# Math without pencil and paper

For someone who is physically unable to use a pencil and paper, what would be the best way to do math?

In my case, I have only a little movement im my fingers. I can move a computer mouse and press the left button. Currently I do very little math and when I do I use MS Office but this is a very slow way of doing it and will break my thinking process.

EDIT: Thank you for the answers. I have a lot to try. I won't accept an answer since it's dependent on the disability but thank you all.

• What level of math are you referring to? The amount needing to be recorded varies depending on what one is doing. Proof writing can in some instances be done entirely verbally. One can learn to do a great deal of mental calculation, but there are easily limitations on what can be accomplished under ordinary circumstances. I know of mathematicians with physical limitations who use scribes to help write for them. I expect that they are provided to students of the university via the office of disability services, or to faculty via departmental funds. Jun 20, 2016 at 16:35
• Math for a computer science master. I did the last year of highschool completely in my head but it's easy to lose track. Jun 20, 2016 at 16:40
• "Do Math" if a pretty broad range, what sort of math, and do what to it? Jun 21, 2016 at 10:41
• How did Solomon Lefschetz do math? See books.google.com/… Jun 21, 2016 at 20:38
• Can you use your mouth to hold a pen? I had a disabled student that wrote with his mouth - the best student in the class AND his penmanship was the best I've ever seen. Jun 22, 2016 at 19:14

(Background: I have a chronic pain condition, and it is extremely painful for me to do any sort of repetitive fine motor activity, including writing and typing. I earned an undergraduate degree in math with quite little handwriting, and I'm currently a graduate student.)

Unfortunately, I've never been able to find any one solution or approach to replace the work that many people do with pencil and paper, but there are lots of partial solutions that work for different situations and types of mathematics. Here are some of the ones that I use and which, based on your description, might be applicable.

1. Getting really proficient at mental math. I think you've already done a lot of this, so all I'm going to say is that I can do a lot more math in my head than I would've believed was possible pre-disability. It just takes a lot of practice. Of course, it doesn't work for everything; some calculations just require remembering too many things at once.

2. Speech-to-text. If you have clear speech, then a speech-to-text system is potentially very helpful. (I use Dragon NaturallySpeaking.) Maybe you already use one for general writing and computer use. If so, then you have almost certainly noticed that they are not designed for mathematics (or, for that matter, computer science), so some additional software is necessary in order to do math. I use a system based on NatLaTeX to dictate all of my formal mathematics, including anything in that I'm going to turn in for my coursework. Basically, NatLaTeX defines a speakable form of many common LaTeX commands, including everything you need for most mathematical expressions. Using a custom vocabulary in Dragon, I can dictate a plain text file containing this NatLaTeX source. I then use in scripts from the NatLaTeX project to transform my dictated text into actual LaTeX source, which I can then compile into nicely typeset mathematics using a standard LaTeX compiler. (Actually, I use a batch file to automate the process.) Just as a note, I have made several modifications to NatLaTeX in order to optimize it for mathematics (the original author was a physicist) and to adjust for changes in LaTeX. Feel free to contact me if you want a copy of the modified scripts. I do eventually intend to post them somewhere online, but I need to spend some time updating the documentation first (and that's really hard to justify spending time on it while I'm preparing for comprehensive exams!).

The big advantage to a dictation system like this is that you get to go through the process of formally presenting your work, which is really helpful for checking understanding and practicing proof writing. Disadvantages include a steep learning curve and not being able to see your work (typeset, at least) in real time. You also have very limited choices in text editors because Dragon will only "play nice" with a couple of them.

3. Programming languages and computer algebra systems. When you actually need to do calculations, plot functions, etc., it is hard to beat a good computer algebra system. There are lots of choices out there, and I think the choice of which one to use ultimately comes down to personal preference and perhaps compatibility with whatever assistive technology you use. Of course, you still face the problem of how to handle getting input to that computer algebra system. Typing with one finger on an on-screen keyboard sounds rather slow and tedious. Here are a few alternatives you might want to look into.

