Fuzzy string matching using cosine similarity - <AUTHOR> http://blog.nishtahir.com/2015/09/19/fuzzy-string-matching-using-cosine-similarity/ <comment><USER2> Many people comment that there are nicer string-distance measures.

However the real advantage of cosine distance is that you can perform dimensionality reduction (see e.g. page 38 of [1]). This allows you to work with very large documents efficiently and fuzzy. It also allows you to create efficient data structures for finding similar strings and much more.

[1]: [http://web.stanford.edu/class/cs246/slides/04-lsh_theory.pdf](http://web.stanford.edu/class/cs246/slides/04-lsh_theory.pdf)

</comment><comment><USER3> That is a simple but very effective method. I am using it for my projects with this tiny library I've written earlier this year (C++). It uses Cosine Similarity + Trigrams and is quite fast.

[https://github.com/felixguendling/guess](https://github.com/felixguendling/guess)

<comment><USER4> NLP neophyte here: where do trigrams fit in? Do you base your vectors on trigrams instead of characters?

</comment></comment><comment><USER5> This is an experiment that everyone who learned about cosine distance and vector space information retrieval probably tried, just to see what would happen.

"Fuzzy string matching" isn't really the right term. Of course string distance will give a better fuzzy string match. But this finds words with similar structure rather than words that are similar in the traditional sense. Perhaps there's no obvious application, but it gives results that are very interesting in and of themselves.

It might serve to partition search space for another more expensive algorithm.

<comment><USER6> > It might serve to partition search space for another more expensive > algorithm

This is what we're doing. We're working on approaches for performing nucleotide sequence alignment using approximate time series representations of vectorised DNA sequences. There are some really elegant lower bounding similarity search methods for time series which generate no false negative alignments, allowing use of more expensive alignment algorithms later on in the search to prune out the false positives.

We've tested a few transformations including Haar wavelets, DFT and PAA and implemented an indexing structure in C++.

Preprint: [https://www.academia.edu/12575290/Alignment_by_numbers_seque...](https://www.academia.edu/12575290/Alignment_by_numbers_sequence_assembly_using_compressed_numerical_representations) (slightly outdated... Email me for more info)

</comment></comment><comment><USER7> I am not thinking this through but since cos is monotonous between 0 and pi/2, you can avoid the square root by squaring the numerator and still have a good similarity indicator. This is going to eke out some performance with a short tweak.

<comment><USER8> It might not be a good idea to just square the numerator. Doing so would give an advantage to words with a small norm, i.e. a small number of letters. Unless you actually want that to happen of course.

You could alternatively square the thing in it's entirety, this causes problems if a.b is negative but if you're only using positive coefficients (as in this case) then that can't happen, but even if it could then it's pretty easy to check if it is positive or not.

<comment><USER7> That is what I meant. I wanted to point out that sqrt is more expensive compared to squared and you could just square the whole thing without affecting the results too much.

cos(theta) and cos^2(theta) are pretty close and monotonous between 0 and pi/2:

[http://www.wolframalpha.com/input/?i=cos%28theta%29+vs+cos%2...](http://www.wolframalpha.com/input/?i=cos%28theta%29+vs+cos%28theta%29*cos%28theta%29)

</comment></comment></comment><comment><USER9> Dimensionality reduciton is correct - the cosine similarity on a character/word level really just provides a metric for measuring the "anagramness" of two words. But "Mary" and "Army" would have a perfect similarity.

</comment><comment><USER10> This is nice, but a better measure of string similarly is an edit count based measure. See Levenshtein distance for example: [https://en.m.wikipedia.org/wiki/Levenshtein_distance](https://en.m.wikipedia.org/wiki/Levenshtein_distance)

<comment><USER11> On the other hand, computing Levenshtein distance is O(n^2), while the cosine approach is O(n).

<comment><USER12> This very topic was discussed yesterday. For a given word, you can construct an automaton that matches that word within a certain edit distance in O(n) where n is the length of the word. Since the result is a DFSA, you can also see if a string matches in O(n). (See the work of Schulz & Mihov.)

Moreover, rather than going over each word in a dictionary to find out whether that word is within a certain distance, you can compute the intersection language of the Levenshtein automaton and the dictionary automaton (which is generally faster, intersection is O(mn) in the number of states of the two automata).

For an implementation of this and more, see:

[https://github.com/danieldk/dictomaton](https://github.com/danieldk/dictomaton)

</comment></comment></comment><comment><USER13> Postgres users might find Teodor Sigaev's smlar [1] extension useful for this kind of thing. It implements cosine and other similarity metrics.

[1] [http://sigaev.ru/git/gitweb.cgi?p=smlar.git;a=summary](http://sigaev.ru/git/gitweb.cgi?p=smlar.git;a=summary)

</comment><comment><USER14> Would this be able to find near matches within substrings, e.g. a search for "house" in "this is my hiuse typo" would bring back a near match result?

<comment><USER5> No. The character distribution of those two strings is massively different. You'd have to tokenize the long string into words first.

EDIT: If I understood correctly and the vectors are normalised, it would match 'house' to 'hiusehiusehiuse'. Which may or may not be desirable.

<comment><USER15> The vectors are not normalized. That is:

        from __future__ import print_function
        from collections import Counter

        def cosine_similarity_unweighted(a, b):
            a = Counter(a)
            b = Counter(b)
            dotProduct = sum((c in a) * (c in b) for c in a)
            magnitudeA = sum(1 for c in a)
            magnitudeB = sum(1 for c in b)
            return dotProduct / (magnitudeA * magnitudeB)**0.5

        def cosine_similarity_weighted(a, b):
            a = Counter(a)
            b = Counter(b)
            dotProduct = sum(a.get(c, 0) * b.get(c, 0) for c in a)
            magnitudeA = sum(value**2 for value in a.values())
            magnitudeB = sum(value**2 for value in b.values())
            return dotProduct / (magnitudeA * magnitudeB)**0.5

        print("Scores for 'hello' and 'holl'")
        print("unweighted", cosine_similarity_unweighted("hello", "holl"))
        print("weighted", cosine_similarity_weighted("hello", "holl"))

gives the output:

        Scores for 'hello' and 'holl'
        unweighted 0.866025403784
        weighted 0.925820099773

The article gives:

        holl        index: 0.9258200997725514

which means the article used letter weights.

That said, the scores for your example are the same, because each of the letters in 'house' is used only once (the 3 in the numerator cancels out the 3 in the denominator):

        Scores for 'house' and 'hiusehiusehiuse'
        unweighted 0.8
        weighted 0.8

If the query had instead been 'houss', you would see a difference in the similarities:

        Scores for 'houss' and 'hiusehiusehiuse'
        unweighted 0.67082039325
        weighted 0.676123403783

</comment></comment></comment><comment><USER16> much better than difflib sequencematcher</comment>


Thoughts on user growth and product - <AUTHOR> https://blog.jonlu.ca/posts/thoughts-on-product <comment><USER2> Hi Jon, I was a PM on Growth at Pinterest many, many moons ago (~2013-2014?) so it was cool to see your post and read through your observations on Growth at Pinterest. A couple points I might add that might be of interest to your readers would be that the biggest changes can be technically driven, and the biggest changes are sometimes the smallest. I was at Pinterest from 2013 - 2017 and iirc the two biggest experiments in terms of percentage user signups and retention were the conversion of unauth pages to react [1] and the change from "Pin" to "Save". They were both roughly equal magnitude in impact (again, iirc), and completely outsized all other growth experiments shipped in those respective years. Two very different projects, approached from very different angles, that ultimately lead to a faster, and more understandable, product.

