It is not only letters that she will identify. She likes numbers too. So much so that we have to stop at every letterbox on our walks, and when we leave, she always says 'more?'
That's lovely, but I can't help wondering: does she actually understand what the numbers mean? Or does she just know that a certain shape is called a 'three'?
When we do the washing, I have tried asking her to hand me one or two pegs. I usually get one peg, and no prompting or demonstrating what 'two pegs' means can get her to respond with two. If I say, 'How many is that? Let's count! One, two, three...' she seems to understand this as a ritual, and when playing with her toys will spontaneous point repeatedly at them saying 'two, fwee, fwee, fwee' or something along those lines.
I thought she might have grasped the concept of 'two' the other day, when I pointed to two balls and said, "How many?" and she said, "Two!"
Then I thought about it a bit more and showed her one ball, and said, "How many?" and she said, "Two!" And I showed her six balls and said, "How many?" and she said "Two!"
So nope, no concept of numbers whatsoever. The closest she has to an understanding of quantity is being able to identify "more" and "no more".
Which got me wondering: what is normal toddler numeracy development?
Number Sense
It turns out children have an innate 'number sense', but that it is not really operational at birth. At some point the number sense switches on, and a child will start to be able to systematically and correctly identify 1 vs 2 objects. It can take months before they are then able to distinguish 1 vs 2 vs 3 objects, then 1 vs 2 vs 3 vs 4. During this stage, 20 objects are much the same as 35 objects. Then sometime round the age of 3, they suddenly become much better at distinguishing different quantities - they seem to develop an understanding of the concept of precise numerical quantities in an abstract sense (Wynn). It makes no difference if the children can count to ten by rote, until they develop their number sense, asking for six objects is just as likely to get you ten or three.
So when does a child's number sense switch on? The answer is that the age varies by about two years. Some children can accurately understand numbers before the age of three, meaning they can produce the correct number of objects as high as they can count. Others, at age four, are still struggling to accurately produce two objects on request. It seems that the number sense process, then, normally starts sometime between ages 2-4 (Gunderson and Levine). Hence why, at 17 months, Bethany cannot hand me two pegs on request.
When parents say their 1 year old can count, what they typically mean is that their one year old can say something like: 'one two three four...' but probably not that their 1 year old can successfully identify and hand them four objects. The number of numbers in a child's counting sequence probably is a good indication of their memory for language rather than their number sense.
Number Sense vs Counting to Ten Using Symbols
A recent study looking at how 4-7 year old Indigenous children in remote communities in the Northern Territory respond to numeracy tasks if they don't have words for the numbers concerned. These children spoke the Indigenous languages of Walpiri and Anindilyakwa, in which children are taught words for 'one' 'two' 'few' and 'many', but are not yet privy to adult ritual words which include the numbers up to twenty. The non-English speaking Indigenous children were compared to English speaking Indigenous children, and asked to complete various numeracy tasks that did not depend on knowledge of English terms. For example, children were asked to match objects to the sounds made by banging two sticks together, or shown a number of counters being placed on a mat then (after the mat was hidden) asked to 'make your mat like hers'. Hence if the sticks were banged 4 times, the correct response was to produce 4 counters, or if 8 counters were placed on the mat the child had to remember and replicate this on their own mat. The study found that children who only spoke Walpiri or Anindilyakwa did as well or better than the English-speaking children on numbers up to 9. The real noticeable difference was based on age, rather than language, with older children doing noticeably better than the younger children (Butterworth et al).
This goes some way towards upsetting one hypothesis that children need to develop symbolic representations for numbers (eg. words, symbols, or signs) and without them will remain oblivious to numerical concepts. Number sense seems to have a certain innate quality that is not dependent upon language. The strong influence of an innate number sense has been found on maths ability right through schooling - where people who can distinguish more quickly between different quantities using a non-verbal test easily tend to have stronger maths skills as measured in the classroom (Libertus). Reaction time and accuracy on these kind of tests have been found to explain up to 20% of the variance in maths ability throughout schooling, even after controlling for age, vocabulary size, intelligence, memory etc. (Libertus).
On the other hand, there are correlations between the amount that parents talk about numbers and children's development of number sense. A 2010 study looked at the frequency with which parents mentioned numbers in front of their children and their children's development of early math knowledge, and found that parents that engage in more number talk tend to have children with better early math knowledge (Levine et al). In a follow up study, forty-four children were followed from 14-30 months, with data video-taped in the home for 90 minutes every 4 months while children and caregivers went about their normal activities. The tapes were then evaluated not only for the frequency of the number words spoken, but how frequently the words referred to a tangible quantity the child could see, and whether the numbers counted were less than four, or four or more. (Four was the dividing line because the studies by Wynn mentioned above seem to suggest that once children acquire an understanding of how to identify 4 objects, they have the principle of numeracy, rather than just being able to differentiate between very small exact quantities). Then, at almost 4 years of age, the children were tested for their understanding of numbers (Gunderson and Levine).
