# Subitizing

If you didn’t already know this, it is interesting. Humans have an inherent knowledge of numbers,
typically up to about four. Try this with a bunch of pennies. Have a friend drop a few pennies in front of you. If there are 4 or fewer, you recognize the count instantly. It doesn’t matter how they are arranged (assuming all are visible), you just know. At six or seven there’s a pause. You find that you first have to pick out a 3 and a 4 and add them. You’re now calculating or counting. That’s cheating. It was not natural knowledge.

Also interesting is that the Japanese Soroban, abacus has 4 lower beads for units and a single upper bead for a five. If you have 4 set on the bar, to add one you zero the 4 and bring down the 5 bead. It’s a mini carry. The point being you see the 4 bead count instantly without cognitive load.
Animals too have this sense, although more limited. Most can tell the difference between one and two. There are human cultures where there are no names for numbers above the Subitizing range.

And perhaps this is why we are most comfortable manipulating numbers below five. This is the idea behind compliment arithmetic which converts adding a 7 to subtracting 3 and adding a ten. And that feels easier to most because 3 is within your Subitizing Power

If mental calculators had designed our number system, they might have been wise to choose a base 5 system. More verbose, true but easier to calculate with. If you think about it, this is what the Soroban is doing.

https://en.wikipedia.org/wiki/Subitizing

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I like this topic. Savants are an interesting case in this because it is believed that some savants have a much higher subitizing limit, ranging from 15+ till a 100.

Rudiger Gamm was actually tested and he could count around 100 dots in 1 second instantaneously. However, he is not a savant and it is possible that he used a technique for this but I don’t see how one could train himself to count 100+ items in 1 second.

I’m not sure that having a base no greater than the subitizing range + 1 is that desirable compared to having a good array of factors, particularly well-chosen (low) prime factors. That all four of the numbers in the subitizing range divise 10 (z) evenly is one of the benefits of dozenal.

Hexal could come a lot closer to prioritizing the subitizing range, but I think it comes at the cost of failing to take advantage of our ability to work with larger multiplication tables, instantly recognize far more than 6, ↊ (z), 10 (z), or even 14 (z) symbols as well as to add and subtract them again, instantly, with practice.

In practice I’m not even convinced that the soroban has any particular advantage for anzan. My understanding is that fast practitioners of anzan are doing the carry (if any) and complement (or number to be added) on the current rod instantaneously. So effectively, they may as well not be visualizing beads, since the advantage of beads (they can move) is only necessary on a physical soroban, but (say) images from a [0, 9] shape list, or even just the arabic numerals, given that the anzan masters themselves have proved that such can be done with quite abstract imagery (columns of visually identical beads).

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When it comes to the subject of lightning calculation or working huge numbers, I don’t know what’s the best. I don’t do that sort of thing. I am interested in doing practical arithmetic reliably and fluently. I have played with a soroban but don’t have much skill with it.

In my work and my interests I do a lot of calculating, some with numbers some with symbols. I don’t need lightning speed, I need fluency. I do not want to interrupt my train of thought. For practical work one rarely needs more than three decimal places. I do a lot of “back of the envelope” in my head. The whole industrial revolution was done with slide rules. If a calculation does need refinement, then I work with a mental estimate and refine later on the computer.

This is ordinary practical math. Skills that ordinary people should and could have regardless of their natural talent. For that purpose, the Soroban works very well. An ordinary person of average intellect can reliably do arithmetic all day long.

I want to think and read fluently. I don’t want to have to look words up in the dictionary or pause for numbers. I do 2x2 multiplication in my head smoothly but not fast. Nobody here would be impressed but other people are amazed. They wont even attempt it. They think I’m freaky. Well I am but this is basic competence like literacy. We should all finish high school with this skill.

I love it that there are lightning calculators among us but the urgent need is for a numerate population. People today won’t even start to think about numbers. A pop test I like to surprise people with.,. “quick! no calculator! What’s 4.8735/0.98158 ??. Quick now, that’s an easy one…well?..”

It’s revealing. They recognize the outlines of a long division problem. Don’t even remember how that goes. Their eyes glaze over. If you pose this question to an engineer, a scientist or any “numerate” person, the answer comes and quick as a flash “About 5!”. But most wont even focus long enough to spot that simple relationship.

