you to actually write some words for your research paper, as opposed to just showing a bunch of lines of equations with some descriptions beside them. (That would be a bad paper.) You should exhibit all of the important steps you took in solving your problem, and you should provide some (a) description as to what they represent, as well as (b) some motivation for why you did things the way you did. Remember: the goal is that a classmate could read your paper and discover what you did and learn something.

I do not want to put a specific "page limit", either a minimum or maximum, on this project. I would say that, on average, a final write-up will be anywhere from 3-6 pages. This depends on how much math you need to show, and also on how detailed you make your descriptions. If you have a lot to say, go for it and write more. If you're concise with your math and skip over some easy details, maybe your paper ends up fairly short. This is all fine. Just keep in mind, though: longer is not necessarily better, and I will not use the length as a factor in your grade; however, I do find it very hard to believe that all of your details and descriptions could be squeezed into a page and a half. (Not impossible, but very very rare.)

Answer 7

An excellent and comprehensive reference on this question: Student Writing in the Quantitative Disciplines: A Guide for College Faculty Paperback by Patrick Bahls. It helped me design a lot of exercises I use in my high-school classes.

Personally, the best tip I've received for giving students helpful direction and feedback in a time-efficient way is to develop your use of rubrics. See this short post for details.

Answer 8

Inspired by Paul Lockhart's "A Mathematician's Lament".

I would argue that what students actually need is play. Play is essential for learning especially at the earliest stages of education. Given that the central element of mathematics is proof, this means that students should form their own conjectures from an early age. They should also attempt to justify them using a verbal explanation (ideally without much mathematical jargon which the child does not understand yet).

Contrast this with how we currently teach students: start by assuming these facts as a given, and then force students to memorise them through rote-and-drill repetition. This is the complete opposite of what we should be doing. Rather, students should firstly ask the right questions, for example by drawing a semicircle, and then wondering what happens to the angle inscribed in it as the point moves. Then they will ponder, struggle, try and then try again, before they might make another copy and discover a rectangle hidden within the circle:

Aha! Since we made a copy of the triangle, each pair of opposite lengths must be the same, so we have a rectangle! Therefore the angle inscribed in a semicircle is 90º!

This is how we make students engage with mathematics, discovering the intrinsic value of truth and beauty hidden within it.

Students need real experience with mathematics through inquiry and through play before they can progress with applying the same methods to harder problems, or manipulating different methods of proof such as proof by induction. In other words, students need to have lots and lots of fun with mathematics.

Even in high school curricula where mathematical investigation and reasoning are given significant attention such as the IB Diploma, used in many international schools around the world, investigation is treated as a subject in isolation, divorced from the rest of the course. Students are guided through a bewildering assortment of topics throughout the course where mathematical concepts are seen as disparate and disconnected. Investigation is limited to a month or two where students start from scratch, particularly those that have come from more traditional curricula. After 12 years of rote memorisation, of seeing mathematics as a game where symbols are manipulated as if by magic, no wonder students come to struggle with mathematics later. If all they have seen are grammar and conjugation drills in their English class, how will they be equipped to write and read stories, plays, novels and poems, just as they will engage with the art of proof?

In mathematics, communication is key and notation is secondary. All of those pesky symbols: $\alpha, \beta, \gamma, \Sigma, \Pi$ can come later. And so students need to get in the right frame of mind as viewing mathematics as a narrative, with an inherent narrative structure of exposition, discovery and conclusion, not as a meaningless bash of symbols. We do the exact same thing with English, or students' native languages.

So for students who have been completely deprived of this throughout their "education", we need to start small. Students will naturally be horrified at the sight of a research paper as it will be shockingly new. Start by using very simple yet perplexing examples: why does a pyramid take up a third the space of the cuboid bounding it? They will very often not know why. Spend as much time with them as possible, no matter how long it takes for them to understand. Tell them that mathematics is like any work of art or music in its beauty and its lack of relevance in the real world. First give them as much space and time to make mistakes, to ask "what if?". Gently prompt them in the right direction when they get lost, which will involve the destruction of many assumed truths that they may hold sacred. You will need to hold their hand (yes, even in undergrad) and then let go when they are ready.

This will mean forgetting what you already know about teaching. Get them to chat to each other, present their work on the whiteboards, get up from your lectern and wander around the room (if you haven't already). And you will give them the power of independent thought and autonomy, divorced from any pressure about passing grades or graduation requirements. Because that is what it means to be a great teacher and a friend, a great mentor and a learner, and what it means to play. Make your classroom as unbounded as the imagination and curiosity of students. And in time they will learn to write.

Answer 9

Most of them just aren't used to having to present their thoughts/work in a way beyond "here's a bunch of equations in a row and here's the final answer in a box".

I find students emulate very well the style of how they've been instructed. Their presentation style reflects the lectures, tutorials, explanations, and texts they have been exposed to.

If we want students to adopt a clearer, less terse method of explanation, then we must not subjugate them to a style opposite of these objectives.