The construction of a classification schema as a foundation for mathematical understanding in young deaf children.

Kritzer, Karen L.

Abstract

While it is known that the mathematics achievement levels of deaf children are substantially below that of their hearing peers, it is not known when or in what capacity these delays begin. It is possible that differences in achievement are also demonstrated in the early thinking skills of these children, for example, the ability to classify. The purpose of this pilot study was to begin to examine the pre-classification skills demonstrated by young deaf children. Findings from this study indicate a possibility that deaf/hard-of-hearing children experience substantial limitations in pre-classification skills as demonstrated through their performance on free-sorting, abstract tasks of the nature used in this study. Limitations in the development of pre-classification skills could impact deaf children's understanding of hierarchical concepts and part/whole relationships thereby influencing their ability to demonstrate adequate understanding of mathematical concepts.

Background

The low performance of deaf students in the area of mathematics has been well documented (Wood, Wood, Griffiths, & Howarth, 1986; Traxler, 2000; Luckner & McNeil, 1994, Ansell & Pagliaro, 2006). Yet while it is known that the performance of deaf students is not up to par, the reason behind and solution for this problem are still unknown. Typically, research in this area has focused on the school age population, including students between the ages of 12 years (Wood et al., 1986) and 19 years (Luckner & McNeil, 1994). These studies have been solely focused on deaf children's mathematics learning in the classroom and/or their abilities to problem solve (Luckner & McNeil, 1994; Ansell & Pagliaro, 2006). An area that has not yet been investigated is the thinking skills, including the ability to classify, that deaf children bring to the classroom with them.

Defining Classification

As defined by Piaget (1962), the first step in classification is an ability to make collections. This differs from formal classification in that membership in a collection is dependent upon perceptions, therefore, the members of a set must be physically present (Phillips & Phillips, 1996). Young children are not capable of classification because they are not yet able to abstract out any one attribute to tie a group together. While it may be possible for them to organize groups that differ by one criterion, (e.g., a group of cards that differ only by color), they are not able to do this if there are multiple differences (Lunzer, 1964). The ability to classify develops through practice in making collections (Phillips & Phillips, 1996).

According to Piaget, there are three sequential levels that one must pass through to develop an understanding of classification. The first two levels are pre-classification skills and include the making of graphic and non-graphic collections. True classification is demonstrated in the third level through expression of the understanding of class inclusion and hierarchical relationships (Phillips & Phillips, 1996).

In graphic collections the items to be sorted are viewed independently. A common approach to grouping at this level is to sort items into carefully arranged spatial configurations. For example, the child may create pictures or designs out of the materials to be sorted. A child functioning at the graphical level will examine similarities between items; comparisons however, are made between only two items at a time. S/he is unable to establish a relationship of similarity between individual items and the whole group. Properties of items are not considered as criteria for membership in a group, rather what the total arrangement looks like is the child's primary concern (Phillips & Phillips, 1996).

Unlike graphic collections, in non-graphic collections items are assigned to piles or groups based on similarity. While this is a more sophisticated level of pre-classification, non-graphic collections differ from true classification in that items still need to be within close proximity to each other and their properties must be directly perceptible (i.e., all items belonging to that group are present) (Phillips & Phillips, 1996). For example, while the child may be able to create a group of "red things," to the child this refers only to those things that are immediately present. The child must be able to deal with universal classes before s/he will be able to make more than non-graphic collections.

Within the level of non-graphic collections, there are hierarchical stages that are achieved. These stages are described as follows: (Phillips & Phillips, 1996):

* Stage 1: a variety of small collections are made based on varied criteria which overlap. For this reason, established groups are not mutually exclusive. For example, when sorting a group of colored shapes the child may include a group of "blue things" and a group of "triangles." This becomes a problem if a 'blue triangle' is among the items that the child has to sort. At this stage, all materials to be sorted may not be included in the arrangement.