I would agree fully with this but even if the soroban had 6 beads it doesn’t really reduce its usability, you can sort of get better at recognizing task specific quantities, in the sorobans case because the beads are similar it is much easier to see the entire thing as a shape rather than 3-4 things, and instead see each column as another thing, this tends to be how faster speeds are achieved by manipulating 3 columns at once, though this is only something I have learned after very little research there may be more to it.

I still question whether this is all since that would mean that using a soroban doesn’t carry advantage over visually calculating with pure numbers.

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With training all things are really possible but there are more techniques than we know. It may just be possible to subitize in shape nodes, I have done this randomly out of amusement and it has had oddly strange success. A kind of simple example of this is if you see a square of 12 dots , in a 3x4 layout. You can pretty much instantly without counting know, that there are 12 such dots inside. If you also form an octagon with the odd dots , which you can do pretty much instantly depending on their position you would know that there are 8 dots to form this shape. At-least seems to work well for me, though I don’t have much of a need to bolster subitizing like this.

My friend once trained himself to subitize in groups of 9’s rather than just 4’s without any techniques.

100 items in 1 second may also be 25 items in 0.25 seconds or 10 items in 0.10 seconds, which seems more doable if the dots are spread out.

I went ahead and looked a bit more for what Rudiger Gamm was saying. He mentioned a lot of ‘training’ methods in his book which emphasize activating both sides of the brain along with a bunch of techniques and tricks I am actually slightly doubtful on his training methods, but they seem to work for him. Just to make sure I might get his book in German and read through it to see for myself.

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I think one has to consider the huge investment we already have in Western mental techniques. It may not be cost effective. I tried the abacus for a while and it’s fun but the Japanese work long and hard to become proficient. If your basic math skills are poor then it might make sense.

I think it has a big advantage in that one is physically engaged with the beads. The Japanese take it very seriously. They have worked out which finger movements are most efficient to use when. It’s very refined.

Numbers to me are very natural and real in my head. I see them as having a context of relationships to other numbers. I suspect that for abacus practicioner, the abacus is their mental language. That’s how they see numbers and their relationships.

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I have tried the abacus and have at-least gotten slightly more proficient than without it. Later on however, after I had tried just visually calculating along with numbers it out-beat my ability to use the abacus, just makes me wonder if there is some ‘unknown’ advantage to imagery aids for calculating as this would imply that visual imagery devices in general can enhance cognitive capacities in different aspects over not using imagery.

I have not got an answer though as to why the abacus would beat ‘visually’ manipulating numbers if both were done to mastery.

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A good way to know if Rudiger used techniques for subitizing is to test him with random items. The thing with subitizing is that it doesn’t matter what kind of objects you are looking at, you can still “know” the amount instantly. If he can’t count other random items instantly then you know he used techniques to count random dots and only practiced counting random dots. Unfortunately, we will never know the truth unless Rudiger reveals it.

I don’t know my subitizing limit. I’ve mentioned in my other posts that I might be a savant because of my odd memory feats and calculating ability but unlike some other savants, I do not “know” instantly the amount above 8 items. However, because of my visual memory, I can count more items afterwards in my afterimage. If you flash a picture of random items in front of me for a second, the picture remains for some time clear in my mind and I can actually count them in my mind, up to around 12 items.

This is true but he may also have a system to deal with a lot of patterns, nothing is truly random after all, it would still be a good way to test since his time wouldn’t be constant across different random orders of dots, which would be a strong indicator.

From what I have seen about him though its almost certain that he uses some techniques and trains.
Even his calculating ability is quoted to be acquired after his early 20’s, which is very much different to not using techniques.

That is interesting, for me it is a pretty hard rule that whatever I look at lasts at high detail for a few seconds, decreasing in time the more objects there are unless I actually remember it or connect/encode it, or even recall it but I can oddly recall memories I have forgotten if I focus long enough on them.

I for example remember that someone mistakenly called me with a phone number but not the number they called me with, if I then spend 15-45 minutes thinking about it I can recall the digits in the phone number, not exactly a useful ability though. The only time it has use is if some random cue helps me recall the details instantly which happens rarely.

In your case though does this mean you can blink at a paragraph and then read the words up to a certain number? Most people would struggle past 3-4 words with this. Generally if I learn a lot of things in a short period my digit span ,visual span and verbal span pretty much doubles until I stop learning, perhaps a practice effect. I can’t exactly blink at a paragraph and read it though.

