Numerical Cognition and Development

Numerical Cognition and Development




Numerical Cognition and Development

Many researchers have sought to identify how children learn and understand numbers. They are interested in determining the age at which children become aware of numerosity and when they develop numerical sense. Children less than one year old seem to have an understanding of the sense of numbers, although they may not know what the numbers represent. I think that it is important for people to understand about numerosity. It is critical to know how children develop their knowledge of number systems later on in life. This will involve identifying the factors that can lead to better numerical understanding. This will help in determining why some people tend to have poor knowledge of numbers and other related concepts such as mathematical calculations later on in life. This is despite the fact that they may not experience in other areas such as languages. Therefore, the main question involves determining how and when do children begin to understand numerosity, and which factors can hinder their numerical cognition.

A Featural Analysis of Preschoolers’ Counting Knowledge

I. Main Question

The authors seek to find out whether the knowledge that preschoolers’ have of counting is restricted to their capability to carry out normal procedures. The researchers intend to identify the importance of the application of procedures and principles as they relate to children’s skills in counting. Their research follows studied indicating that children know the principles used in standard counting methods. They identified five essential and non-essential procedures that people use when counting. The essential procedure is that a person allocates only one number word to an item. The non-essential procedures include counting from left to right, beginning to count from the start of a row rather than from the middle, and counting adjacent objects one after the other.     

II. Methods

The researchers used an equal representation of thirty participants aged three to five years old, with ten participants representing each age. Girls and boys were equally represented. All the participants were students in the same preschool. The researchers identified nine problems, which they presented to the participants. Four problems, using 48 items were counting errors. They included omitted words, errors with double count objects, skipped objects, and extra words. Four problems, represented by 24 items were correct but unusual counts. They included counting in reverse from right to left, starting to count in the middle, double point and non-adjacent objects. The final problem was a normal correct count.   

The researchers used a puppet to find out how the participants understand correct counting, using the essential and non-essential procedures identified. The researchers used a vacant room in the school, where they brought individual children to participate in the research. The children would count ten chips, which the researchers had placed in a row, and they would listen as the puppet counted. They praised the children when they finished their counting task, and they informed them what the puppet would do. They also made the children know that the puppet would make mistakes when counting. After the puppet finished counting, the researchers asked the children whether they had noted any mistakes made. They used ANOVA and Newman-Keuls to analyze the results

III. Results

            The participants were more likely to reject all error counts made compared to the normal correct counts. They were also more likely to reject counting in reverse and double point counting, which are two correct but unusual counts. The three-year-old participants rejected fewer counts than the older participants did. All the participants were most likely to reject counts on skipped objects and were more accepting of the normal correct counts. Most of the five-year-old participants rejected any counts that did not recognize the word/object connection. The result differed between small set and large set trials. On small set trials, the participants were less likely to reject counts in reverse direction compared to other correct but unusual counts. On large set trials, they only discriminated between normal correct counts, double point counts, and counts made in reverse on one side and the error counts on the other side.

            Children who counted correctly were most likely to identify and reject counting errors. Children who understood the procedures for correct counting and considered them essential were more likely to identify any violations made. Four and five year old participants were more likely to identify and reject counts that went against word/object connection. They were more likely to distinguish the essential and non-essential procedures. The participants understood that it was essential to allocate one number word to one item and they knew that it is unessential to count in the normal direction

IV. Conclusion

The researchers concluded that preschool children are able to carry out the necessary procedures in counting even before they know the principles behind it. The children are able to perform the necessary procedures at an early age, but they learn how to identify some principles gradually. They continue to develop counting skills as they develop and grow. This is shown by the differences in levels of understanding concerning knowledge of essential and non-essential features in the different age groups. However, they noted that the counting skills that preschool children have are not restricted to the normal counting procedures.  

Children’s Acquisition of the Number Words and the Counting System

I. Main Question

The main question in the research concerns how children learn what number words mean. The study examines how long children take to know number words and the time and method they use to learn what different number words mean. The study seeks to find out how children develop their meaning of number words. It will investigate whether children develop this meaning in stages or whether it happens immediately. Children learn numerosity at an early age but they do not know how to relate it to the specific word. Children take a long time to understand how the system of counting characterizes numerosity. Knowledge of number words is essential in understanding the counting system. An awareness of numerosity is necessary in understanding what number words signify.