[1] [https://medium.com/pinterest-engineering/driving-user- growth...](https://medium.com/pinterest-engineering/driving-user-growth-with- performance-improvements-cfc50dafadd7)

</comment><comment><USER3> Some wise advice here. I relate to the thought on cyclical user growth.

Some execs refuse to hear that Q3 is regularly going to be a dead space no matter how aggressive the company gets, for example. It's a challenging work environment when there's no room for a down month or a holiday-season slump.

</comment><comment><USER4> For me, the real growth question with Pinterest is how come they rank so well on Google? They dominate result for many image-related queries despite being a content aggregator. Everything on Pinterest inherently lives someplace else on the web, no? How come those aren't what I'm seeing? The site seems to have become an SEO powerhouse despite - at least for me - never being the result I want.

<comment><USER5> I believe Pinterest use machine learning to generate labels for images for SEO purposes, which I imagine has an enormous impact.

<comment><USER6> Honestly I'm surprised they're not penalized by Google, given that pretty much any search result page does not contain any of the words in the search result snippet, in my experience.</comment></comment></comment>


Why Titles Matter a Lot if You're a Blogger - <AUTHOR> http://www.bothsidesofthetable.com/2013/09/07/why-titles-matter-a-lot-if-youre-a-blogger/

<comment><USER2> How is this new, or news? Headlines and titles have _always_ mattered in the media. Even in the world of blogs, haven't we seen folks like Buzzfeed and Upworthy demonstrate the value (sometimes dubious, though) of catchy headlines?

On a lighter note, here's a Twitter parody account of Techmeme Headline writing: [https://twitter.com/nottechmeme](https://twitter.com/nottechmeme)

</comment><comment><USER3> Simply useful article, embodying its own advice: Sort of intelligent SEO advice for ordinary types. It's an updated analogue of the kinds of reasonings my (hard-nosed news-editor) journalism professor used in his teaching us hard news.

</comment><comment><USER4> TL;DR:

Because it affects clickthrough rates. There ya go. On with your day.</comment>


Apple to Target Combining iPhone, iPad and Mac Apps by 2021 - <AUTHOR> https://www.bloomberg.com/news/articles/2019-02-20/apple-is-said-to-target-combining-iphone-ipad-mac-apps-by-2021 <comment><USER2> This is weird. It's the sort of thing that management dreams of but which companies like Apple should realize is a bad idea. They definitely need to provide tools for making app families work smoothly across each platform, but even more importantly they need to be segmenting their devices more, not less. The latest iPads are a great example of a device that has enormous potential which is being hamstrung by an OS and dev platform that doesn't really provide a tablet-focused experience.

And then the Mac is losing what should be its primary audience through unwanted innovations and otherwise stagnant hardware, and a failure to recognize the importance of catering to the power users who might want an actual escape key, multiple types of ports, and a keyboard that doesn't feel like it came off a rejected tablet accessory.

I can see an argument for fragmenting the laptop world into Pro/developer hardware and consumer hardware. But Apple seems to have got the needs of those groups mixed up. Do Apple's own software engineers love the newest Macs I wonder?

<comment><USER3> _And then the Mac is losing what should be its primary audience through unwanted innovations and otherwise stagnant hardware,_

While I agree with most of the post, the "Mac is losing people" narrative isn't supported by the numbers. Apple just reported its highest revenue quarter for Macs in its entire history.

The industry has spoken, statistically, no one cares about PC software. Writing Mac software even less. I would rather live in a world with cross platform iPad/Mac apps using Apple's SDKs than Electron apps.

<comment><USER4> Apple's revenues are a crappy statistic because the majority of their revenue and profit both come from consumer-level content consumption devices. For all the talk of "Microsoft Office on your iPhone" there is still nothing to supplant the hidden iceberg of small-to-Medium-to-Enterprise businesses that are only able to function because of hacked-together VBA macros in an Excel spreadsheet stored on a Synology box somewhere.

PC software _matters_ because it's still the only platform that enables scrappy non-expert _go-getters_ to get things done. Despite the announcement of "real" Photoshop for iPad - Hello? Binary 8bf plugins? Where are you?), Apple's continued devotion to the Walled Garden concept and associated prohibition on extensible iOS applications - combined with the high barrier- to-entry to develop for Apple's ecosystem means that the PC will always be relevant.

As an aside, it's kinda funny that VBA macros copypasta'd together by individuals with little to no formal SE or CS training provide more utility- per-MB-of-RAM-consumed than Electron apps.

<comment><USER3> I did specify "Mac Revenues"....

<comment><USER5> And I think someone followed up with a critique on frame. I'd agree for a different reason. People would want to know how much slowdown there is in userbase gain and how much of the revenue is made from getting existing users to pay more.

</comment></comment></comment></comment></comment><comment><USER6> The article hits on this, but I think it probably makes sense for Apple to do this as they plan to run all their hardware on Apple chips in the near future. That said, I think it will likely result in lazy UX that's "good enough" for multiple device types.

<comment><USER7> > I think it will likely result in lazy UX that's "good enough" for multiple > device types.

This is already the case: the Marzipan apps shipping with Mojave are _shockingly_ bad (Home, News, Voice Memos). Standard keyboard shortcuts don't work, drag-and-drop is inconsistent, and it's clear that they never considered standard Mac features like multiple windows while designing them.

I worry that people will look at these badly-ported, out-of-place apps and say "Apple are doing it, so why can't we?" rather than "Even Apple could only come up with this, so let's not."

<comment><USER3> Yes they are bad - but they are also a technical preview. Apple said as much. Would you rather Apple come out with the framework in a year without getting real world feedback?

<comment><USER8> The News app says it's version 4.1.1. It's part of a shipping release version of macOS. I don't know why any regular user should come to the conclusion that this is some sort of technical preview. Why wouldn't they think that it's simply terrible software?

<comment><USER3> We aren't suppose to be regular users. We are suppose to be technical. We discuss technical issues on HN and do stuff like watch the WWDC keynote.

<comment><USER8> Absolutely. I just think these apps should never have shipped as part of an OS release. They are demo apps for the Marzipan technology and should have shipped as part of an SDK.

<comment><USER3> Yeah that or just label them as beta to set expectations. Unfortunately, Google has ruined the term beta for users. Maybe put them in a folder called "Prerelease".

</comment></comment></comment></comment></comment></comment></comment><comment><USER9> Why? Has anyone asked for this?

I want my desktop apps to be dense and feature rich professional applications. My phone apps are relatively simple tools.

I don't understand why.

<comment><USER10> Personally, I would love if my phone apps were dense and feature rich professional applications, too.

A lot of times I want to do something and it feels like the apps are holding me back just for the sake of being excessively simple. I've had this feeling on both Android and iOS

<comment><USER11> Seriously asking, was this sarcasm? I can see feature rich iPad apps... but is the phone meant for more than quick glances?

