Hot answers tagged

17

First, make sure they know that: The purpose of exams is to test students' knowledge and understanding. The burden of proof is on their side, that is, blank/unreadable sheets work against them. The teachers might choose to decipher some of the messy work, but this choice is intrinsically unreliable, erratic and may produce unfair results. The teachers ...


16

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 ...


15

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 ...


15

To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some point, mathematical sentences where written without symbols; the introduction of letters to denote mathematical object improved quite a lot the depth and ...


15

It's hard to tell what will be the changes in 10 years. Maybe we will all have jacks in our heads. Then again, some things change slower. (Where's my flying car?) Most of the reasons for getting rid of worksheets have been around for 20-40 years and they still have high demand. They still have various advantages. Most importantly, if you look at the ...


12

I'd suggest Halmos' "How to Write Mathematics"


12

What do we understand as mental processes? All of us (Math teachers) dream of entering the brain of our students, see what's happening and adjusting some connections... However, the thing is that their brains are a kind of black box for us. So the only way we actually have to be sure they have catched some mathematical concept (or argument, or property, etc....


12

I wouldn't do that. The parenthesis in use are also used for legal expressions within equations. So you can end with one line containing the same parenthesis meaning different things, what looks like a seed of confusion to me. If you prefer, you can replace the GeoGebra parenthesis with some others i.e. <>, or [] etc. to keep the notation similar to ...


11

One professor (who I greatly respect) did as follows: He made the papers a part of the exam, the second part being the talk/presentation. The papers had to be in some particular conference format (there were style files at the conference page). There were two dates in which students could present, and it would be organized in a mini-conference form (20 ...


11

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 ...


11

Besides mastering the material in the course one thing that the students have to learn during studies is to communicate mathematics in written form. Almost nobody comes to university and is able to write clear proofs or mathematical arguments. You need to communicate that this is part of what they have to learn. You may grade as harsh as you like if you are ...


11

I found this an effective teaching technique. I take a topic they know, and find a Wikipedia article discussing that topic. If you are specifically focused on proofs, as opposed to more generic descriptions, you can find many proofs in Wikipedia. E.g., of Sperner's Lemma, or Euclid's proof of $\infty$ # primes. Then I project the text in class, and have ...


10

The way I choose to combat this is to make my -> Homework Guidelines <- very clear from the start of the semester. I expect the students to follow the guidelines and have the disclaimer that "If your turned-in homework takes too much effort to read, it will not be graded!" Since I have instituted my guidelines, the homework assignments have been ...


10

There are probably many tips on this; I remember some discussions of font choice happening here. Let me give only one: take your time. One easily feels that writing on the board is too slow, and is thus inclined to deform letter, take shortcuts in sentences, and use abbreviations. Most of the time, these are mistakes because it makes the meaning ...


10

The problem with $(4x+7=6x+2)-6x$ is that there is no subtraction operation that involves subtracting a term from an equation. Subtraction involves subtracting a term from a term. So the correct notation is $(4x+7)-6x=(6x+2)-6x$


10

Speaking as someone who has taught college precalculus several times, I have an intense dislike for the way that Geogebra writes this step. In my opinion, it is very important to emphasize to students that we are subtracting 6x from both sides of the equation, which means of course that we are subtracting 6x from two different expressions. If we write "-6x"...


9

Set up a grading schema, awarding points for grammar, spelling, correct use of the technical terms. Central should be points for correct, logical overall structure, and enough explanations to lead the reader from one step to the next. You can publish the grading schema for their guidance, and perhaps write up an example of how their work should look like. ...


9

Blackboard writing 101: always break the chalk (to reduce the likelihood of squeaking) Other points will depend on how much space you have available. Working consistently from left to right helps, although an alternative is working right to left (assuming you are right-handed and have a fair amount of space) to keep your body away from what you've most ...


9

Taking a break from a convoluted computation... To me the key limiting factor is space. I can spread out several sheets over the table and have various bits and pieces directly visible. I cannot do this to the same extent on my tablet or my computer. Therefore I also use loose sheets not something that is bound together. One might argue I could set up a ...


9

Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction. There is also the "Using What You Are Proving" (maybe similar ...


8

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 ...


8

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. 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 ...


8

This is not the type of resource you request, but if you seek examples of excellent expositions geared toward the general public: The Best Writing on Mathematics series (Princeton link). Now five ten years: 2010, ..., 2019. All edited by Mircea Pitici.          


8

My favorite little tidbit on this topic is due to Jean Pierre Serre: One should aim to be "precise, yet informal". Here, "informal" means not using symbols and notation, i.e. not using "formalism". Regarding the steps suppressed in mathematical writing, this is very sensitive to the audience one is addressing. Routine arguments that can be checked by a ...


7

For all of the community college algebra classes I teach, I certainly make proper mathematical writing the number one priority; which is not to say that I have students compose everything in English writing. It's already an overwhelming challenge for students to get the algebraic grammar and syntax right, such that I already feel like there's not enough time ...


7

I have offered papers for a variety of classes. For most students, the idea of writing a math paper is completely foreign to them. As a mathematics educators, we should encourage the learning of all of the mathematics out there, and I have taken this past opportunity to offer historical papers. I think it would otherwise be difficult for mathematics students ...


7

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, ...


7

Textbooks tend to try to use the language that would be familiar to readers. In natural speech the use of "likely" is much more prevalent than that of "probable". Interestingly, though, according to Google N-Grams, that hasn't always been the case, with "probable" and "likely" starting nearly level in popularity in 1800: And yes, as you mentioned, as ...


7

Let students see what you're writing as you're writing. Otherwise, they're bored while you write and must hurry to read when you finish. Instead of facing the board, stand with your side to the board and your chalk hand in plain view, sort of like a "Wheel of Fortune" host. This posture also makes it much easier to talk to your students and not to the ...


7

(Quick note: I am not a math teacher. I am a CS teacher with a bit of dysgraphia myself. However, my insights here actually came from when I used to be a high school music teacher, where I was practically forced by the subject matter to tackle this problem with my beginning theory, AP Music Theory, and advanced theory classes. Also, there are some nice ...


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