• If you have head and neck movement, a head pointer is one option. I sometimes use one that actually marketed as a gaming device called TrackIR. (Gaming peripherals are much less expensive than assistive technology peripherals!) You can use this to type on an on-screen keyboard or to interact with the input panels found in many computer algebra systems. For mouse click, you could use a switch with your finger or a dwell/hover click.

• Eye tracking is a technology that has recently become much more affordable due to relatively new consumer-level devices marketed for gaming applications. Just a couple months ago, I got an Tobii EyeX eye tracker, and it has been great! I use it with a simple mouse emulation script and Dasher to write code in Sage. Dasher is really cool, by the way, and quite good for text entry with low-precision input devices like eye trackers. It's also much faster than most on-screen keyboards. Just as a warning, the EyeX is not intended as an assistive device so you do have to do a some software configuration in order to make it do what you want it to do. But you're studying computer science, so I don't think that should be a problem for you. (FreePIE, OptiKey, and Dasher are all good pieces of free software to look up and consider for use with the EyeX.)

The big advantage to computer algebra systems is that you can do all of the numerical calculations and tedious symbolic manipulations without needing to ever write down a single number. The disadvantage is that hiding all of the details can sometimes hinder your conceptual understanding.

Those are the main strategies I use. I hope something here can help you out, too.

EDIT: There has been a request for some examples of what NatLaTex can do, so here are a couple of different examples pulled files on my hard drive.

Discrete Math Example:
NatLaTeX Input (dictated with Dragon)
Given a poset "(P, precedes)", a collection of linear extensions "{calligraphy R } equals left curly brace precedes sub one, precedes sub two, low dots, precedes sub k right curly brace" is called a realizer'' of "P" if "precedes equals intersection of sub {i equals one } to the k precedes sub k", where each relation "precedes sub i" is interpreted as a set of ordered pairs and "intersection of" is set intersection. Equivalently, "{calligraphy R }" is a realizer of "P" if, for all "p, q in P", "p precedes q" if and only if "p precedes sub i q" for all "one less than or equal to i less than or equal to k".

LaTeX Output
Given a poset $$( P , \prec )$$, a collection of linear extensions $${ \mathcal R } = \{ \prec_1 , \prec_2 , \ldots , \prec_k \}$$ is called a realizer'' of $$P$$ if $$\prec = \bigcap_{ i = 1 }^k \prec_k$$, where each relation $$\prec_i$$ is interpreted as a set of ordered pairs and $$\bigcap$$ is set intersection. Equivalently, $${ \mathcal R }$$ is a realizer of $$P$$ if, for all $$p , q \in P$$, $$p \prec q$$ if and only if $$p \prec_i q$$ for all $$1 \leq i \leq k$$.

Analysis Example:
NatLaTeX Input (dictated with Dragon)
begin theorem [Monotone Convergence Theorem] Let "left curly brace f sub n right curly brace sub {n equals one } to the infinity" be a sequence of nonnegative measurable functions with "f sub one less than or equal to f sub two less than or equal to low dots less than or equal to f sub n less than or equal to f sub {n +1 } less than or equal to low dots" and "limit of sub n f sub n equals f" (pointwise). Then, "f" is measurable and @begin{equation} limit of sub {n right arrow infinity } integral f sub n d Greek mu equals integral limit of sub {n right arrow infinity } f sub n d Greek mu equals integral f d Greek mu @end{equation} end theorem

LaTeX Output
\begin{theorem}[Monotone Convergence Theorem] Let $$\{ f_n \}_{ n = 1 }^\infty$$ be a sequence of nonnegative measurable functions with $$f_1 \leq f_2 \leq \ldots \leq f_n \leq f_{ n + 1 } \leq \ldots$$ and $$\lim_n f_n = f$$ (pointwise). Then, $$f$$ is measurable and $$\lim_{ n \rightarrow \infty } \int f_n d \mu = \int \lim_{ n \rightarrow \infty } f_n d \mu = \int f d \mu$$ \end{theorem}