<comment><USER10> It was not sarcasm. Sometimes quick glances can involve advanced tasks too.

Or sometimes I'm on the go, with no tablet or computer and I still want to do something advanced, even if the phone screen is small.

Besides there's no reason why I shouldn't be able to hook up my phone to an external display/mouse and get a full desktop experience. (Samsung is ahead of the pack in this, but the apps are still "mobile" versions that are not as fully fledged as their desktop counterparts)

<comment><USER11> Cool, thanks.

</comment></comment></comment></comment></comment><comment><USER12> What worked and didn't work with Microsoft's attempts to do this with Universal Windows Platform?

Was it just lack of active phone / tablet marketshare, or was there a fundamental technical reason it didn't take off?

<comment><USER6> Speaking for myself (as a huge fan of C# and WPF), I didn't feel like there was enough of a phone/tablet/Store/Xbox market for UWP apps that made it worth restricting myself to UWP. I feel like it was a case of trying to make one tool useful for all jobs instead of choosing the right tool for each job.

</comment></comment><comment><USER13> Now we know why the new keyboards are so awful. They want to delight users by removing it completely for the new touch-only interface.

<comment><USER14> Sigh, I really hope you're wrong. I can't stand the new keyboards (sticking to a last-gen Macbook Air) but the only thing worse than those keyboards are the official iPad keyboard folios from Apple. I'd seriously rather use the onscreen keyboard...

On the other hand, if we had to rely on keyboard cases, there might be better options than what we have today - e.g. I've heard a lot of good about this Razer _mechanical_ iPad Pro keyboard although it's obviously very bulky: [https://www.macworld.com/article/3115739/input- devices/razer...](https://www.macworld.com/article/3115739/input- devices/razer-mechanical-keyboard-case-review-luxurious-keys-for-ipad- pro.html)

</comment></comment><comment> matthew-wegner This is already sort of here, and badly. If you're on Mojave, open up the built-in News.app.

Resize the window (it's terribly slow). Mouse hover states aren't even there. Bouncing over-scroll is still present. The speed at which Apple can successfully make this not-awful is probably a good indicator of them trying to make it ship for all developers.

</comment><comment><USER15> This is one of those corporate directives from upper management that was made in a vacuum. Those three devices have different access patterns, user bases, and user abilities.

</comment><comment><USER16> Yesterday there was an article on HN about audio glitches on Macs, [https://news.ycombinator.com/item?id=19196354](https://news.ycombinator.com/item?id=19196354), with comments (including mine) about Apple products going down hill. I think this idea is a glaring example of how Apple has lost their way. They are exploiting a market - Apps - instead of adding value and building good products.

I seem to remember Microsoft do exactly this same kind of thing as they were on their downward trajectory?

<comment><USER17> Microsoft managed to turn it around with a new CEO and a new market category. I wonder what Apple's next venture will be, and if it will be as successful as Microsoft's cloud and subscription-service products.

</comment></comment><comment><USER18> This is great. For those here commenting "why? My iPhone is very different from my iMac", an example: I am writing an iPad app for a client that has a lot of custom Metal compute code. It is great that I can basically run that code on my iMac Pro as well, for rapid prototyping, and because having a desktop companion app makes a lot of sense. Right now though I have to be very careful how to share code between the two apps, mainly because UIKit is not available on the Mac. Unification would solve that headache for me.

<comment><USER19> What people who haven't written Mac apps need to realise is that many of the UI APIs are not even remotely shared between macOS (AppKit) and iOS (UIKit). Even the most basic concepts like colour are not handled uniformly.

Having UIKit available on macOS doesn't mean you port over your iPad GUI and call it a day; you would still write a desktop GUI. But by having the same API means that your business logic and glue code doesn't get infected with quite so much platform-specific crap like UIColor vs NSColor.

</comment></comment><comment><USER11> I understand the desire to push computation to a single device, but I don't agree with it. It is very rare for a device that does 'everything' to do anything 'well.'

In software, Eclipse is the perfect counter argument. It does nothing well, imho. Sketch is the opposite, singular focus = quality experience.

My phone serves a purpose and I expect apps to be bite sized experiences and quick data points.

My computer serves a completely different purpose. The applications should be equally focused on that purpose

<comment><USER20> I think it's also a bit (just a bit) due to our need to focus on a single domain. It's rare to have your brain ready to navigate the mathematical realm of all universal abstractions in a single setting (can happen but not common for the mere mortal I suppose).

Meanwhile people are doing [https://blog.adafruit.com/2019/02/14/the- spudwrite-single-pu...](https://blog.adafruit.com/2019/02/14/the-spudwrite- single-purpose-user-device-for-creating-writing-made-with-e-paper-mbed-and- stm32f401-cortex-m4/) in their backyard.. says something

</comment></comment><comment><USER21> A lot of people are focusing on the user experience in their commentary here, which I totally get, and I would argue is the most important part of this initiative. I personally have a lot of open questions about how a user interface will work across different paradigms (touch and cursor). The first Marzipan apps (News, Stocks, and HomeKit) have lots of weird corner cases and sharp edges.

Specifically, some of the "elder statesmen" of Apple commentariat see Marzipan as a threat to the old Mac indie app ecosystem. I get that, and I think that there are definitely some aspects of the old paradigm that are at risk (ie, AppleScript integration, etc).

However, I'm giving Apple the benefit of the doubt here.

1) Apple is unmatched in their ability to design user interfaces. It is entirely possible that they will come up with a solution for cross-paradigm interface that works, or come up with a system to "translate" an interface for multiple target paradigms.

2) Apple has a history of successfully pulling off large transitions like this. In fact, they might be the _only_ company that has done this successfully multiple times. (That is, architecture change and UI framework migration).

On top of that, I think all the doubt and suspicion has to be weighed against the following upsides:

1) The upside for the Mac is _enormous_. The sheer amount of software that could immediately become available on the desktop would probably be the worlds biggest inflow of software to any operating system, ever. And, it is, for the most part, _good_ software.

2) This is mostly a move on behalf of developers. I get the "but no one asked" responses, but the developer story here is probably one of the major reasons Apple undertook the work (that and ARM).

3) This is Apple's answer to Electron for the Mac. It will undoubtedly be better in user experience, performance, and resource usage. (We will have to wait and see whether it is successful, though).

Stephen Troughton Smith [0] has had one of the most refreshing takes on Marzipan that I've seen. He maintains an open source tool [1] that will get your iOS app to run on Marzipan.

[0] - [https://twitter.com/stroughtonsmith](https://twitter.com/stroughtonsmith) [1] - [https://github.com/steventroughtonsmith/marzipanify](https://github.com/steventroughtonsmith/marzipanify)

</comment><comment><USER22> I've been working on an IoT mobile app suite for about 4 years now for a client. While the app has evolved significantly over the years, one thing remains constant, Only one platform is getting top billing, the other is getting the leftovers. All the designs and workflows and copy are solved on the phone first, and tablet gets a cursory glance and some changes if things don't look right. In fact at this point many of our menu views are simply modal popups on tablet so that we don't have to put any mental effort into solving view layouts.