II Method

The researcher used twenty participants aged two and three years old, comprising of nine girls and eleven boys. She divided the participants into two groups. The first group of 14 participants consisted of six girls and eight boys. She tested them for seven months with sessions lasting five to eight weeks. She tested the other group of three girls and three boys for two months. She identified four tasks, which included ‘give-a-number’, ‘color control’, ‘how-many’, and ‘point-to-x’. She used toy animals and a puppet as objects. The ‘give-a-number’ task involved identifying the number words that the participants knew. To determine this, the researcher asked the participants to give a puppet a number of items ranging from one to five. She placed the children in four groups depending on their results.

The ‘how-many’ task sought to find out whether children knew the theory of the cardinal word. The researcher asked the children to count sets of items from two to six. This determined whether the children knew the number word they had mentioned last. The researcher determined the participants’ knowledge of color by showing them four different balls and asking them to identify the colors. Those who failed to identify the colors were asked if particular items in the study were top or bottom. The researcher chose the students who identified the four colors, the words ‘top’ and ‘bottom’ and those who were successful in giving one item in the first task to participate in the point-to-x task. The researcher showed the participants cards with two pictures of different items. She asked them to identify the number of items in the card. She assigned the first three tasks to the participants on the first day and the last task three days later.  

III Results

The researcher asked 392 questions to determine the participants’ knowledge of color and recorded only 12 mistakes. In the ‘how-many’ task, the researcher classified the participants into three groups and recorded their mean highest correct scores. Scores for the three groups were 4.8, 5.7, and 5.6. All the groups in ‘give-a-number’ task were successful, and only one student in each of the four groups tailed. Eleven children knew the cardinal meaning of the numbers four and five. Six of the eight participants succeeded at identifying six items. The participants who understood the basic meaning of number words solved the give-a-number task and point-to-x task by counting. The researcher followed four of the six children in group 1 level for seven months. Only one child in the group learnt counting during the time and the researcher placed him in groups 4 after 5 months. The researcher observed the remaining three children for four months and she placed them in the second group. The researcher followed four participants of group 2 level for seven months. The researcher only placed one of them in the third group after seven months. She placed the other 3 in the fourth group after an average of 4 months. The researcher only observed two children in the third group and one of them moved to group four   

IV Conclusion 

Children determine the distinctive numerosity of counting words from an early age, even before they know the basic meaning of the numbers. Children learn the basic meaning of large numbers in stages. Knowledge of the basic word principle enables students to determine the meaning of the number words. Children learn differently depending on their development levels. It takes about one year for children to understand the relationship between numerosity and the system of counting. This is only for children who already understand the basic meaning of the number word one.

The Enigma of Number: Why Children Find the Meanings of Even Small Number Words Hard to Learn and how we can help them do better

I. Main Question

The main question in the research is whether there are different systems for representing small and larger numerical sets. This follows studies indicating that children find it easier to identify the small numerosities that they do not need to count. Children find it hard to apply the knowledge gained on number words. They are not able to relate specific words to their quantities. The researchers seek to find out whether structuring information differently will lead to an improvement in learning and number acquisition. The way that people structure information involves determining how they phrase the words. When talking to children, it is sometimes more effective to consider phrasing the sentence in a way that they will understand than in observing grammatical structure. For instance, children respond appropriately when someone tells them to give him ‘balls three’ instead of ‘three balls.’ Doing this places the object before the number, and it leads to greater comprehension. Children are already familiar with the objects but they do not yet have a clearer understanding of how this relates to number words.  

II Method

The researchers used 56 participants ranging from the ages of 30 to 40 months old, with the mean age being 35.7 months. Thirty participants were female and twenty-six were male. The researchers divided the study into four stages, which included familiarization, pre-test, training, and post test. In the familiarization stage, the researchers used images of common items and they asked the children to identify them. There were ten slides with each having three images. The pre-test stage examined whether the children would be able to identify sets of objects based on numerosity. The researchers used twelve slides of different sizes, with each having images of common items. The slides were divided into two sets. The first slide contained sets of two, four, or six objects and the second one had three, five, and seven objects. They identified two conditions for the training stage, which included the feature-to-label (FL) and the label-to-feature (LF) conditions, and assigned half of the participants to each condition. In the first condition, the researchers began by showing the participants the items before showing them the number label. In the second condition, the researchers first showed the children the label before introducing the item.

III Results

All the participants passed the familiarization stage. The participants’ results in the pre-test stage were above chance, although they did not have much understanding of numerical concepts. The children did not discriminate between the two conditions in the pre-test stage. The two groups in the pre-test stage did not show much difference. Post-nominal structure construction is more useful in enhancing learning compared to the normally used linguistic construction. The participants in the FL condition registered improved post-test results but LF participants did not show any improvements.  