* Stage 2: a large number of small collections are made. None of the criteria overlap and no items are left unassigned. At this stage, each item commonly gets its own group. For example, given colored shapes to sort, the child may establish a group of blue squares, red squares, yellow rectangles, and so on.

* Stage 3: items are grouped based on one criterion. For example, given colored shapes, the child can sort the items by shape or by color.

* Stage 4: items are grouped based on multiple criteria. For example, the child who can sort shapes based on color is also able to re-sort the same items based on shape. At this stage, the child realizes that a grouping does not cease to exist simply because its members have been rearranged.

At the class inclusion level, children are able to deal with subordinate and superordinate classes. They can compare specific subclasses with superordinate classes. For example, a dog (subordinate class) is also a pet (superordinate class). Children at this stage can view an object as being a member of multiple classes simultaneously. They can also conserve wholes while making qualitative comparisons between the whole and its parts. For example, there are more children in school than the set of girls or boys considered independently (Phillips & Phillips, 1996). The ability to understand and make use of hierarchical relationships is the first meaningful form of classification used by young children and is essential for the understanding of part/whole relationships.

Findings from studies by Piaget indicate that an understanding of class inclusion develops around the age of 7 or 8 years (Inhelder & Piaget, 1970). A more recent study indicates the development of this cognitive skill at an even younger age. As part of a study examining young children's understanding of part-whole relations, Sophian and McCorgray (1994) examined the performance of 4 to 6 year old children on their ability to solve Piagetian-type class inclusion tasks. Their findings indicated that 5 and 6 year old children successfully answered class inclusion questions but 4 year olds did not.

The Role of Language in Understanding Classification

Language may play a substantial role in children's acquisition of classification skills. According to Vygotsky (1962), early in language development words are perceived as merely properties of objects. Words are not symbolic representations and their attributes may change. There is evidence of this in a story Vygotsky tells of a child who first used the word quah to refer to a duck swimming in a pond, next this word was used to refer to any liquid, then a coin with an eagle on it, and then any round coin-like object. Vygotsky refers to this as a chain-complex. In a chain-complex, attributes are used to link individual items, usually in pairs (1962).

The appearance of the first generalized concept in a child's lexicon is a substantial milestone. For example, early in language development, a child may learn the word flower then the word rose. Initially there is no superordinate relationship between these words as both exist on the same plane. However, when flower becomes generalized, the relationship between these words changes in the child's mind. This vertical transference of knowledge allows for concept development (Vygotsky, 1962).

For all children, first words often consist of labels for objects in the environment. For example, the child learns that shoe is the label for the thing that one wears over one's sock (Mervis, Johnson, & Mervis, 1994). For hearing children, awareness of the asymmetry in linguistic relationships develops as the child is exposed to additional vocabulary to define the shoes s/he wears on different days. For example, one day mommy may announce that the child must put his/her sneakers on to go to the playground, while on a warm day sandals may be more appropriate. In this case, the young hearing child learns that the category of shoe can be more finely disseminated and specifically defined. The hearing child learns that an asymmetrical relationship exists between shoe and sandal as the second word fits into the category defined by the first. In addition, the child learns that the relationship between sandal and sneaker is symmetrical as both are equivalent subordinate categories under the same superordinate term shoe.

Simply knowing words may be enough to encourage young children's development of a categorization schema. Use of similar words to describe objects indicates awareness of a commonality between them (Mix, Huttenlocher & Levine, 2002). For example, a blue shirt and a blue car have something in common; the property of color. Similarly, mathematical word knowledge may apply to young children's acquisition of mathematical thinking. Three dogs and three trees also have something in common, the property of number.