• This is a fantastic first post; you clearly know what you're talking about. Welcome to MSE! Jun 22, 2016 at 3:06
• It is an amazing testimony to human ingenuity and courage to see a professional mathematician refuse to be hindered by disability. Excellent post. Jun 23, 2016 at 1:39
• I think first time I got in tears after seeing a post on StackExchange. Both for the OP and answerer here Jun 23, 2016 at 5:39
• Sample of her (math part ) code with minor editing: $${ f_n}_{ n = 1 }^\infty$$ $$f_1 \leq f_2 \leq \ldots \leq f_n \leq f_{ n + 1 } \leq \ldots\, \lim_n f_n = f$$ $$\lim_{ n \rightarrow \infty } \int f_n d \mu = \int \lim_{ n \rightarrow \infty } f_n d \mu = \int f d \mu$$ Jun 27, 2016 at 23:01

Voice input takes much load off the keyboard. Google search by voice is possible, as it converts spoken matter directly to text. Even individual letters can be spoken into. Mouse left click can be sparingly used to make corrections for documents. About math symbols I do not know, but believe may be possible after some trial.

I produced the following:

a x squared + bx + c = 0

when I spoke to/into Google search microphone: $$a x^2 + bx + c =0$$

EDIT 1:

Encouraged by your response, may I go on with a suggestion.

All characters on a standard keyboard have an ASCII code but not all their corresponding voice inputs (with standard spoken English) translate to math symbols e.g.,

back slash, caret or power, greater than, curly opening bracket, square closing bracket, exclamation and interrogation marks

do not automatically print symbolically as:

( \ , ^ , > , { , ] , ! , ? , )

So a coding exercise for someone here or in Google Company likely implementable with an optional Voice MathSymbol Switch/Selector from keyboard appears to be an urgent need. This can be directly embedded as Latex input format.

Voice input $$\rightarrow$$ Math symbolic equation output. It may be doable.

This may help not only Jeroen but even the vast majority of us more luckily endowed :) ..

EDIT 2 :

Just googled MathTalk.com Dragon Naturally Speaking in Anna Kirkpatrick's moving answer here. This can symbolically convert polynomials, powers, trig, fractions, Greek characters and what have you with adequate speed.

MathTalk

• Just as clarification, I don't use MathTalk. I use NatLaTeX. I choose NatLaTeX over MathTalk for a couple of reasons. MathTalk is a set of DNS commands/macros, so you have to pause before and after each command. I wanted the flexibility and power of interfacing directly with LaTeX. In particular, I didn't want to end up in a situation where my research needed notation that wasn't supported. (With NatLaTeX, I can add new commands with a single line in a config file.) They're both valid options with pros and cons, though. I would guess MathTalk has a much kinder learning curve, for example. Jun 24, 2016 at 5:36
• Thanks,seen it, I referred to it only generally. You and Jeroen pioneered in indicating the possibility and projecting a need to all. Can I ask you to include a few Latex lines of output you made? Also, if someone is inclined to organize in putting together a package of what you did on a server, I am on my part willing to donate. Jun 27, 2016 at 14:46
• I added some LaTeX examples to my original answer. Hope that helps / shows what you wanted to see. Feel free to send further questions my way. And thank you for pointing out MathTalk as an option. I had kind of forgotten about it, to be honest. Jun 27, 2016 at 22:33

If I were unable to use my hands, I personally would attempt to use a sophisticated voice to text interface like Dragon Naturally Speaking http://www.nuance.com/dragon/index.htm, which includes addons that specifically permit you to voice-type into IE or Firefox. Why would that capability be particularly useful? Because the second portion of my strategy would be to then go to https://sagecell.sagemath.org/ and use that (Sage) free alternative to Maple, Mathematica, Matlab and the like. Sage is a full symbolic math processor with beautiful graphics output and, if you are saving your abstract math work or sharing it with other, you can save the work you are doing only in pdf form or output LaTex and the like to get high quality mathematics documents.