I'm sure management would love to know they could get a mac app for free as well, but I wouldn't imagine it would look like a native mac app.

</comment><comment><USER23> Well it would definitely be nice to have more serious applications for the iPad and perhaps some new mobile productivity apps crossing over to the Mac.

But the UX is so fundamentally different that I'm afraid the majority of the applications will be pretty awful on the Mac as mobile is less information and feature dense while having a smaller screen.

What could be really great is having feature dense UI's on the larger iPad Pro's since they do have the screen real-estate but often it's underused.

</comment><comment><USER24> Touch and Desktop mouse/keyboard are different enough to each have their own UI...im guessing this just is another universal binary project...

</comment><comment><USER25> I have been using an iPad exlusively for 3 years and this makes sense to me.

Once they add this (and a mouse to iPad); then we can have the best of both worlds

<comment><USER26> Most likely you will get the worst of all worlds.

Desktop is driven by a pointer (mouse/trackpad) and keyboard, both with high degrees of fidelity. iPad and iPhone only have touch. (iPad has a keyboard as an afterthought, and the support for it is inconsistent even within the OS itself).

<comment><USER25> Can you give examples of: "keyboard support is inconsistent within the OS itself"

I have used iPad Pro exclusively for last 3 years for python coding and office work and not sure what you mean by it.

<comment><USER26> Unfortunately I've given up on my iPad, so I can't give you precise answer. So, anecdata only.

As I tried to work on it the way I work on my laptop, I would continuously run into an impedance mismatch. What I could tab to I now had to touch. What I could reach with my mouse/trackpad, I now had to touch. What I could select with my mouse/trackpad, i now had to touch. Multiple gestures and keyboard shortcuts ingrained into me by my laptop usage failed on the iPad because the software was never built to be controlled via keyboard.

MS tried doing combining desktop and mobile twice, and it failed. I have no idea why Apple thinks they can do it with any degree of confidence.

</comment></comment></comment></comment><comment><USER27> If Apple continues as in the past they will deliver great APIs, SDKs and User Guidelines which allows developers to build great apps which feel good on both iOS and macOS. I trust them that they this comes true and I'm excited to see the possibilities which their multi-platform app development tools will unleash for the Apple ecosystem.

<comment><USER28> > _great APIs_... _great apps_... _feel good_... _I trust them_... _I 'm > exited_... _possibilities...will unleash_...

That sentence has too many happy adjectives - something which seems to be endemic in the Apple world - and could have been lifted straight from an Apple advertising campaign, the latter in no small part due to the use of the word _unleash_ which thrives in that context.

<comment><USER27> Go ahead and compare Apple's Cocoa framework with the MFC or Android framework. Build apps on all these platforms and you'll see what I'm talking about.

</comment></comment></comment><comment><USER29> Assuming of of the reasons for this is to make Mac apps work on ARM.

</comment><comment><USER30> Granted they are a different company, but this was a huge failure for Microsoft. Different devices have different strengths and weaknesses.

</comment><comment><USER31> More likely to have apps ported from iOS to macOS than the reverse IMHO

<comment><USER32> Yes, and we are already seeing it. The "Home" app on macOS right now is clearly reusing a lot of iOS UI elements. It works, although it's tough to imagine a more native Mac app wouldn't be a better UX.

I think there's probably a case to be made when the app you're building is "mobile first," so you (as the developer) inherently do not want a richer, more feature-filled UI on the desktop. In that case, there's value in having a single UI to maintain and for that UI to be familiar to the user in both places.

<comment><USER3> You act as if it is some great conspiracy that the Home app is a port from iOS. They outright said that it was a proof of concept for the cross platform framework they are working on.

<comment><USER32> My apologies if that's how it came across. My intention was to support the parent comment's assertion by pointing to a live example supporting his claim.

Honestly we already discussed this when Marzipan was announced. I guess the news here is just the years they're targeting? Regardless, a lot of the comments here are worried about what the headline implies, which is much more sinister than "a fast and clean way to make an iOS-focused UI app also work on the Mac." And there are already live examples on macOS that actually work fairly well!

</comment></comment></comment></comment><comment><USER33> I hope they've learned from that joke Marzipan.

</comment><comment><USER34> This can be done using WebAssembly and swift web.</comment>


GitHub.com font changed - <AUTHOR> https://github.com/github <comment><USER2> One of the most useful options in Firefox is the checkbox "Allow pages to choose their own fonts, instead of my selections above"[1], also available as browser.display.use_document_fonts in about:config. When this is unchecked, Firefox will ignore all fonts specified by web pages and just use the much- more-likely-to-be-reasonable choice of default font you have made; and you can then disable web fonts, which will make pages load faster.

I don't think allowing websites to use specific font names in CSS was ever a good idea. It would have been better to limit them to predefined family names, which would map to fonts the user could customize. In any case, this allows me not to care about the whims of website designers and to always have text in a readable font. Only one issue: one of the unfortunate trends in web design is to use web fonts for icons. Not allowing websites to specify different fonts from the default also happens to break the icons. This isn't enough of a big deal for me to disable the option, but it's an annoyance worth knowing about.

[1]: [https://clbin.com/bFSlHF.png](https://clbin.com/bFSlHF.png)

<comment><USER3> What happens when sites are using font's for images?

<comment><USER4> Then it breaks. I can deal with that -- there aren't very many sites that decide to abuse fonts that way, in any case.

And who knows, maybe one day SVG will actually reliably render correctly in browsers, and people will use images when they want to display images

<comment><USER5> The problem isnt so much SVG rendering, its that you can grab an icon font with 400 icons that look great, its easy to add them to places beside text, like in menus or on buttons.

You can use SVG, but there is no CDN-hosted SVG sprite with 400 icons to replace things like FontAwesome or Ionicons, so you would have to find all the SVGs you wanted. Then you have the challenge of getting them the right size and position beside text in very tight spaces.

<comment><USER4> You mean like this: [https://css-tricks.com/svg-sprites-use-better-icon- fonts/](https://css-tricks.com/svg-sprites-use-better-icon-fonts/)?

<comment><USER3> Are there any frameworks like bootstrap that include SVGs?

</comment></comment></comment></comment></comment></comment><comment><USER6>
        apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"
        Helvetica, arial, nimbussansl, liberationsans, freesans, clean, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"
        sans-serif

Seems like they're just trying to use whatever decent sans-serif font is available.

</comment><comment><USER7> Viewing it on OS X, it looks like they're using the San Francisco font. A little jarring if you're used to seeing the old typeface 20+ times/day, but it looks good.

Fortunately, the transition seems to have gone a bit better than Medium's: [https://medium.com/design/system- shock-6b1dc6d6596f#.j5z5g5g...](https://medium.com/design/system- shock-6b1dc6d6596f#.j5z5g5gy6)

<comment><USER8> You would think that a site as large as Medium would think of actually testing their site on Windows before deploying.