The study found that number talk referring to present objects ("You have five fingers." or "How many bees? Let's count them: 1, 2, 3") was significantly more important for the development of number sense than number talk in the abstract (eg. "I'll be five minutes" or "Let's play hide and seek, I'll count to ten: 1, 2, 3... etc"), and further that number talk that discussed 4 or more present objects was more important for developing the principle of counting than number talk limited to three or less present objects (Gunderson and Levine).
How can we reconcile these results with the innate number sense displayed by Indigenous children? Well, one thing that should be noted is the way in which Gunderson and Levine tested the 4 year olds' numeracy. They gave them 16 questions that involved presenting the children with two sets of objects, such as 3 squares and 5 squares, and then asked them either to 'Point to 3' or 'Point to 5'. Unlike the tests in the study with Indigenous children, this test inherently requires an understanding of verbal symbols for numbers as well as a 'number sense' for the concepts behind those symbols.
Bearing this in mind, we might conclude that perhaps the toddlers whose parents did a lot of number talk in Gunderson and Levine's study just had better vocabularies generally or other advantages conferred by socio-economic status (which would probably also relate to the amount of educational number talk). However, the study controlled for parental socio-economic status (including education), parent's general talkativeness, and the child's vocabulary, and still found that specific and concrete number talk to toddlers was significantly correlated to those children's ability with the number task at age 4. The number talk about present objects still accounted for 15.7% variation over and above those other factors (Gunderson and Levine).
The hypothesis I would suggest might follow from these studies is that children have an innate number sense, but that the confidence with which they learn to match their innate sense of, say, 'fiveness' to the word 'five' and to the symbol '5' depends on the extent to which these connections are reinforced by hearing and seeing the words with the examples present. Hence, when they come to be tested with the question 'Point to 5', a child who has developed a strong association between the symbol and the concept will be more likely to give a correct answer than a child who has a weak association. Correctly identifying a 5 requires a working 'number sense' but an incorrect answer does not show that the number sense is not yet working, but perhaps just shows that the association with the symbol is not yet strong enough for the child to perform well on the test.
Of course, unless school curriculums have changed greatly since when I went to primary school, I daresay classrooms are not teaching basic numeracy in terms of matching counters to the number of sticks banging together, but are heavily focused on teaching and assessing the use of spoken and written arabic numerals. It is therefore no surprise that kindergartners who score highly on the kind of test administered by Gunderson and Levine tend to have an ongoing advantage in terms of understanding maths throughout their schooling (Gunderson and Levine). I also suspect that children who enter school with the skills to do well on these tests are explicitly or tacitly given a message that they are 'good at maths' and acquire a motivational advantage through confidence compared to the children who are given the message that they are not good at maths. This view is reinforced by other studies which have found that nonverbal calculation abilities are less sensitive to socioeconomic status than are number problems in 'story form' (eg. 'Joe had three pegs and then he went to the shop and bought five more pegs. How many pegs does Joe have?') (Jordan et al).
One issue not explored by the Gunderson and Levine study that I would love to know the answer to is whether number talk at the early toddler stage (when number sense is perhaps just starting to develop) is in itself important to the results, or whether that parents who engaged in early number talk continued this number talk as the child grew older and more receptive to it, and it was this later absorption of information that was important? After all, if Bethany cannot conceptually understand what I am doing when I count out 5 pegs, how is that different (from her perspective) to me counting to 5 in the abstract?
Gender
Statistically, boys do better at maths than girls. Significant differences have been found between boys and girls on number sense tasks as early as kindergarten (Jordan et al). In particular, kindergarten boys seemed to have an advantage on nonverbal calculation tasks. This may suggest that number sense generally develops earlier or more strongly in boys. Also, any non-verbal tests devised to test early numeracy require good spatial reasoning, and spatial reasoning tends to be stronger in males (Jordan et al). It may be that number sense is itself linked to spatial reasoning, which is a crucial skill for correctly understanding, storing, retrieving and comparing different quantities of objects. It would be interesting to know whether activities that develop spatial reasoning skills assist in the development of number sense.
Early Numeracy Skills
Drawing on this discussion, what skills are needed for early numeracy?
Number sense - an awareness that sets of objects can be grouped by the quantity in the set. There is a good chance that when and how this switches on may be genetic, however, it may be that the right environmental conditions for fostering a strong number sense have not been measured.
Counting skills - This has been described as mastering a set of five principles, known as:
- The one-one principle: each object in a set is counted once and only once.
- The stable-order principle: when counting, one must use a set list of symbols in order - if you count 2, 4, 6, 3, 1, it doesn't work.
- The cardinal principle: the last number you say is the number of items in the set.
- The abstraction principle: abstract things can be counted as well as tangible objects.
- The order-irrelevance principle: when counting objects, it doesn't matter which object you start with or end with, the answer is the same.