IV Conclusion 

Information is necessary for human development. Children improve and hasten their learning process when presented with information. Information structures such as introducing an object to a child before assigning a label will help to shorten the learning process. Post-nominal construction leads to lesser discrimination. Children are more likely to prefer smaller number sets based on how a person presents information to them. Proper linguistic construction ensures that the children are comfortable with large number sets as well.

Children’s Understanding of Counting

I Main Question

The main question in the study concerns how children understand abstract concepts in relation to numerosity and whether they understand the cardinal word principle. Although children may understand how to count objects it is not clear whether they use the same understanding to count abstract ideas such as sound.

II Method

The study involved two experiments. The first experiment evaluated how children performed when counting actions and sounds and how they counted objects. The researcher used eight participants aged two and a half years old, eight three year olds, and eight children aged three and a half years old, which he classified as Age I II and III respectively. All the children were from daycare centers and there was an approximate representation of boys and girls. The investigator identified four conditions for the research, which included sound, object, jump, and cave. The researcher arranged the objects and asked the children to count them. The researcher asked the participants to count the objects as she transferred them from where they were to a place where they could not see them. This represented the cave condition.

To capture the abstract element the researcher captured four sounds, which included an elephant roaring, the ring of a doorbell, splash made in the bathtub, and the beep sound of a computer. The children counted the sounds as the researcher played them from a tape recorder. The second experiment investigated ‘the give-a-number’ task and it was performed after the end of the first one. The researcher asked the participants to give a puppet a certain number of toys. The researcher asked them to recheck whether they had given the correct number of toys, which they did by counting. The researcher divided the participants into grabbers and counters, depending on how they gave the puppet the toys. The counters added up the number they would give, while the grabbers snatched a handful of toys and gave them to the puppet. .  

III Results

Older children counted correctly most times compared to the younger ones. The mean correct count for age I II and III were 0.67, 1.22 and 1.64 respectively. The children found it easier to count the small set sizes, with an average of 1.33 compared to 1.02 of the larger sets. The children performed better in the object and cave conditions, than they did on the sound condition. Older children showed less discrimination between large and small sizes of the sound condition. Almost all the children showed some ability in counting non-objects. They were able to count non-object trials in at least 25% of the cases. Three of the children in age I did not understand the idea of counting non-objects. Older children showed greater ability to count non-objects and the performance was 38%, 60%, and 79% for ages I II and III respectively.

 The participants were more likely to apply the cardinal principle when referring to non-objects. They were more likely to perform a recount when asked how many objects they were in a set. They could not perform the same objects when counting non-objects. Older children were more likely to apply the cardinal word principle. In the second experiment, there were four ‘counters’ and they all belonged to age III. They added or removed the objects to suit what the researcher asked them. They did this until they achieved the required number of toys that the puppet needed. All the remaining children were grabbers. They all gave the correct item when the researcher asked them for one object but the results differed when applying larger number sets.

IV Conclusion

Children begin to understand the cardinal word principal when they are above three years. Children count better in familiar contexts than in new environments. Children will often use ordered lists when counting even if these lists go against the normal order. They will have an idea of how the numbers follow each other, even if they will not know all the numbers. They begin to have an understanding of the abstract nature at an early age.

The literature review has revealed, and made some issues about clear concerning numerosity and number systems. Children begin having an understanding of numerosity when they are very young. They are able to apply the knowledge they have learnt when counting objects. As they grow older, they become more aware of abstract concepts and they can apply their understanding of numerical systems. Children understand the order of numbers and they will apply it even if they do not know all the numbers. They have different understanding of what the last number means, representing a conflicting view of the cardinal word principle. Young children who have not developed a clear understanding of numerosity will usually assign an object a given number. Therefore, his understanding of the last number concept applies to the number he has assigned the object, rather to the real number that the object represents. Although children understand number words, they may not always know how to assign them in different concepts. This weakness is mostly because of cultural influences, which determines how parents and teachers introduce numbers to the children. Introducing the objects before allocating and labeling it will lead to greater comprehension. Children will be in a better position to identify how many objects they are using this approach. This is different from telling the children to identify a certain number of objects, by starting with a given number.


Briars, D., & Siegler, S. R. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20 (4), 607-618

Ramscar, M., Dye, M., Popick, M. H., & O’Donnell-McCarthy, F. (2011). The enigma of number: Why children find the meanings of even small number words hard to learn and how we can help them do better. PLoS ONE, 6 (7), e22501

Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155-193

Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220-251

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