</comment></comment><comment><USER9> HORRAY! You can use a Chrome extension to put the font back :)

[https://chrome.google.com/webstore/detail/the-old-github- fon...](https://chrome.google.com/webstore/detail/the-old-github- font/gklibcblhkjlfhfelnbelngjciflknhp/related)

[https://github.com/rreusser/the-old-github- font](https://github.com/rreusser/the-old-github-font)

</comment><comment><USER10> Goddamn, I hate Roboto, it is way too fat by default.

<comment><USER11> The Bold Roboto [1] just looks awful on my machine. Arial [2] looked much better, and if they want to use the system font, Ubuntu [3] also looks much better.

[1] [http://i.imgur.com/zfThWXP.png](http://i.imgur.com/zfThWXP.png)

[2] [http://i.imgur.com/qngKbGq.png](http://i.imgur.com/qngKbGq.png)

[3] [http://i.imgur.com/77vDhCd.png](http://i.imgur.com/77vDhCd.png)

<comment><USER12> Roboto... that's what got me here. I guess Segoe isn't too bad on Windows, but Roboto is just tragic. Wow. Really bad.

</comment></comment></comment><comment><USER13> Bold markdown is basically broken on OSX+Chrome, see before & after screenshots here, which I tweeted at Github: [https://twitter.com/MrOlovsson/status/752843094765236224](https://twitter.com/MrOlovsson/status/752843094765236224)

</comment><comment><USER14> You can alias fonts in Linux, i.e., replace one font with any other. Font Manager is a nice UI for this. I never found anything like it for Windows or OS X.

</comment><comment><USER15> How do I get the old font back? Chrome/Windows font is terrible!

<comment><USER16> [https://github.com/rreusser/the-old-github- font](https://github.com/rreusser/the-old-github-font)

</comment></comment><comment><USER17> Segoe UI on Windows. Roboto looks way better.

</comment><comment><USER18> The idea behind this change seems to be centred around using a modern font no matter what device the user is on while not sacrificing performance. The trade-off here is brand consistency, but if the alternatives are to use an archaic font or a slow webfont, I think this approach wins.

</comment><comment><USER19> _Firefox hack:_

There is an extension to override/inject CSS for specific domains: [https://addons.mozilla.org/en- US/firefox/addon/stylish/](https://addons.mozilla.org/en- US/firefox/addon/stylish/)

Old styles can be grabbed from the: [https://github.com/rreusser/the-old- github-font/blob/master/...](https://github.com/rreusser/the-old-github- font/blob/master/extension/content.css)

After applying the following config: [http://i.imgur.com/hGffN8I.png](http://i.imgur.com/hGffN8I.png) GitHub look is back to normal.

</comment><comment><USER20> Anyone know what it was previously? I'm writing a user stylesheet to change it back.

<comment><USER21> Helvetica on OSX, Arial on Windows.

<comment><USER22> They also changed the size from 13px to 14px.

</comment></comment></comment><comment><USER23> Yeah, I'm getting Cantarell rendering on Gnome, it's no San Francisco.

</comment><comment><USER24> Looks terrible on chrome on windows arrrgg why did it have to change

<comment><USER25> It's using your system fonts. If they look terrible, it's not Github's fault…

</comment></comment><comment><USER26> previous was a lot better.

</comment><comment><USER27> and there I was trying to fix my Firefox (clearing caches, cookiest, etc.) ... XD</comment>


Can 1/3 and 1/3 = 2/6? It seemed so - <AUTHOR> http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/ <comment><USER3> The units are missing, and I think that's a key factor here.

Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions.

The student came up and wrote 1/3 + 1/3 = 2/6.

What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those tables).

The teacher then demonstrates an entirely different formula: 1/3 (of the students at a table + 1/3 (of the students at a table) = 2/3 (of the students at a table).

The confusion comes because no one calls out that they're talking about fractions of different things.

Edit: There are a whole range of exploratory questions you can follow on from here as well.

Imagine if the tables have different numbers of students or if there are more than two tables. Helping students navigate these types of ratio transformations is why keeping track of units is so important. Otherwise, things can get hairy for the students very quickly.

<comment><USER4> You're right, but the tricky part is, how do you explain that to children who are _just_ being introduced to the concept of adding fractions, without leading them off track? That's _hard!_

> The confusion comes because no one calls out that they're talking about > fractions of different things.

But she did: "When thinking about fractions, it's important to keep your attention on what the whole is. [...] you're thinking about the two tables together."

Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:

 _" 1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at_ both _tables "_

 _" 1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the_ first _table "_

Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition _much_ more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.

\---

The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the _same_ table...", and move on. Ignore the confusing example and refocus on the simple one.

<comment><USER5> The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.

<comment><USER6> Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?

<comment><USER7> This hits the nail on the head:

1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.

<comment><USER8> And then it's easy to make them see how her answer was right in its own way by adding another unit, (person at table a+b and people at table a+b) and show why it may be harder to works like that for now.

I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!

</comment></comment></comment></comment></comment></comment><comment><USER9> This is a lovely (edit: having been in similar shoes, also terrifying-in-the- moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students.

Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students.

It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates _are_.

Some other lovely examples:

3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.

x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.

\sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.

There are many more.

<comment><USER10> > x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.

That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

<comment><USER11> It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse.

Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.

<comment><USER12> > It's not because 'function powers' make sense, yhey are just iterated > function application.

To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!

<comment><USER11> Indeed!

</comment></comment></comment></comment></comment><comment><USER13> The difficulty isn't with fractions. It's about understanding what "+" means.

Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie").

His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way?

Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway.

Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).

<comment><USER14> That's my favourite explanation so far. That's how I would put it to the student:

Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples:

* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.

* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.

The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3"

The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6"

What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.

<comment><USER9> We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us.

Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)

</comment></comment></comment><comment><USER15> As others have already pointed out, the question is ambiguous without more information. _Always_ include the unit; bare magnitudes could mean anything! The student was describing the _mediant_ [1], which was the correct solution when the problem is interpreted with forbiddenvoid'a units[2].

"Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence)

[1] [https://en.wikipedia.org/wiki/Mediant_%28mathematics%29](https://en.wikipedia.org/wiki/Mediant_%28mathematics%29)

[2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)"

[3] [https://en.wikipedia.org/wiki/Farey_sequence](https://en.wikipedia.org/wiki/Farey_sequence)

[4] [https://en.wikipedia.org/wiki/Ford_circle](https://en.wikipedia.org/wiki/Ford_circle)

[5] [https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree](https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree)

[6] [https://www.youtube.com/watch?v=0hlvhQZIOQw](https://www.youtube.com/watch?v=0hlvhQZIOQw)

</comment><comment><USER16> I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging.

It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: [https://en.m.wikipedia.org/wiki/Missing_dollar_riddle](https://en.m.wikipedia.org/wiki/Missing_dollar_riddle). That's an absolute nightmare.

<comment><USER17> Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids' intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2.

You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you've invented averages!

No way someone would have come up with that approach on the fly, though.

<comment><USER16> Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: [https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...](https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode)

<comment><USER9> And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples:

1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules.

1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular.

15% + 20% - who knows? 15% of what? 20% of what?

(1,2) + (2,4) - can't be done.

a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.

It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.

<comment><USER18> > (1,2) + (2,4) - can't be done.

are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors

> a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.

what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)

<comment><USER9> Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition.

The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.

I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".

