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Problem Solving.

Published 25 April 2021 at Yours, Kewbish. 2,814 words. Subscribe via RSS.


I was working on a physics problem earlier today: something to do with a ferris wheel, centripetal acceleration, and calculating the range of something dropped at a certain point on the wheel. It wasn’t particularly difficult, but it’s a good example of the ways I’ve had to wrangle with existing tools in order to work the way I’d like to. Normally, when I complete practice problems, I don’t bother writing out full equations or even a good amount of work, and just stick to a couple notes for formulas and things I’ve moved around dashed out in Chrome’s address bar. The rest of the arithmetic part ends up being bodged together in the omnibar’s built-in calculator, which I find is more than good enough for most problems. When I’m doing practice for myself, I’m just focusing on if I can get the formula and the problem’s givens somehow into the correct answer, and I know that if I properly tried, I could get the steps down anyhow.

However, this specific problem was for a friend who wanted a written solution, which meant that I had to work an actual diagram out. I didn’t fancy trying to ASCII art one together in Vim or something, so I turned to whiteboard.team, a free, no-account-needed (which is actually great, since most of the time, my boards are temporary and just for sketching out ideas and such) online whiteboard app. I’ve found the lack of a cloud-sync or saving option while online not to be a problem at all - again, I’m generally only keeping boards for a couple minutes for a screenshot. I sketched out the little wheel, slapped together a poorly-formatted, rather terse solution, snapped a screenshot, and sent it off. Close board, and go about the rest of my day.

Chemistry labs, physics problem sets, and calculus reviews have a couple things in common besides a general feeling of dread, accompanied at times by procrastination: they involve manipulating equations and working with data in ways that often don’t work entirely well with existing tools. For example, with the physics problem example above, I’d have loved to get a way to format equations with at least proper subscripts and integrate a quick calculator in the whiteboard, so I wouldn’t have to keep switching between a myriad of new tabs with nothing but half-baked expressions in the omnibar and a Unicode reference page to copy out special characters.

I think tools nowadays tend to do either one thing very well à la the Unix philosophy, or several things half decently - something like the spread of things like Notion and other one-size-fits-all-problems software. On this specificity spectrum, I’d still rather my tools tend towards the ‘one thing very well’ side - I’d like to have the power to do more complex things when I need to rather than be handrailled in. However, the problem with having a wide toolkit of highly specific programs tends to be sprawl, where the programs you need to have open to do one given thing increases as the required functionality gets split across more and more apps. As well, as bemoaned by most of the HCI / developer community, most tools lack a centralized standard, or even expose their internal API, to move information from app to app in a non-manual way. Especially when apps are very seemingly disconnected, requests for connections and integrations are very (understandably) niche, and end up serving only a couple users. Why can’t I get a way to move equations between document formats, by, say, exporting a Word equation to clipboard for evaluation in Desmos or Chrome or whatever calculator, and then automatically copying the answer back into my doc? I doubt Word and Desmos / Chrome would ever build a proper two-way integration (rightfully so, but that’s besides the point), so I either have to resort to manually repeating things like this, or have to change tools (well, apparently I can evaluate functions in Excel, but that’s still a bit of a hurdle).

Maybe it’s the influence of HCI Twitter and the amazing tools for thought I’ve seen prototypes for on my timeline, but I’ve been thinking a lot about a possible ideal mix of all these problem-solving softwares, or at least a combination of features that’d personally benefit me best. With the amount of apps and things I’ve tried out, I’m surprised no one’s thought about integrating the best of each of these tools, but hey, ‘software to automatically evaluate and nicely format my equations in a way that’s acceptable by my teachers’ is a pretty niche specification. Consider this thought-chain a speculative dream of what could have been (and hey, what might be, if I ever decide to take a stab at integrations myself one day).

Desmos Comes Close

I’d like start by extolling the virtues of Desmos, specifically focusing on their wonderful graphing calculator. With labels for notes, arbitrary point insertion, variable assignment, and immediate cell evaluation, it’s become an invaluable tool, especially for formula- and repetition-heavy work. Though I sort of neglect the graph itself for most of my problem solving (save for in maths), I’m pleasantly surprised at how full-featured and flexible the calculator itself is. Returning to the physics workflow example, I can pre-define constants and pre-program common formulas, while keeping changeable variables for mass, velocity, and other values that change between problems. This feature alone has saved so much time - instead of looking back for intermediate values or estimating them and losing precision, I can simply define them and reuse them in future calculations. Because Desmos also calculates the expression value immediately, I can get the convenience of the Chrome omnibar calculator with added annotation and manipulation powers.

