Subitizing: what is it?

[Chris1]

Subitizing is the direct perceptual understanding of the numerosity of a group. Literally, it means instantly seeing how many.

The latest research appears to confirm that babies can subitize. Many studies suggest that babies can distinguish quantities up to three (Starkey & Cooper, 1980; Starkey,Spilke&Gelman1990;Strauss & Curtis 1981) This is shown by habituation studies. So long as the numbers are within their subitization range they do not seem to be dependent on the pattern in which the objects are arranged. If a baby is habituated to three items arranged in a triangular pattern (s)he will still treat them as the same old three if they are represented in a straight line. This contrasts with their reaction to numbers over 3. Tan and Bryant(2000) found that these can be recognized if, and only if, the pattern remained the same.

The Shichida Dot Cards show quantity randomly along with regular representations. I have not seen the complete set of Shichida cards(there appears to be a lot of secrecy about the Shichida methods?) but the images that i have seen indicate that fixed patterns and repeated grouping within the subitizing range might be used.   

[KL]

This is very interesting. Karma for yet another great post!

You mentioned that they can't subitize beyond 3 unless the pattern remains the same.  I wonder whether it matters if you orient the dot cards in a different direction.  DomanMom, what's your experience with this? Do you always show the cards in the same orientation?

As to whether it is really true that they can't subitize on random patterns, I guess we'll hopefully be able to find out before too long - Little Math dot patterns are completely random. In fact, virtually everything can be random - pattern, shape, color, size, relative size, etc. (We are targeting to launch beta in the middle of January.)

How do Shichida dot cards achieve randomization of patterns?

[Chris1]

Shichida sites with illustration of products

http://www.shichida.com/dtl_shohin.aspx?scd=123-600

http://www.shichida.ne.jp/torikumi/index.html    Maths card illustration bottom left hand column.

Shichida- Maths main sets-Random Dot cards, Variety Dot cards and Organised Dot cards-these sets all have a corner missing suggesting that Shichida considers orientation to be important. The illustrations show the use of various colours and dot size to promote recognition of quantity.

Chris.          Missing corner might simply aid removal of cards from box.

[Tanikit]

This is interesting - I believe that animals though are also able to subitise up to at least three.

Adult humans though are supposed to be able to do so up to about 8-10 (12 at a push) and with practice possibly up to 20. I know Doman suggests that you show the dot cards up to 100 in a random orientation (that is why he uses square shaped cards as they can be turned in any direction) If it is not random then how can it be subitising - you are using the pattern then as a symbol of the quantity much the same as the symbols 1, 2, 3 etc are used to show a quantity.

I must say though that just to believe that babies can see 98 from 99 without ever having seen it for myself is difficult and I do have the tendency to believe that perhaps they are learning a pattern (even a random sequence repeated can become a pattern) Despite my reservations about the method I have already started using it on my daughter and will see the results I get later.

[DomanMom]

No, I have never used a consistent orientation and Doman recommends against that. When I first began this method with my son I wondered about that, whether he was truly recognizing the actual quantity or he had just memorized a pattern. But very soon I found out that he really was recognizing the quantity, as I not only had been changing the orientation but had actually been using totally different cards. I have used about a half dozen different sets of cards over the months, from the square ones that I made with red label stickers to powerpoint slide shows (which used a different pattern every time it was shown) to stamped handmade cards to rectangular black ones I printed out on the computer. Using different patterns has never bothered him the slightest, so I know that he is not simply memorizing the pattern on a card because he knows "96" no matter how the dots are arranged.

As for the studies that demonstrate that babies cannot subitize numbers beyond the number three, it may be true that the babies failed these scientists' tests but that doesn't exactly prove that babies cannot understand numbers. I think a big problem with a lot of research that is done on young children is that they are more out to measure what children are doing rather than what they are capable of doing.

For instance, we have probably all heard of the famous test that Piaget performed on preschoolers, which supposedly proved that they were void of a sense of what we call number conservation. Piaget showed a young child two rows of objects and asked him if one row had more, or if they both had the same amount. The child said they were both equal, but then when Piaget took one row and spread the objects further apart (in front of the child), when the question was asked again the child then said that the row that was spread further apart had more. Piaget then concluded that children do not develop "number conservation" until they are seven years old.