<comment><USER16> How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.

<comment><USER9> I also didn't encounter it until I was relatively older. Our school has been pushing to introduce more rigorous language and definition in our 7-12th math program. For some students, it really seems to help. For others, it's really hard.

</comment></comment></comment></comment></comment></comment></comment></comment><comment><USER19> A great opportunity to introduce multiplications of fractions!

1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too.

When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents.

The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important!

To do this, we need to multiply the fractions together like so:

(Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls).

So in this case we would have:

(1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3).

This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing:

(1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).

<comment><USER20> I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still _feels_ like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation.

Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it).

Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.

</comment></comment><comment><USER21> I think this is a perfect example of why people find math hard. Not because there's something inherently difficult, but because we have teachers like this one in charge of open and curious minds at a young age. You get a teacher who doesn't quite understand what it is she's trying to teach, and this confusion is multiplied over and over again for each batch of students that pass through. Not too long afterwards you get kids who think they don't understand math, when in fact it was a failing of the teachers who couldn't correctly explain it in the first place.

</comment><comment><USER4> Something I missed the first time around—in the comments below the article, the teacher expands on what she actually did. This was originally one long paragraph, but I've added paragraph breaks for readability:

\---

> At the time, I let it go so I could think and regroup myself. There was more > than fractions to think about: Is it possible for both equations to be true: > 1/3 + 1/3 = 2/6 and 1/6 + 1/6 = 2/6\. This was a long discussion that kept > their interest, in which I learned that many (most?) of the students didn't > think of fractions as numbers. That was another hurdle. I kept going back to > what we know about adding whole numbers.

> Then I took another approach, once I was sure that they understood that 1/3 > and 2/6 are equivalent, so how can I add a number to itself and wind up with > a sum that's the same as one of them? In all of these discussions, students > would change their thinking. We looked at adding on a number line. I used > pattern blocks to explore the same problem. I kept talking about keeping our > attention on what was 1 whole.

> My, this all takes time, and the time is important for students to develop, > cement, and extend their understanding. What I didn't do, that I've been > thinking about now, is to make it part of a writing workshop on persuasive > writing and have them choose a conjecture and write. I think I could spend > most of the year on this with students.

</comment><comment><USER22> I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express.

The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."

It's hard when the language required to express the ideas they're having is just a little to advanced. And nothing about this stops when you get older. The "just a little outside your knowledge" keeps stretching on forever.

<comment><USER23> >The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."

I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.

<comment><USER24> > Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not > because of any intuitive reason, but because it's disallowed by the 'rules' > of fraction addition - that's it.

That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6\. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.

The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.

Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3\. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.

<comment><USER23> My only point is that analogies are flawed. The abstraction that they provide hides complexity that at some point will leak out.

>There is an intuitive reason why 1/3 + 1/3 != 2/6.

You and I have different ideas of what 'intuitive' means. It's 'intuitive' once you understand the rules of fractions and what they mean. It's not so easy to derive this rule if you're working in the space of real world things.

And sure, I agree you can ad hoc extend the analogy of tables to bring it in line with the underlying mathematical rules, but then your analogy is no longer as simple as it was. The complexity is leaking out of the abstraction you had it under.

>The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents

Sure, using abstractions and analogies is a powerful way of teaching. All I did was point out that analogies have limits and at some point they can become detrimental to understanding the fundamental concepts.

This is a common complaint by Physicists when doing public lectures on Quantum Mechanics and then having people extrapolate from the metaphors to derive incorrect physical rules (e.g. faster-than-light communication from a shallow understanding of quantum entanglement).

>because it deals with an improper translation of the English into mathematical notation.

It isn't just about the improper translation to English. It is also about the improper mapping of fractions to real-world things. 2 girls out of 6 kids in a table maps nicely to the fraction 2/6\. But even though 2/6 is equivalent to 80/240, the latter is a little harder to map to a table of 6 kids and 2 girls - don't you think?

>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.

I disagree with that in context of learning how fraction operators work. The fourth-grader logically extended the analogy that they were given because conceivably, there could have been an operator defined that matched their intuition, for example, let's call it '@' and define it (not rigorously) as "a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for combining tables of boys and girls and getting the fraction of girls to match the fourth grader's intuition. The fourth-grader is learning fractions for the first time, and that operation could have conceivably existed - so the only reason they were wrong is that they haven't been told what the rules of fraction addition are and NOT that they misunderstood the analogy. The problem is that the "+" operator does not work that way because it isn't defined this way as per axioms for fractions.

<comment><USER24> I still don't agree. Sure, there are limits to particular intuitions, but all of the rules make perfect sense with real world quantities.

For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.

The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real- world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.

The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.

</comment></comment></comment></comment></comment><comment><USER25> In the comments, an educator references an excellent article related to this confusion, "When Can you Meaningfully Add Rates, Ratios, and Fractions" that implicitly suggests some pedagogical approaches: [https://flm- journal.org/Articles/11019C10CF34E90DC5866E53E90...](https://flm- journal.org/Articles/11019C10CF34E90DC5866E53E905E8.pdf)

</comment><comment><USER26> When you want to add fractions of a whole together.

         1     1     2
        --- + --- = ---
         3     3     3

When you want to add wholes together and make a new fraction.

         1  +  1     2
        </comment><comment>--- = ---
         3  +  3     6

</comment><comment><USER27> From a 'not currently in front of the class' perspective, it's pretty clear what the student meant by 1/3 + 1/3 = 2/6\. They were taking + to mean something like a general 'and' or combining action, not in the strict sense of standard fraction addition. It even has a name in mathematics, 'Farey addition'.

The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.

</comment><comment><USER28> I happen to have a math teacher masters though I myself do not teach (but I do help with the program of a tiny, tiny reform school). This teacher here had painted themselves into a corner and it's hard to get out of it. Do not explain fractions with things you can't change the denominator of.

Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15\. This is all play and very smooth.

To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.

</comment><comment><USER29> One of the commenters gave a great suggestion:

> I would use bar models to show how the whole changes. First, draw one bar > (table) with 3 students inside; label the bar one whole, star one student > and label that student as one third of that whole. Do the same thing beside > the first bar model, again showing the bar as one whole, starring one > student and labelling that student as one third. Then, push the two bar > models together (I'd use a Smart Board) to show a new whole: the two bars > together with 6 students in the one bar is labeled as the new whole. Point > out that this is a NEW whole, with a different # of pieces. Now they would > see the 2 starred students in the one new bar made up of 6 total students, > so the "old" 1/3 student becomes the "new" 1/6 student once the whole > changes. This is like a name change, once the size of the whole changes. You > can also show them that by cutting each "student" in half, you would get an > equivalent fraction of 1/3 = 2/6 of the first one whole bar. So 1/3 = 2/6, > not two 1/3's = 2/6.

[http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it- seeme...](http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed- so/#comment-27347)

</comment><comment><USER30> Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality.

From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.

<comment><USER9> Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?

<comment><USER30> Introducing line segments as alternative representations of numbers at this point feels very natural, and is already implied by most circulum with the standard 'number line'. As you say, we don't do anything with that until much later.

Let's leverage that early on!

