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.
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).
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 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.
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.
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.
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.
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.
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.
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
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
There’s no fault. The interest is natural and appropriate. I appreciate your contributions.
I’m not here to be a topic Nazi. Let the thread go where people are interested. I will chime in now and then to try and draw attention to my main interest but if people aren’t interested, I get ignored and the thread goes down another path. That’s fine. That’s a natural flow of conversation in a group. Disagree with me, ignore me it’s all fine. Tell me to shut up in my own thread - I’ll be in your face.
I am often wrong and need correction. That’s why I’m here. Just be kind and civil when you do so. - Thank you
I hate that people are getting dumber too. IQ scores have dropped all around the world since at least 1980 and I am talking first world countries like Norway, France, Britain, Denmark and even the Netherlands.
Here in the Netherlands, arithmetic/mental math is not even seen as important anymore and they have removed the arithmetic test from the exams which is stupid as hell. Students with learning difficulties like dyscalculia are on the rise and they decide to remove its possible prevention from the exams. In my girlfriend’s class, almost nobody is decent at mental arithmetic. She said one girl even cries when she sees simple arithmetic. This girl might have dyscalculia but what makes me mad is that it possibly could’ve been prevented if she was taught arithmetic better and kept being stimulated to do so. The same is going for dyslexia.
Hello again.
In the past I taught mathematics, and I often thought about ways to improve mental calculations and mathematics in general, but since I stopped doing that I never have a way to test my theories.
I was very fast at arithmetics in the past, but I stopped working on that and now I am quite slow.
In particular I had two ideas:
Use much shorter words for numbers, only one-syllable words, that are difficult to confuse. English is not so bad, but still you have “seven” “hundred” “ninety” “eleven” “thousand”, etc. In other languages words are much larger.
Rely more on imagination and less on paper. Imagine the numbers. Do everything on the brain. Even add colours to the numbers. I use this theory in other areas, like drawing, or language learning, and it works for me.
I had friend who was a clinical psychologist. Phd in Psychology. She was always showing me these magazine articles health, diet. The usual stuff, wild unsupported claims. Magic berries, Kroil Oil. I couldn’t understand why someone with her level of education would pay attention to this stuff. So I asked, how much statistics did you have to know for your degree. None!!!
This means you cannot function as a scientist or a scientific medical practitioner. You cannot assess evidence. You don’ t know whether a paper’s results are strong or barely significant, the sample size, the design of the experiment - even though it’s your field you cannot make a judgement. You can’t tell solid work from ■■■■■■■■.
Well the world can get by without psychologists, but she votes! Fortunately her position on climate change is the “right” one. But she couldn’t defend it. If pressed, the most she could do would be to quote authorities. It’s luck and emotion. We are in a liberal community. That’s what all our friends believe. The worst is that she has no clue how badly adrift she is.
I love working with pencil and paper. I love the sensation of graphite laying down on the paper. I like to sketch too but I’m not much.
But we part company there. Verbal flow is my main carrier. It’s the rhythm and sound. The length of the phrase is no extra burden. In fact if it has a better flow, I prefer it. I have a persistent auditory loop. Sometimes I acquire things I wasn’t really listening to. I hear them in the various languages that they speak or the accents that they had. But it is not strong at keeping track of positions, I am dyslexic and that was always my hobgoblin. I don’t carry powers of ten around like baggage. I use a series of pockets or positions each one corresponding to a power of ten and accumulate intermediate values there. I might at one point have 25 in the tens pocket which is later normalized by carrying the 2 into the hundreds pocket. That’s why I need the fingertips.
Often the numbers just kind of get together, almost in a conjugal manner, and the answer just forms. I don’t know how. Truth is, there’s a lot going on in here that I don’t know about. I have no idea how I speak English. In writing, I might deliberate but in normal speech I put no effort into forming the sentences that emerge from my mouth. Even here at the keyboard with time to write, I don’t know what I am going to think next.
I love working with paper. But pencil is only for drawing, never ever for writing. I am very picky with the paper and the pens I buy.
I do not think I am.
I need to pay attention when I speak, especially when I am teaching, because I have to use languages that are not my mother tongue (such as English, among others), and I need to concentrate a lot. Also to avoid repetitions and fillers like uhm, ah, eh and y’know. Sometimes in the effort to concentrate I close my eyes, and some people find it strange. So I have to add an extra effort to keep my eyes open.
When I was in Engineering school (I also did that) I used to do calculation and arithmetic “by approximation”. I was not so much concerned with the precise result, but to get as close as possible in a very short time. I was very good at that, even calculating areas under a curve, or solutions to equations up to few decimals. That power disappeared with time.
Another area related to visual memory is that for some years I worked very much in digital imaging, and I was able to describe colours on the screen in terms of Red, Green and Blue with some precision. This was when I was a teenager, an age in which I spent endless hours locked in my room creating images with the computer.
I am not sure why talking about this here can be useful to anyone. At least it serves to compare ways of thinking and using imagination.