Language and Deaf Children

Since the majority of deaf children are born to hearing parents who have varying levels of signing skills, many deaf children experience insufficient access to language. This could influence their ability to formulate linguistic hierarchical relationships, thereby encouraging them to perceive words as merely labels or properties. For example, while a mother may engage her deaf child in conversation regarding the kitty that lives at a friend's house, such conversation may never refer to the kitty as an animal or a pet. Without exposure to multiple terms to refer to the kitty the child may have a difficult time developing a mental category for domestic animals. Since s/he also does not hear the words animal or pet applied to the doggie that lives in his/her own home, the child has no means by which to formulate a relationship between dogs and cats, or to develop an understanding of the superordinate term pets that would include multiple types of animals. Lack of understanding regarding the hierarchical relationships within words could have an impact on deaf children's conceptual understanding of the world around them, thereby influencing the incidental or experiential knowledge deaf children possess when they arrive at school.

Language and Mathematical Thinking in Deaf Children

Lacking an adequate language base, a school age deaf child's ability to classify could be limited to pre-classification skills such as the making of graphic and non-graphic collections. S/he might not have developed an understanding of class inclusion which includes an understanding of part/whole relationships. This lack of understanding could have a substantial impact on deaf children's academic learning, specifically in the area of mathematics.

When solving mathematics problems, difficulty related to limited classification skills may be expressed in two ways: 1) if a child perceives words as properties, then the numbers appearing in a problem will be viewed as no more than labels; 2) if a child lacks an understanding of part/whole relationships, s/he will view all numbers in mathematics problems on the same plane, namely as separate individual units. In both cases, the child will be unable to establish a relationship between the quantities represented by the numbers in the problem.

A potential example of this is found in a study by Ansell and Pagliaro (2006). In this study, the following was one of nine story problems presented on videotape signed in American Sign Language (ASL) to deaf children: There were 11 children on the playground. 7 children went home. How many children were still on the playground?

Of the subset of children who, based on an ASL assessment inventory, were expected to have understood this problem, less than 50% applied a viable strategy for solving it.

A plausible explanation for the deaf children's low success rate in solving this problem is that for either of the two reasons stated earlier, they were unable to perceive the relationship between the two quantities presented in the problem. Therefore, 1) the number words 11 and 7 may have been perceived as labels, or properties, rather than as concepts reflecting quantity or 2) these children may not have developed the understanding of class inclusion which is crucial to developing a viable strategy for solving this problem.

In either case, it is possible that the children who were unsuccessful at solving this problem perceived the "11 children" and "7 children" as two mutually exclusive groups. This being the case, they were then unable to comprehend that the group of 7 children was actually a part of the whole group of 11 children initially on the playground.

Classification Skills of Deaf Children

In general, limited research has been done in the study of classification skills in deaf children. Research by Furth (1961) examined the classification skills of 180 deaf and 180 hearing children between the ages of 7 and 12 years. Children in this study participated in three classification tasks; sameness, symmetry, and opposition. No receptive or expressive language was required for participation in these tasks. For the sameness task, children were given two lids with simple figures drawn on them. One lid contained two of the same figures the other contained two figures that were different. The child was instructed to identify the lid on which the figures were the same. For the symmetry task children were shown two cards with drawings on them, one drawing was symmetrical, the other was not. The child was instructed to identify the symmetrical drawing. On the opposition task, four round discs differing in size were placed in front of the child. The experimenter would point to either the largest or smallest of the discs. The child was expected to choose the disc opposite in size to the one chosen by the examiner.

Findings from this study indicated that while deaf and hearing children performed similarly on the first two tasks, the deaf children had substantial difficulty with the category of opposites. The rationale for this, according to Furth (1961), is that due to language exposure the concept of 'opposite' is more familiar to hearing children than deaf children. Children tend to learn the words for extremes for example, hot and cold, good and bad, tall and short, before they can characterize dimensions of those extremes. Not having exposure to a full language model, most deaf children miss out on these conceptual variations and need to specifically be taught the concept of opposition.