Although many studies now prove this experiment to be a poor measurement of three- and four-year-olds' mathematical abilities, and that children really do have a developed sense of number conservation long before age seven (or even three) it is an example of how scientific experiments can be used to demonstrate things that may not be entirely accurate. I recently read of how this same "number conservation" experiment was repeated on two- to four-year-olds, except this time they used M&M candies rather than marbles. Even the two-year-olds consistently chose the bottom row (which had six M&Ms close together) rather than the top row (which had four M&Ms spread far apart). The child certainly understood which one had more, all he needed was a little motivation (more candy for him!).

If we did a study of the reading abilities of your average one-year-olds, we could certainly conclude that most one-year-olds cannot read and most kids do not read until the age of five or six. But that does not prove that one-year-olds cannot read, simply that most don't. All in all, young babies may not be able to pass that particular scientist's test but that doesn't mean that they can't understand math, perhaps, as in the experiment with Piaget, it was the test that was lacking, not the babies.

[Chris1]

Extract from research on Subitizing

"Experiments 1 and 2 investigated whether infants ’ numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 vs. 32 discs, but not 16 vs. 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children, and many non-human animals."

"the error in numerosity representations is
proportional to numerical magnitude, and therefore discriminability between two
numerosities depends on their ratio"

"More recent studies, testing infants’ small-number
discrimination in two-dimensional displays with strict controls for total contour length or
in three dimensional displays with strict controls for total volume, have found that infants
respond to the latter variables and not to number (Clearfield & Mix, 1999; Feigenson,
Carey & Spelke, 2002).."

Could this explain why babies are unable to give a verbal response? Are they comparing/estimating quantity and responding to gestural cues?

Chris

[KL]

But very soon I found out that he really was recognizing the quantity, as I not only had been changing the orientation but had actually been using totally different cards. I have used about a half dozen different sets of cards over the months, from the square ones that I made with red label stickers to powerpoint slide shows (which used a different pattern every time it was shown) to stamped handmade cards to rectangular black ones I printed out on the computer.

That's great to know!

I wish we had more personal anecdotes from other parents, cos ultimately, as you point out, results from studies is one thing, but actually experience of a child's abilities is another. Often time, I find that both are right, and the differences are due to differing circumstances and factors.

Thanks for the extra research quotes, Chris!

[joan]

But very soon I found out that he really was recognizing the quantity, as I not only had been changing the orientation but had actually been using totally different cards. I have used about a half dozen different sets of cards over the months, from the square ones that I made with red label stickers to powerpoint slide shows (which used a different pattern every time it was shown) to stamped handmade cards to rectangular black ones I printed out on the computer. Using different patterns has never bothered him the slightest, so I know that he is not simply memorizing the pattern on a card because he knows "96" no matter how the dots are arranged.

Hi DomanMom,

Yes indeed it's great, Hunter's achievements are an inspiration to us. Have you tried just put a handful of cheerios in front of him? If you do, will he be able to instantly recognize the correct quantity?

Thanks,
Joan

[THen]

Hi!  I'm just trying to clarify what I think I understand y'all saying.  

So, Chris was reading a study that said infants can tell if one group has more or less (if the difference is great enough) and that they can tell if there are 1, 2 or 3 things.  

We were wondering if the children who learn with the dot system could actually recognize how many items there are, not based soley on a known pattern they had learned.  

DomanMom says her son was able to tell how many itmes there were no matter what pattern they were in.  And that perhaps the test just don't show the intelligence of the young children.

Am I right so far?  

So, children who are taught to recognize numerousity can learn it, even when the pattern presented to them changes.  Right?  

DomanMom, if your son was given a handful of raisins scattered on a plate, could he tell you how many there are?  That would be pretty random.  I was just wondering, as I use very random patterns (that usually don't repeat themselves) to teach my son and was hoping that was alright.  I suppose, if a child learns a pattern, but can use that to do calculations, that would be fine, too.  

Thanks Chris for the interesting topic!

Oh, I see joan has just posted a similar question, but I will post anyway.