Text labels have weirdly helped me think through problems and substitution more effectively than just trying to keep formulas, values, and all the next steps in my head. Part of it is that it’s a lot easier to see where you are in a problem when everything’s written out, especially when determining next steps. Visually seeing possible substitutions and figuring out what’s next is plenty simpler when I’m able to view both my notes and the calculations in one place. Previously, when using just Chrome, I’d have to tab between calculations and formula notes, and it was honestly an information overload, as I had to keep each tab open to preserve intermediate values and could never find the work I was looking for.

Tables have also been especially useful in chemistry, with all the ICE tables we need to compose for our calculations. Calculating K values is an area where I find I need the guidance and actual visual information of a table, since there are way too many values to tackle to lay one out mentally. They’re also super useful for applying formulas to a range of values without having to create a new Excel workbook for a problem that’ll take a couple minutes - a past overhead that Desmos now obviates. While these tables are nowhere near as powerful as Excel cells, they’re more than enough for calculating intermediate values and sketching out work.

Images, however, is one way Desmos falls short of what I’d ideally need. I generally find that I need to diagram things for physics (or, at least, it’s good practice), and trying to diagram anything other than a circle or a line is a lot more work than I’d like. Sure, I can quickly draw something out in Vectr and upload it, but I often find I’d rather just go without and edit an explanation in in post, or forgo the diagram entirely. It’s no fault of Desmos’s - the entire point of the calculator is that it specializes in graphing and scientific computations, and it’s gone above and beyond what I need from it. However, it’d be amazing to figure out a way to combine both the power of the calculator and, say, something like Vectr or even a rudimentary version of whiteboard.team. Most of the drawing functions are redundant and overcomplicated, but even a box, circle, line, and arrow set of functions would be amazing. Oh well - it’s outside of the scope or purpose of Desmos anyways.

Free-form Structure

I find that my work in solving these science problems feels sort of like sketching, albeit perhaps on a digital canvas, rather than physical paper. When I’m trying to work at my speed of thought, the most friction-free experience is generally with text, but diagrams and laying things out spatially can help sort out how to manipulate the formulas and such. I use the term manipulate, since it really does feel like you’re slotting things in and moving terms around.

Highly limiting mediums like ASCII take away the aesthetics side of things, where I almost feel like I need to make something pretty and presentable. However, I generally don’t want to make diagrams in ASCII, especially when there are more abstract shapes and arrows to work with (physics won’t work, though chemistry might, since it’s more text heavy) Especially when using more rudimentary tools that impose a certain structure, it feels that tools are pushing you against the decision paralysis presented by more complex and full-featured tools. That’s why I tend towards quick scrawling even when solving problems - I’d rather check that I can understand and produce the answer, rather than fuss over how to get a subscript in another subscript1. Tools like Vectr are great for producing more polished drawings, but I end up spending much too long fiddling with fonts to get everything on a page to look nice and cohesive. whiteboard.team’s sketching tools are quicker to work with, and while everything comes out looking a bit like an MS Paint attempt at modern art, it works for sorting the problem out quickly.

I guess this sort of touches on the presentation side of things, and the purpose of whatever I’m sketchnoting. If it’s something just for me, scribbling down a couple boxes with unlabelled arrows is generally enough for me to get the gist of what I need to do. However, if (like in the scenario I shard in the introduction) it’s something I need to present for marks, I’ll put more effort in. This ’effort spectrum’ is something that sort of changes the given toolkit I use for a given purpose.

But also, compatibility

Speaking of changing the tools I use for a given purpose, I also have a minor point with file formats and such. Word, at least in my program, is the de-facto standard for documents and homework, so I’m required to work with .docx files - even if I don’t have to directly edit in Word, I need to produce something Word-parseable in the end2. Working with Word equations is not that painful of an experience - while my classmates seems to collectively agree that handwriting chemistry equations is much easier than typing them out, I find that working with digital equations is as fast as and as convenient as working on paper. Perhaps it’s just a matter of medium preference.