</comment></comment></comment><comment><USER31> "If we reduce 2/6, we get 1/3, so 1/3 + 1/3 = 1/3, that seems weird, what did we do wrong?"

</comment><comment><USER32> This is why I taught my kids to label units.

( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different.

To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ).

To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.

</comment><comment><USER33> I think the problem is the teacher moved from one type of "fraction problem" to another kind.

The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils.

The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem.

When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units _also_ need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units.

I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).

</comment><comment><USER34> I would say it was a mistake to introduce the concept of adding fractions from two different "wholes". Instead, teach that in order to add fractions, you first have to get everything into the same "whole".

Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing.

Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept

</comment><comment><USER35> Use pie graphs to show the original tables split up into thirds of one girl two boys.

Then show two ways of combining two tables.

(>-) combined with (>-) is a table of six: 2 girls, 4 boys show all boys and girls moving from tables with 3 seats to the same table with 6 seats.

Now show a table with three seats, one occupied by a girl. Show another table with same configuration. One girl moves from one table to join the first table. 1/3 full + 1/3 full (at the same table) is 2/3 full.

</comment><comment><USER36> I solved this by then introducing the kids to slices in a pie. Visually, the kids can see that 1/3 of a pie plus 1/3 of a pie is not the same as 2/6\. Fractions are often introduced as "parts of a set" which is great for the first day. It's easy for kids to see and understand, but kids are more familiar with trying to slice the birthday cake for all the kids at the party. After the kids realize there are _two_ kinds of fractions, we spend time at each problem thinking through is this a set (which we'll return to in ratios) or a whole (which can't be expanded) and visually representing fractions both ways to check their answers. Adding units is absolutely important, but when I teach it, that's what I add in fourth or fifth grade when I'm teaching them to reduce.

</comment><comment> 08-15 That's what happens when you try to "build an intuition" for a concept instead of defining it.

Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? _That don 't make no sense!!_

The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that _don 't_ correspond to integers. The question isn't so much "What _is_ half a bottle of water?", it's " _How_ do you calculate with one half (a bottle or whatever)."

Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)

<comment><USER9> As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult.

I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.

About a third work their way through it fine.

About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.

<comment> 08-15 > About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2

...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.

A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (a _c) /(b_c) = a/b, is it? Of course, she never asked _why_ a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.

I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.

</comment></comment></comment><comment><USER37> This is fantastic material. I'm going to present this quandary to my kids. Some of my kids struggle at math, so it will help them understand that math is a language and, like any language, it has abbreviations that sometimes lead you down a confusing path until you spell out more. Some of my kids are good at math and it will help them relate with others.

</comment><comment><USER38> I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction?

While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require _measurement_ with _units_ to define a new inclusive set.

My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms.

Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.

<comment> GuB-42 Bad idea!

They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills.

EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.

</comment></comment><comment><USER39> My take: Adding proportions is kinda "semantically incorrect". It's well defined as a value, it just doesn't mean what you think it means, akin to thinking that union of two sets is written A + B (which would be the set of pair-wise additions of elements). Now, dovetailing the equation with units that signify concrete quantities (as suggested by others) circumvents this by making it meaningful, at the cost of no longer working with the abstract concept of proportion.

An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)

</comment><comment><USER40> > What would you do now???

I would probably start crying, to be honest.

But in all seriousness, this is the kind of thing that makes me dread teaching. I mean, I know what is wrong, I just can't wrap my head around how to explain it, specially without going into stuff that would be too advanced for these particular students.

</comment><comment><USER41> It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction:

3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7

Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).

On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.

<comment><USER42> 2/4 + 1/3 would _clearly_ be 5/12.

You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3.

Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise):

1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12

</comment></comment><comment><USER43> 1/3 + 1/3 does equal 2/6.

1/3 of one table + 1/3 of one table = 2/6 of two tables.

</comment><comment><USER44> I would express this in a diagram with a dotted line around the set under consideration. When dealing with thirds, make a diagram containing three objects with their respective properties, and end by drawing a dotted line around the set. When switching to sixths, add three more objects, _erase_ the line around the first three, and draw a new dotted line around the whole set of six. Then go on to sevenths by adding one more object, erasing the dotted line for the six and drawing it around all seven.

If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.

</comment><comment><USER45> As the concept of fairness is innate in primates including humans in the 4th grade I wouldn't stop and correct the students but try and re-frame the problem in terms of a problem where fairness was involved and then immediately the ratios and the units would come into more clear focus. As an example I might say that if I promised to give you 1/3 of the pizza as a reward for cleaning your room. And then how much should you get if I had 1 pizza. How about 2 pizzas? Things like that.

</comment><comment><USER46> Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer.

In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4\. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8

<comment><USER46> And now I feel stupid for obvious reasons

</comment></comment><comment><USER47> It's 1/3rd of something. what that something is matters. Fractions express proportions of a whole, so you can't add and subtract them without taking into account what whole they refer to.

</comment><comment><USER48> Her first example, "proportion of a six-pack of water" is actually very complex, with a lot of potentially distracting details. Nowhere does she mention that's what her fractions are measuring, either! The whole lecture was simply a mass of detail with no reference to the underlying concept it was putatively intended to explain. No wonder the children were confused. Why not start with cutting a pie? Guess that doesn't lead to nearly as much busy-work.

</comment><comment><USER49> I would have said that a whole table is 3/3 and that both tables together are 6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good idea to explain that the "whole" isn't all pens and bottles but the group of 3 or 6 or 12 or whatever.

Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six- packs can you make from 15 bottles?

</comment><comment><USER50> Not that I would have thought of this on the spot, but ...

The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.

</comment><comment><USER51> Have the kids stand up and count out numbers.

Now they know how many kids are in the room.

Then have them go and stand in two different corners. Count one corner.

Then have them stand in three different corners. Count one corner.

Finally the four corners. Count one corner.

Then have the two tables distribute their students to the corners and figure out these elusive fractions.

Have each corner be a Pokemon or something. That should do the trick.

</comment><comment><USER52> Don't use tables of the same size:

1/3 + 1/4 = 2/X

There would be no confusion as to what X should be.

And as someone else said, use units:

1/3 of table A + 1/4 of table B = 2/7 of both tables.

</comment><comment><USER53> [http://iconicmath.com](http://iconicmath.com)

We probably haven't come up with the best visual abstractions yet, eh?

</comment><comment><USER54> Wow, I had no idea this blog existed - I loved Marilyn Burns's children's math books as a kid, awesome to see her still in action.

</comment><comment><USER55> That's the same as saying: if you put 1/3rd pizza slice on top of an other 1/3rd pizza slice, you get 2/6 pizza. Layered math.

</comment><comment><USER56> When you put two tables together, each table has 1/2 of the total number of kids. By having 2 tables, now you have 6 kids, so:

the original 1/3 in the first table becomes 1/6

the original 1/3 in the second table becomes 1/6

so adding the two tables means 2/6 are girls

She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)

</comment><comment><USER57> This reminds me of a homework problem I did in numerical methods where I made a small mistake in the program but still got an answer that was super close the the actual answer out of pure luck. The professor spent a while looking at it knowing it had to be wrong before he ran out the decimals and showed that it didn't converge.