[DomanMom]

Yes, we have done math problems with Cheerios, raisins, beads, candies, etc. If I lay two piles of objects down and ask him which one has 17, or which pile has the square root of 81, or which one has 29 divided by 29, plus 7, minus 3, he will pick the correct pile. He has not, however, ever been able to simply look at a dot card, or pile of candies, or whatever, and shout out how many there are as rainman did. It's strange but, everyone I've talked to so far who has done the Doman math says this is the case, that their children can solve the answers to amazing calculations (pointing to the correct answer, whether there be two possible answers or ten) but they are not yet able to verbalize what they know. I wrote a blog post about it here, it's a little long, but it might be useful to helping with understanding this perplexing mystery about verbalization.

http://worldsbesteducation.blogspot.com/2008/11/math-mystery.html

[Chris1]

 I recently received a reply from [email protected] regarding this perplexing mystery.

 I asked the following question -  Do you have details of any children aged 3,4 or 5 giving verbal responses
 to challenging maths questions?

 This is the reply that i received-    Absolutely, but under different conditions.

 3 "RIGHT BRAIN" CONDITIONS

 The ability to subitize in large quantities happens when the left brain is
 disabled:

 1) during the first three years of life before the left brain (conscious,
 linguistic, logical brain) has emerged
 2) when a student is in a meditative alpha wave state (left brain is
 active in beta)
 3) when the left brain has not developed "normally"--as in the case of
 some savants (see Rain Man, toothpick count
  <a href="http://www.youtube.com/v/vqbXPfaN_VM&rel=1" target="_blank">http://www.youtube.com/v/vqbXPfaN_VM&rel=1</a>)

 CHUNKING AND THE MONTESSORI METHOD

 The brain becomes less able to subitize larger quantities as the child
 grows and develops, but can still subitize.  100 can still be subitized by
 the LEFT BRAIN of a child after three if the items are grouped, or
 "chunked." This is how Maria Montessori taught math to preschoolers.

 For example, instead of teaching 41 as:
  *****************************************

 41 is presented and recorded as:
  ********** ********** ********** ********** *

 When children learn math this way, they enjoy learning and through that
 develop early math and reading abilities.

 SUBITIZATION AND LITERACY

 We subitize all the time. If you think of reading as subitization, then
 word recognition of a large word such as "compassion" is simply a
 subitization of 10 units.

 MENTAL MATH

 The easiest way to teach children mental math calculation, however, is
 through the abacus.  As soon as a child can develop photographic memory,
 they can mentally image an abacus in their minds and use it to calculate
 anything, very accurately.

 So, many methods combine for a wholesome, holistic use of the faculties of
 the right and left brains--that work together to subitize and much, much
 more.

 I hope this answer is helpful!

 I think the answer to my question was no?!! 

[Chris1]

Yes, we have done math problems with Cheerios, raisins, beads, candies, etc. If I lay two piles of objects down and ask him which one has 17, or which pile has the square root of 81, or which one has 29 divided by 29, plus 7, minus 3, he will pick the correct pile. He has not, however, ever been able to simply look at a dot card, or pile of candies, or whatever, and shout out how many there are as rainman did. It's strange but, everyone I've talked to so far who has done the Doman math says this is the case, that their children can solve the answers to amazing calculations (pointing to the correct answer, whether there be two possible answers or ten) but they are not yet able to verbalize what they know. I wrote a blog post about it here, it's a little long, but it might be useful to helping with understanding this perplexing mystery about verbalization.

http://worldsbesteducation.blogspot.com/2008/11/math-mystery.html

Hi Elizabeth,

Have you tried showing Hunter a quantity card and then asking him to place an item over the matching numeral on a 1-100 number grid?  Can Hunter write the correct answer or use a number line or grid to indicate the answer?     
 
Chris.

Taken from a Shichida site-     

I was surprised to find that my daughter can solve subtraction problems. But I have noticed that when I asked her orally, for example, "What is x plus y?" she answers incorrectly. However, when I write the formulas for her, she writes the correct answers. It is the same thing with subtraction. Also she can solve two digit plus two digit problems quicker than me! Orally, she can't solve anything.

[lekeebaugh]

I am very intrigued by this discussion.  I read somewhere else on this post that children who do learn to do subitization do not retain it when they get older.  I was wondering doman mom how old your child is who can see large quantities of objects and if any one know of older kids who can do it?  I assumed that once a child could do it that they would always be able to do it but what I read earlier seems to indicate otherwise.  If any of you know more about this I would be grateful for your insights.  Thank you.

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