Now, I should emphasize that I have not implemented the solution I envision (thankfully not requiring it), so I am glibly describing a process of software installation and learning curve/troubleshooting that will be challenging. That being said, I believe this approach would accomplish your goal of hands-free math work in a visual symbolic and numeric calculation environment.

• @BrevanEllefsen Thanks for nod of approval. I was just looking over the Sage for Undergrads textbook again (written by Gregory Bard at gregorybard.com/sage_for_undergraduates_bw.pdf.zip) and forgot to mention Sage also gives you a free cloud service (so you can save and share work) and is a Python programming environment also (I used to do software design back in the 80's and look forward to trying Python). The pdf capability is through print from browser, not directly in Sage. I really enjoy having access to Sage myself. Jun 22, 2016 at 23:56
• It would depend on the class I suppose (36 years since I last set foot in a university class; can't remember if I was allowed to use my TI-58C in Networks I/II--sure it wasn't allowed in Calculus or DiffEQ). It might be difficult for the rest of the class if you were speaking commands into a voice to text interface (though Anna Kirkpatrick above mentions using Dasher with Sage--I tested the online version of Dasher yesterday and it is pretty good, a text entry system without keyboard, kind of active tag cloud of stochastic linguistic character sequences as it were). Jun 23, 2016 at 18:23
• This is a great idea but unfortunately rather hard to implement. (I have tried. Multiple times.) There are several issues, but the biggest one is that code just isn't formatted like English text. Dragon won't (easily) output camel case, words with underscores, periods not followed by a space, etc. There are limited config options, and of course it's not open source, so you can't edit the core language model in fundamental ways. I'm sure one could get basic functions working eventually, but be warned: Dragon doesn't like code. Jun 24, 2016 at 4:56
• @AnnaKirkpatrick I am not surprised to hear you found many practical difficulties in actually implementing a Dragon-->Sage solution (the process of engineering always involves iterative design/test cycles). I was interested to read that you had tested Dasher with Sage to provide an alternate path that overcame the Dragon deficiencies in the code talk department (and your use of Dragon with NatLaTex, another real world adjustment of the concept of writing mathematics). Be warned: I may have questions about critical sets for Sudoku and general graphs in the near future (grin). Jun 24, 2016 at 19:43
• @Ian It depends on the course and the school. For whatever it's worth, I have used speech to text tools on many exams, including using NatLaTeX for math exams. Now, there's no computational advantage in writing by speech, so that is a big difference compared to using Sage or similar. The noise issue can be easily fixed by taking the exam in a separate room, and any reasonable disability services office should be able to help the student arrange this. Jun 27, 2016 at 23:07

You mentioned that you use MS Office and that this is slow.

The other answers have already said good things, but here is maybe a strange suggestion: Learn LaTeX. Install a nice plain text editor and a version of LaTeX. Yes, LaTeX code can be a bit intimidating, but with time it is not that hard to read. The nice thing about it is that you are able to share what you have written with someone else (the PDF file). It is fairly easy to edit. This, of course, might not be the best idea for actually doing mathematics.

Just a thought.

• You wouldn't need to "read" the latex (when, eg, verifying what you wrote), only write it. See tex.stackexchange.com/questions/48714/… Jun 20, 2016 at 21:04
• WYSIWYG latex: codecogs.com/latex/eqneditor.php Jun 21, 2016 at 2:28
• Latex would be pretty difficult if you can only move the mouse and click the left button, I think. Jun 21, 2016 at 4:39
• @littleO Some latex editors have clickable commands in the toolbar. I think I've seen some with visuals of the symbol you're clicking, but I agree it would be pretty difficult. If you only use it for the odd formula though it might be ok. Jun 21, 2016 at 10:41
• @littleO: There are several accessibility tools available for people with disabilities. Jun 21, 2016 at 11:04

You may want to use SymPy, "a Python library for symbolic mathematics". If you use SymPy Live, you don't even need to install anything in your machine.