</comment><comment><USER58> I would try to explain semantics of addition (without the word "semantics" of course). I would say, in the first group 1/3 is the girl so if we take out one boy and replace it with another 1/3 (another girl), the resulting proportion is 2/3\. Once we mix both groups, that is not what "plus" is.

</comment><comment><USER59> I would explain that the "+" sign is not in the right place in the equation and write

(1+1)/(3+3) = 2/6

instead.

</comment><comment><USER60> Many students at beginner university level math courses do the same kind of mistakes, albeit with larger numbers: 10/13 + 7/21 = 17/34\. Decimals are much easier to deal with imo, but mathematicians does not like them because they are inexact.

</comment><comment><USER61> The moment you add three more students, the first girl is no longer one girl out of a group of three students, she is one girl (out of the two girls) out of a group of six students.

Same with the second girl.

So, once you add 3 more students, 1/3 becomes 1/6.

1/6 + /6 = 1/3.

</comment><comment><USER62> "+" is the wrong mathematical operation for combining fractions of distinct sets.

The right operation is a weighted sum:

1/3×1/2 + 1/3×1/2 = 1/3

The "1/2" is there because each of the two sets that we are combining is the same size.

</comment><comment><USER63> Really great anecdote! That first plus sign that Addison wrote is actually more of an "And" instead of an arithmetic "plus"

</comment><comment><USER64> My favourite: I got 1/3 on the first test and 1/3 on the second test. So in total I got 2/6\. So 1/3 + 1/3 = 2/6.

</comment><comment><USER65> I think that one key issue is that the students are confusing addition with the more general concept of considering two things together.

</comment><comment><USER23> This is a great example of the downside of over-relying on metaphors and analogies to teach mathematical concepts. The reality is that fractions are not like groups of pencils or tables of girls and boys. They are rigorously defined mathematical constructs that occasionally can be mapped to real-world things (and usually with severe constraints). 1/3 + 1/3 isn't 2/6 because it doesn't follow from the underlying axioms that define rational numbers - <USER2> not because of anything else.

<comment><USER66> Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere.

The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working.

Much better (imho) than teaching math as a purely formal and transcendent topic that happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught).

Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.

<comment><USER23> >We didn't invent fraction out of nowhere.

But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy.

>The moment where you have to stop relying on intuition is a pretty delicate matter

I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.

<comment><USER67> > You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of > boys and girls (because it does) but because it's against the rules for > adding fractions.

It is pedagogically superior to choose the route implied by the comments about this being a type error.

That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.)

For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following:

1\. "1/3+1/3=2/6" doesn't have any units, but it must _somehow_ map to operations with units.

2\. _If_ unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table.

3\. 2/3 does not equal 2/6, so unitless math can't map to both operations.

4\. macspoofing said that 2/6 is wrong.

5\. Therefore, unitless fraction addition _implies_ addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy.

If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless <-> unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.)

I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors.

Edit: clarification

<comment><USER23> >That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.),

Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out.

Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.

</comment></comment></comment></comment></comment><comment><USER68> 1+1=50.8 My forth grader understood that this equation makes the same mistake. Hint: units of inches and mm must be made explicit.

</comment><comment><USER69> If you make it extreme, then its easier to see the folly / error -

500/1500 students + 1/3 students = ?

</comment><comment><USER70> related: notation as a tool of thought

[https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=tr...](https://dl.acm.org/doi/pdf/10.1145/358896.358899?download=true)

1/3 relates to one of the tables

2/6 relates to both tables

</comment><comment><USER71> You're taking a weighted average, not adding the fractions.

It takes a while to learn what that is used for.

</comment><comment><USER72> No cat has nine tails. Every cat has one more tail than no cat. Ergo, every cat has 10 tails.

<comment><USER10> Half a loaf is greater than nothing. Nothing is greater than God. Therefore, half a loaf is greater than God.

</comment></comment><comment><USER73> The and operation could be logical so the result of anding with one self is a mop

</comment><comment><USER74> > It's hard to think and teach at the same time!

So true...

You do not see anything when you are at the blackboard.

</comment><comment><USER75> This kind of thing fascinates me. I've always felt that you don't really understand a thing, even a simple thing like fractions, until you can find and resolve all the paradoxes that arise in using it naively.

My take when reading the 1/3 + 1/3 = 2/6 equation is that Addison was right but the "+" here is a different operation than the "+" of 1/3 + 1/3 = 2/3\. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2/3, and in the other case we are adding two wholes and counting how much of the new whole is included.

People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory.

The answer "what would you do now?" is the same one I would try to ask the students. First I would stop, confused, and write 1/3 + 1/3 = ? elsewhere on the board, and then wait for the students to see why I'm confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2/6 and 2/3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you're adding parts of the same whole (set or group or table of kids) and in another case adding the wholes.

From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2/4, 1/2, and 0.5 are all "the same thing". Bullshit! It's no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that's obviously false unless we've introduced the narrow context in which it's true? When used as unitless "coefficients" or bare numbers that exist only to scale another value that represents a real quantity _of_ something, _then_ 1/2 and 2/4 are the same. _This_ is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself.

I'd spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for.

Of course we want to be able to have "+" be well-defined, especially in elementary school, and the students understand that! We've now found a perfectly intuitive meaning of addition for which our standard "+" definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, "introduce a suitable notation" for the setwise addition of "marked" sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently.

I saw some mentions of units in the comments, and yet 1/3 + 1/3 = ? is still ill-defined if we know only that "1/3" means "one out of three students". What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?)

Of course what most of us will actually do as the teacher in this situation is freeze and think "oh no, I've made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!". Because that's what we've learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.

</comment><comment><USER76> Apologies but I don't see the mystery here?

A fraction describes a proportion ie. the amount of something(s) in relation to another amount of something(s) thus, if we double all of the things in a fraction the proportion and hence the fraction is equivalent.

</comment><comment><USER77> 1/3 and 1/3 is 1/6 ;)

</comment><comment><USER78> This is pretty simple. Put two tables together, 0/6 girls. Now put only one girl: 1/6 put the other girl: 2/6

</comment><comment><USER79> One third of one half, plus one third of the other half, is one third of the whole?

<comment><USER80> Yeah the writer didn't have the words "of one half" or "of the other half", therefore it seemed a lot more confusing than it should have.

</comment></comment><comment><USER81> This is exactly why third-level teaching is so much easier.

</comment><comment><USER82> I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.

<comment><USER16> A lot of people are at home right now trying to teach their kids who are also at home.

<comment><USER83> I'm surprised I haven't been seeing more discussions about education recently on HN and everywhere else.

<comment><USER16> I suspect we skew young enough on here that most of us don't (yet?) have kids.

<comment><USER84> I very strongly suspect that's not true.

<comment><USER16> It would be interesting to see a demographic survey of HN commenters/contributors. Wonder if one has ever been done?

</comment></comment></comment></comment></comment></comment><comment><USER85> Javascript:

console.log((0.1 + 0.2) === 0.3) // => false

C#:

(0.1m + 0.2m) == 0.3m // => true

see here: [https://dotnetfiddle.net/zVnrNQ](https://dotnetfiddle.net/zVnrNQ)

Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.</comment>