However, one thing that I definitely find tedious is editing equations and working with calculations in the same workflow. When building a lab report, for example, I have plenty of sample calculations to transcribe, and generally data points for these equations source directly from raw data. I usually keep a separate Excel sheet with formulas I can easily drag around and apply to groups of cells, but it’s a pain to try to copy around end values and work with calculations within Word itself. If I’m not Excel-ing things, I keep a Chrome window open, and it’s sort of frustrating to jump around to do calculations. It also introduces a new surface for error - I don’t want to talk about the number of times I’ve mispasted a value or missed an exponent by one.

It’s relatively painless to fix things up if this wrong value happens at the end of a calculation, but when it’s an intermediate value that other calculations depend on, reworking all my equations is an especially tedious task. This is when I tend to crave the variable recalculation abilities of Desmos, which immediately redoes all depending calculations, and Excel, where I can use cell references to keep values ‘bound together’. LibreOffice (/ Word) seems to have a variable function to define numbers and reuse them in the document, but I haven’t found a way to run formulas on them, or integrate them in equation blocks.

I don’t think it’s worth making a point of introducing a new file standard for something like this (relevant XKCD), and I also don’t think it’s worth bothering to ask if I can submit in a PDF or whatever quirky file format an ‘ideal’ software would kick back. I wouldn’t mind having to work with an intermediate file format, as long as it’s fully integrated and exportable to Word (or LaTeX). I think these are pretty lofty goals, however, so I think I’ll just have to live with the monotony of redoing a full set of equations four times over because I accidentally messed up in the first two sections - but that’s a story for another day.


Whenever I’m using only one of the above tools, it feels like I’m missing something. With Word, I wish I’d be able to calculate and substitute variables in equation blocks themselves. With Desmos, I lack the image and diagram manipulation features that help visualize problems at times. With whiteboard and other diagramming software, I lack the ability to format and present my work later, and also don’t get any of the variable manipulation that streamlines my entire calculation workflow. I think my ideal scientific problem-solving software would be something like a combination of:

  • Desmos’s variable system and instant calculations
  • whiteboard.team’s / Vectr’s diagramming capabilities
  • Word’s layouting, formatting, and text hiding, and
  • (maybe, would be nice to have) Vim’s keyboard-driven philosophy

I could replicate several parts of this system on physical paper, but again, I’d be losing out on the variable calculation and the digitization aspects of my workflow. For me, it’s also a lot easier to type things out, both in terms of remembering the solution work and efficiency (I type much more quickly than I can write). I’d also prefer to figure out a toolkit digitally that can continue to work in the future, if the digital aspect does end up being a requirement sometime in university.

In terms of developing things like this, I’m not entirely sure, but it might be possible to wire together Word extensions, a Desmos API, and some sort of diagramming API to make inserting and formatting work into Word more streamlined - I haven’t done much research into the possibility. I think this might have something to do with the whole Bring Your Own Client idea, where there’s a need for more software to open up to remixing and inter-app data flow first. I’d like to investigate possibilities and other tools that’re easier to mix together in the future as well, especially tools that are simpler and more text-based. Developing a toolkit for scrap work and raw calculations is still a higher priority for me than Word integrations, but the whole file format thing is still a struggle, especially since most of the work I do is for school, and I need to follow assignment requirements.

I’ve been putting off writing this post for a while, and I think I’ve sort of lost the rhythm of writing weekly. Part of it is the homework (that they did say was supposed to decrease roundabout now, but oh well), and part of it was a lack of coherent thoughts. I’ve went back and built a list of some ideas for future posts just now, so hopefully I’ll be able to get back to writing more regularly. It’s interesting how writing and outlining something like this sort of cements all the ideas and makes it clear what I personally think, so I guess that’s a nice hidden benefit of writing. Writing this post actually inspired a different one that might come out in the near future, but we’ll see how much time I have with school and such.

  1. I use LibreOffice, since booting back to Windows is too much effort to bother with on a regular basis, and Word on Wine was both a pain to set up and to work with, so some of the instructions for Word didn’t apply. In the end, apparently you’re supposed to wrap the part of the equation to subscript with a set of curly braces. You can then apparently nest to whatever depth you’d like to - a bit of a late discovery, but hey, late’s better than never. ↩︎

  2. Apparently this changes to LaTeX in university, so there’s something else to consider. ↩︎

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