For example, suppose you want to expand $(x+y)^9$ without computing the binomial coefficients manually. Using SymPy Live, we get

>>> x, y = symbols('x y')
>>> expand((x+y)**9)
9      8         7  2       6  3        5  4        4  5       3  6       2  7        8    9
x  + 9*x *y + 36*x *y  + 84*x *y  + 126*x *y  + 126*x *y  + 84*x *y  + 36*x *y  + 9*x*y  + y


Or, perhaps you want the 7th order truncation of the Maclaurin series for $\tan$

>>> x = symbols('x')
>>> tan(x).series(x,0,8)
3      5       7
x    2⋅x    17⋅x      ⎛ 8⎞
x + ── + ──── + ───── + O⎝x ⎠
3     15     315

• The brackets do not line up for me... unfortunately it is too small an edit which stack does not like for some reason. Jun 23, 2016 at 10:45
• Many universities have an administrative departments for people with different issues, Stress, long time illnesses, sometimes there are also some anti discrimination laws, that must/should be adhered to, on my university we have several multi handicapped people, doing fine, and apparently have fine social connections and networks with all other "types" of students. Jun 27, 2016 at 7:51

I'm all in favor of assistive devices, but consider your feet. As someone who struggles to overcome their own disabilities I can tell you that the simplest workarounds are the most reliable.

Wolfram Mathematica is nice. It pretty prints all the math by default and the input format (after adjusting the keybindings a little) is much more "direct" than latex (i.e. entering "3^4" gets you $3^4$). Aside from the easy input, you also get the whole suite of computer algebra functions and symbolic manipulation.

I'm very ignorant on the topic, however I would use a programming language probably. Something like Python would be good. Or also Matlab, which is probably easier and you'll find maybe more documentation about Matlab rather than Python, regarding Maths.

With Python you can do any mathematical computation, at least any you may need from elementary school to university and maybe further to research.

I suppose that you can use one of those keyboards which are displayed on the monitor and you only have to click the letters, in this way you can program. You could write your own programs that you could run in the future, saving time.

Apart from that I have no idea to be honest

• Programming is what got me here. I'll finish my bsc in software engineering in 2 weeks, working as you described. I'm thinking about doing accomputer science master after this. Jun 20, 2016 at 16:38
• "With Python you can do any mathematical computation" in principle this is true, but in practice Python is insanely primitive from a pure math perspective. Imagine trying to do commutative algebra in Python... Jun 21, 2016 at 4:52
• There are libraries like Sage that help with mathematical programming in Python. Jun 21, 2016 at 5:27
• @goblin: Python is primitive from a pure maths perspective, but at least it's much more sophisticated than Matlab... Jun 22, 2016 at 21:57
• @AtticusStonestrom I personally find Python easier to use and more flexible! There’s tons of material online and people make all sorts of tools that you can simply use without having to make stuff from scratch! Julia is also amazing but since it’s newer it has a smaller community which means it has fewer people making all sorts of tools. Yet it’s growing very fast which is good! May 17, 2021 at 23:45

Check out Efofex. In particular, Efofex Equation was designed to type equations without breaking the thought process. Most of the symbols can be entered with single keys or shift and a key (you can use sticky keys if necessary), and there's a fairly complete graphical entry system for everything except basic operators (+, -, *, /, =). If you need it for school, check out their empower program. Their customer service is uncommonly good, too.

It's very individual what works, and I don't think maths is very different from a lot of other subjects in presenting challenges for (partially) disabled people.

Your level/what you want to do probably also matters, the rest of my answer is somewhat geared towards a high level, but it might be useable in any case (I have no experience with your kind of situation).

Some people prefer scribling away on paper, others prefer lying on a bed/couch with their eyes closed and just thinking about stuff. Both approaches requires that you afterwards write up your results nicely.

A thing that sometimes works for me is "scribling" in the air, but just moving my hands+fingers gives me another sense of things, even though you have limited movement of your fingers, the movement of your hands might give you something.