Only two studies have been done regarding the classification skills of preschool level deaf students. Best (1972, 1975) utilized a classification task to observe young deaf children's ability to classify picture cards into similar groups. Different levels of classification were required including: 1) classifying identical items (cards in which the two pictures were exactly the same) and 2) hierarchical classification which required knowledge of subordinate classes within a superordinate class, (i.e., recognizing a pig and a horse as animals). Findings from these studies indicated that a slight developmental lag between deaf and hearing children was already evident at the age of 3-4 years. However, the tasks used in these studies based knowledge of classification on an understanding of verbal, not perceptual concepts. Categories were defined by their labels. Such tasks required children to make a forced choice for an item belonging to either of two categories (1975), or to answer a yes/no question (1972).

The studies by Furth (1961) and Best (1972, 1975) both utilized task which required compliant categorization. The categories were initiated by the examiner and the child was required to sort according to these predetermined categories. This type of design does not allow for free-sorting initiated by the child. Therefore, less information was provided on the deaf children's cognitive process as they attempted to organize materials. The present study differs in that categories were not predetermined. Children were presented with materials and instructed simply to group them. This type of task is abstract in nature thereby requiring more inferential thinking.

Methodology

Research Questions

The following research questions guided this study:

1. What classification skills are demonstrated by young deaf/hard-of-hearing children?

a. How do these skills vary based on tasks?

2. What is the relationship between the classification skills of young deaf/hard-of-hearing children and language level?

3. What is the relationship between the classification skills of young deaf/hard-of-hearing children and age?

Sample

Nine deaf children between the ages of 5 and 11 years participated in this study. The children were all students at a school for the deaf and functioned at a variety of cognitive and linguistic levels. Students were not ruled out from participation due to the existence of other disabilities. Viable data were collected from only seven of the nine children. One child cried at the onset of the tasks. He was inconsolable and therefore unable to continue his participation. Due to the existence of disabilities extraneous to her hearing loss the other child was cognitively and linguistically unable to function independently to complete the tasks and was therefore unable to continue her participation in the study.

Data Analysis

This study examines data based on a qualitative analysis of students' performance on three tasks related to classification. Data were coded categorically according to the following specifications (Phillips & Phillips, 1996):

1. Graphic Collections: grouping based on perception or spatial relationships as demonstrated through the creating of pairs or pictures out of the materials to be sorted. In each case, spatial arrangement was a critical aspect of the child's attempt to sort.

2. Non-Graphic Collections: at this more sophisticated level, some attempt was made to sort materials into groups which were not dependent on spatial arrangement. These groups however may not have been mutually exclusive. Non-graphic collections were further delineated by the stages as previously defined.

In addition, data were analyzed descriptively to investigate patterns in children's performance including the varying cognitive strategies used to accomplish these tasks.

Procedure

Each child met with the researcher individually in a room set aside within the child's school environment. The interview was conducted using either American Sign Language or Sign Supported Speech depending on individual children's linguistic needs. In three cases, a classroom aide was also present for the interview. Prior to this interview, the Language Proficiency Profile was completed for each child by his/her classroom teacher.

Language Proficiency Profile. Children's expressive language competency was measured using the Language Proficiency Profile (LPP; Beko & McKinnon, 1998). This language assessment tool has been used in previous research for a similar purpose (Lundy, 2002).

The LPP rating scale was distributed to individuals familiar with the child's communication use and language style. The rating scale includes 56 questions related to the child's language skills. The rater completes the scale by choosing one of five possible responses: unsure, not yet, emerging, yes and past this level. Children receive one point for each emerging response and two points for each yes or past this level response. The highest possible score a child can receive using this tool is 112 points. Hearing children tend to achieve near this score by the age of 4 years; in general, deaf children reach this score at a later age. By the age of 7 years, the scores of deaf children are still increasing; variability however, is greatly reduced (Bebko, Calderon, & Treder, 2003).

It was hypothesized that a high score on the LPP would be positively correlated with the creation of more sophisticated collections (i.e., non-graphic, as compared with graphic collections).