No, it doesn’t work with reading because in order to understand what you read, you actually have to focus a bit with your eyes which takes time. Studies have shown that the faster you try to read, the worse your comprehension becomes.

However, if you were to flash a paragraph in front of me, I might be able to count the amount of words up to a certain point. I also might be able to localize them.

There is this website where they show you random symbols and you need to look at them for a few seconds and later try to localize one of the symbols. I did this with 100 items. I just glance over them a couple of times and just hope they stick in my mind and a lot of them do.

I thought it was a coincidence but it was not. I did this a couple of times and I either get it right the first attempt or the second or third. Sometimes the location of the item doesn’t stick and I am just as clueless as everyone else. I can do it by just looking at the symbols for around 30 seconds, sometimes less or more, depends on the day of course. I don’t use any techniques.

Really? Mine tends to get better with increasing speed but only if I don’t stop fully sub-vocalising the text. If I read quickly at a rate near 500 wpm my comprehension is better than if I keep it at 250-300 wpm. I think its partly because when I speed up i tend to form links more than when I don’t.

You wouldn’t be able to read up to that point if you can localise and count the amount of words up to that point?

I checked the site and tried playing the game :

Can’t really say much to this inaccurate attempt.

That said, I take significantly longer than you, I decided to time myself with a stop watch, I can learn about 15 of those in about 30 seconds without techniques. It’s quite impressive, that you can learn 100 of those in the time it takes me to learn the odd 15, that is pretty much over 3 a second.

I have to do this for the symbols.

Strangely I still remember the odd few symbols at the numbers. Like that your 83 on your last image was my 82 on mine and that 83 was 3 lines on mine.

I don’t actually learn the symbols. I just glance a few times over them. I don’t know which symbol I can recall and which I can’t, neither do I know their order.

It is more like when you have read a lot of books. If someone asks what books you have read, you probably wouldn’t know them at the top of your head but if they showed some books first you could easily tell ones you’ve read and those you didn’t. This kind of memorization is very good in my mind. I can solve wordsearch puzzels without a pencil because of this and sudoku’s.

You wouldn’t be able to read up to that point if you can localise and count the amount of words up to that point?

No, I cannot read them. My mind remembers the shapes of the words. I guess it would be more accurate if I said sentences instead of a paragraph because I cannot localize words in a paragraph of 50 sentences after a 1 second glance. I would at least need 30 seconds of course.

Also, try to read even faster than what you are used to. At one point your comprehension becomes worse and worse. A rate of 500 words per minute is on the higher end for the average reader but not necessarily high enough for comprehension to be lost, this depends on other factors such as memory. My reading speed is pretty average, only about 270 wpm but my comprehension is much better and my memory makes up for it.

Kim peek comes to mind as he could read two pages at once, one with his left eye and the other with his right. What we would read in 1 to 2 minutes, he would read in seconds and he still had a 98% recall.

When I do this subvocalization breaks, which I guess is a fair reason for decreased comprehension. I guess a lot of the research on this was based on speed readers who eliminate subvocalization.

Googling around a bit I got " It took Kim Peek just over an hour to read Tom Clancy’s The Hunt for Red October", which at 270 wpm takes ’ 10 hours and 3minutes’ according to some online estimator. This means Peek was reading two pages at a time at a rate of 2700wpm in total, which is certainly impressive, even more so with his 98% retention.

That is not just impressive, that is arguably super human. If you or I would try to read at 2700 wpm our comprehension would be basically 0. 2700 wpm is 45 words per second. In other words, everything I just wrote in 1 second.

It’s insane.

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I am a very fast reader. I read a great deal. I am limited by my comprehension speed not by my “decoding ability” .

Reading, is not an end in itself. Unless you are performing stunts. What’s the point of blitzing through technical material too fast to even understand it? I’ve spent days trying to get through a single page in a math book. And that’s not the way I want to read fiction either. Do you watch movies and listen to music on fast forward? I can blitz a newspaper article. But those are light on content and the sentences are very predictable.