For a slightly more elaborate summary of these principles, see The Principal Counting Principles by Ian Thompson. I can see right now that Bethany does not have an understanding of any of these principles yet. When she pretends to count she often points to the same object twice and skips objects, her count list is sometimes 'two three three' and sometimes 'two two two', and if I count for her and then say how many, she still says 'two'.
Number-specific symbols - in particular, how to recognise and say the words 1-20, and how to recognise and write number symbols.
The count sequence - memorisation of a specific list of symbols in a particular order that can be used for counting objects (eg. one, two, three, four... etc.)
General language and cultural concepts - General language and exposure to concepts such as currency, shops, dividing an object to share are all used to convey number stories which are an important part of early mathematical learning.
Linear number sense - while an innate number sense has been demonstrated in respect of numbers up to 9, it is not clear whether we innately have a good sense of much larger numbers, or whether this develops by association and an understanding of symbols that allow us to retrieve that information to make estimates. In experiments where children of various ages (starting from kindergarten age) are given a number line that starts in 0 and ends in 100, they are asked to guess where on the line specific numbers would fall (eg. "where is 25?"):
0 <--------------------------------------------------------> 100
A kindergartner will typically spread out numbers 0-20 along about the first half of the line, then group all the other high numbers somewhere randomly up the second end. By contrast, a seven year old will produce pretty accurate placements if using numbers 0-100, but if asked to do this task with numbers 0-1000 will produce a more logarithmic answer, similar to the kindergartners doing the 0-100 task (Booth and Siegler).
Children who have a good linear sense of numbers learn to manipulate the numbers (eg. how to add them together to produce a new number) more easily than those who do not (Booth and Siegler). This linear model gives us a good ability to estimate numerical answers, even when the quantities are much larger than perhaps our innate number sense can deal with.
So what can you do with your toddler if you want to foster numeracy?
You can count objects ('let's count the steps as we walk up them') and identify numerical sets of objects (eg. you have five fingers) - this would illustrate counting principles and teach the count sequence and number-specific symbols, even if the numerical meaning is lost on your toddler at this stage. You can expose your toddler to cultural rituals involving numeracy, such as going to a shop or dividing a cake into pieces to share. You might encourage spatial awareness, talking about objects being 'next to', 'above', 'on top of', 'beside', 'in' other objects etc, and identify that objects can be on their own or in groups.
Product reviews
We have two products that are related to numeracy. The first is the Leapfrog Chat and Count Phone (approx $25 from Big W).
We bought one of these for Bethany as something to give her when she wants to play with out mobile phones. When you press one of the numbers on it, it says the number and shows the number symbol, then shows a visual of that number of objects. The difficulty is that the speakers produce a voice that is not very clear (Bethany does not seem to identify what it is saying, even though she knows the number symbols and what to call them) and generally very young children do not pay as much attention to disembodied voices as to real persons. There is a substantial delay between when she presses the '6' to when the six items actually appear on the screen as they appear through a very slow animation. The quality of the image is very poor - a pixellated grey and black image on a tiny screen. That said, the phone keeps her entertained for short periods of time (like, a couple of minutes maybe once a week), and for those few minutes she enjoys the fact that when she presses a number a matching number appears on the screen, and sometimes that a 'woof' (dog) appears.
The other product we have is a simple wooden puzzle with pieces in the shape of the digits 0-9 (I don't know the brand, it's unmarked). However, it looks just like this and cost $10:
There were no representations of quantities of each number, so I drew the right number of dots on the puzzle pieces and in the spaces where they are placed. Bethany responds to the puzzle with significantly more interest than the phone. She will happily stick at the puzzle for about half an hour, then come back to it several times in a day. She does regularly get frustrated by her inability to get a puzzle piece in, yell out, and chuck a mini-tantrum, but there are other times when she persists with it. At any rate, frustrated or not, she keeps returning for more, and for a couple of months now it has been one of her favourite toys.
References
Booth and Siegler, 'Magnitude Representations Influence Arithmetic Learning' (2008) Vol 79(4) Child Development 1016.
Butterworth et al, 'Numerical thought with and without words: Evidence from indigenous Australian children' (2008) 21(4) Philosophical Psychology 443 (see press release from University College London at l, or pdf of study)
Cho et al, 'How does a child solve 7 + 8? Decoding brain activity patters associated with counting and retrieval strategies' (2011) Vol 14(5) Developmental Science 989.
Gunderson and Levine, 'Some types of parent number talk count more than others: relations between parents' input and children's cardinal-number knowledge' (2011) 14(5) Developmental Science 1021.
Jordan et al, 'Number Sense Growth in Kindergarten: A Longitudinal Investigation of Children at Risk for Mathematics Difficulties' (2006) 77(1) Child Development 153.
Levine et al, 'What counts in the development of young children's number knowledge?' (2010) 46(5) Developmental Psychology 1309.
Libertus, 'Preschool acuity of the approximate number system correlates with school math ability' (2011) 14(6) Developmental Science 1292.
Wynn, 'Children's acquisition of number words and the counting system' (1992) 24 Cognitive Psychology 220.