.Henrik

You might be interested in watching Using Python to Code by Voice.

• For those looking for an Open Source implementation there be Silvius, which was built (from what I gathered from the video), in response to the numerous issues people had trying to replicate what the other speaker accomplished with proprietary software. Apr 10, 2019 at 0:55

I'm not sure what devices might be able to fit a stylus to your fingers (even a modified one that can accomodate your strength and range of motion), but have you considered using a device like

Also I'm wondering if anything software-based has predictive symbols for math similar to Windows 7 'math input' panel.

• Tablet is not an option for me sadly. I tried the Windows 7 'math input' panel with the mouse, it works but my handwriting (mousewriting?) is pretty bad. It's better than MS Office atleast. Jun 20, 2016 at 16:51
• In that case, and reading some of the other comments, I'd suggest you write code for a 'voice-to-math' engine 😊 Jun 20, 2016 at 17:37

Its author had problems similar to that you face.

Edit: This link no longer seems to work. Can somebody explore if this text exists on the internet anymore?

• @user236182 Edited for him Jun 23, 2016 at 12:38

A long time ago I worked supporting disabled students using computers and this is a problem we often had. Also my writing is impossible to read, so I was allowed to use a computer to do exams, including maths courses at university, my wife has Cerebral Palsy and likewise used a computer while doing her university maths degree.

There are two separate problems to solve, do not mix them up!

Firstly the disabled person has to be able to “type” on the computer (and use the mouse), for this there are lots of custom hardware and software solutions, including speech input. (Let’s just say that Stephen Hawking can’t use speech input, so don’t assume that speech input is the answer for everyone.) An assessment by an expert needs to be done so as to advice on the best options. In the UK contact The Access Centre at Hereward Collage or Abilitynet, in the USA try the Trace Centre at University of Wisconsin-Madison.

However this is often a solved problem, as the student will have been using a computer at school, also at least 50% of our clients could use a standard keyboard with no modifications, other then maybe enabling “Sticky Keys”.

Then software is needed to allow “maths” to be done.

This is different than allowing maths to be presented in a nice way. Image learning to solve your first equations using Latex notation! When “doing maths” by pen/paper, people do not work “left to right”, they build up equations in all sorts of order, therefore the software that is used has to be flexible.

For a long time ChiWriter was the chosen package for disable university students dong maths degrees, as it did not force a person to input an equation in a given order. It is no longer on the market, and must have been replaced with something better! Sorry I am nearly 20 years out of date on what is the best software to use.

I am a graduate student in a mathematical field with a chronic injury to my hands. I have recently been using Mathfly to dictate LaTeX math by voice. Please see mathfly.org for instructions on how to install and use it as well as demonstration videos and the chat room. It is free and open source, and well documented.

Unlike mathtalk, mathfly allows continuous command recognition (ccr) which means that you don't have to pause between commands. Mathfly is also completely customizable.

Though not impaired, I recently began working with Accessibility, so my computer and I can communication audibly. Was very challenging at first, but re-running the "improve" process helps, plus this week I found a note and downloaded WSRMacros, for Windows Speech Recognition. Essentially I use it to make boilerplates ahead of time, set one up so I'd say, for instance, "quadratic" and it might output Ax^2 + Bx + C, which appears fairly useless, except that you can then turn around and by voice "select A" and replace it, repeat for B and C. Tedious, but each new macro adds more functionality, and the symbols are integrated into the macro, rather than somehow spoken into the equation for each instance. Maybe you can combine this with the other ideas above, like using Google voice to get a good basic equation on your clipboard. Also, MathML is interesting, it's like writing webpages in HTML, but with specific math tags. These are free and/or open source tools, or included with Windows.

• Good solution, I used Micros like this with ChiWriter, but how do you prove that the macros are not giving a student an unfair advantage in an exam. For example the "quadratic macros" is clearly OK for a 3rd year maths degree student, but not for someone taking a school test when one of the questions is "what is a quadratic equation?".
– Ian
Jun 28, 2016 at 9:18