Unless you are performing stunts, perhaps speed itself is not very valuable. What one really wants in this and other skills like "figurin’ " is fluency. Reading should be transparent and effortless. It should exact a minimal cognitive load. It should not be tiring and one should not have feel the anticipation of “work” when turning to a book. It should be as natural and transparent as your speech. With this, speed comes naturally and it’s often a good index of a person’s fluency.

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As much as this is a valid point, I still can’t forget the sight of going into a university library and seeing that those math books you may have spent a day on for a single page are much more than you would get through if you could just 500wpm to full understanding and comprehension. They were even on the lesser side of books by subject.

There is far too much information in comparison to the speed you can get through it, even only on the hard upper limit of reading speed. Of course speed alone isn’t as valuable as comprehension and understanding but it doesn’t change the fact that it is still lacking. The fact that there is a hard upper limit is usually something you try to avoid because it means no matter how well you can gather an understanding from what you are reading at, it will never be faster than the rate you are reading at.
(of course unless you can deduce any of the rest of the text from the start).

Say that people could read at a rate that they themselves found sufficient, they would never really put much consideration to increasing their reading speed, the next hard limit would be the rate at which they can understand something or even depth and so today’s ‘increase your reading speed’ would turn into ‘increase your comprehension or understanding’.

It’s not really bad to work from the top down, you can also work from the bottom up and increase your understanding/comprehension and lastly your speed. It’s entirely choice which isn’t negative or ineffective in one way or the other. I personally rather pick all of them, so I don’t have to be stuck at any one point and waste time. I think this is rather the case as oppose to people picking speed and thinking comprehension and understanding can go out of the window. As much as their methods may be implying that, they are only ‘still’ using their methods because they are in disbelief that it really won’t be able to at some point maintain comprehension at higher speed.

It’s all better than doing nothing, just imagine trying to understand an entirety of something you can’t even look at in your life time.

What may have been hard and difficult once may become easier or even effortless because you have grown past the difficulty you may have had then. You may also end up having a ‘better’ transparent and effortless rate and comprehension thereafter. It’s kind of the same with memory techniques, it may be effortless to go at a slower rate, yet with more practice, even the faster rate you found difficult once will become almost effortlessly natural. Usually this practice benefits from fighting the difficulty.

Nobody is arguing against slow reading and nobody said you should read fast. I said it myself that I am just an average reader. Everyone knows comprehension>>speed. I rather be able to read a chapter once very slow but know every single detail of it than read it fast and only know a little. I don’t really see a reason for your post.

We were just exploring the high end of fast recognition and subitizing with some little facts here and there.

P.s.
I sometimes do watch youtube videos on x1,5 speed if people talk to slow

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Pretty snippy! You should have said that differently. We might see each other again in this forum. Nor was there much need to be defensive.

And you are most welcome to do so. But I started this thread on Subitizing and my interest is in pedestrian performance. And it’s just fine if I make a comment to keep it in view while the thread drifts. I don’t need you to see a reason for my posting. You brought up Speed Reading. You were welcome to do so but you are not going to completely hijack my thread. I have every right to keep my agenda in view.

My interest here, and I think that of many others is the development of practical skills for everyday life. But as often happens here, the conversation quickly switches to high performance skills. Specialists, savants. And the talk is about how one might learn to do amazing tricks. Is the Soroban the best choice for lightning calculation? I don’t know. I don’t care. I am interested in its application for ordinary people. This is too common in our discussions. Rather than focus
on ordinary practical skills we spend our time amazing about the superstars.

It’s like going to an exercise forum and discussing how top athletes train. Sure it’s interesting but it’s irrelevant to a bunch of flabby middle aged men trying to get healthy.

So while people go off on savants who can read 2k+ wpm, excuse me if I point out that this is not relevant to the average person trying to improve. I don’t object to these comments. I’ll follow along but I have a right to keep my agenda in view.

And another point that is important enough to me that I raise it frequently, is the confusion between speed and fluency. You don’t care? That’s fine. Skip it. I’ll be bringing it up again when I get the chance.

This is not some whim on my part. I consider it very serious. I see around me a population that hates to read even as much as a page and will not even think about numbers. The result is a population that cannot get a grip on the facts and is easily bamboozled. I think we are in very serious trouble.

Of course, if you want a thread dedicated to high end subitizing you could start your own. You’ll have plenty of interest. It is interesting. But here, in my topic, Speed Reading at 2700 wpm is a tangent. But you are welcome anyway

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