People Sort. The first task consisted of 40 figurine dolls which varied by age (toddler, school-age child, adult, and senior citizen) and race (African American, Asian, Hispanic, Native American, and White). The task proceeded as follows: 1) children were instructed to sort the dolls into groups, 2) explain their groupings, and finally 3) asked if there was another way that the dolls could be sorted.

Card Sort. The second task consisted of 36 cards with pictures of everyday objects (apple, bike, ball, boot ... etc.). These cards were part of a 'Memory' game that contained two sets of the same pictures. Only one set of cards were used during this study. The task proceeded as follows: 1) children were instructed to sort the pictures into groups, 2) explain their groupings, and finally 3) asked if there was another way that the pictures could be sorted.

Shape Sort. The third task consisted of 37 shapes which varied in dimension and color (Phillips & Phillips, 1996). The task proceeded as follows: 1) children were instructed to sort the shapes into groups, 2) explain their groupings, and finally 3) asked if there was another way that the shapes could be sorted.

Results

Classification Skills Demonstrated by Young Deaf/Hard-of-Hearing Children

A variety of classification skills were demonstrated by the children who participated in this study. Although the children sorted the items into either graphic or non-graphic collections, no child in this study demonstrated a capability beyond non-graphic collection stage 3. While in some cases children were able to sort the items into groups based on one criterion, no child in this study indicated an ability to classify items based on multiple criteria. Once the items were sorted into groups children were unable to rearrange them.

It was hypothesized that the children's performance would indicate that the three tasks differed in difficulty. This would influence the sophistication of the type of collection the children made when sorting the different materials. The first task, people sorting, was hypothesized to be the least difficult due to the concrete nature of the materials involved. The second task was more abstract as it required working with pictures rather than actual objects and the third the most abstract as only shapes were involved. It was believed that the materials for this third task would be the least meaningful to students thereby making the task more difficult than the previous two. For these reasons, it was believed that more sophisticated classification schemes, in the form of non-graphic collections, would be demonstrated on the first task than either of the other two. As figure 1 indicates, this was the outcome.

People Sort. Six of the seven children organized non-graphic stage 3 collections. These children sorted the dolls into groups based on one criterion. Three children sorted the dolls by family/race (five mutually exclusive groups, one for each race) and three according to age (four mutually exclusive groups: toddlers, school-age children, adults and senior citizens). Two differences worth noting are that one child who separated the dolls by race abstracted out only the African American group and one child who separated the dolls by age only grouped the infants.

[FIGURE 1 OMITTED]

Only one child separated the dolls into a graphic collection. The categories established by this child, while personally meaningful to her, were not unambiguous or generalizable. For example, in one category each doll represented someone the student knew: "this is my sister, my mom, my mom's friend and her baby." Spatial arrangements were a critical aspect of this child's organization. For example, in additional categories the child dolls represented students in school and were separated into a "hearing" class and a "deaf" class. Only spatial location determined if a child-doll was classified as hearing or deaf.

Card Sort. Five of the seven children separated the cards into non-graphic collections. Two of these children set up non-graphic stage 3 collections. These two children classified the items based on one criterion. For example, the food group contained the picture of the apple, carrot, hotdog and all other pictures that contained food items.

Three children set up non-graphic stage 1 collections. A variety of small collections were established based on varied and overlapping criteria. For example, one child set up a category of red things in addition to food. In this case, the groups established by the child were ambiguous. The apple could have fit into either group. Two other children set up pairs which were dependent upon similarity between two individual items. For example, the bike and ball went together because both pictures included grass. In each of these cases, similarities were only found between two items. No association was made between the individual items and a larger group.

Two children organized the cards into graphic collections. These collections consisted of sorting the cards into random pairs (i.e., in each case the child picked up two cards and put them together without looking at them).

Shape Sort. One child organized a non-graphic stage 3 collection using the shapes. This child sorted the shapes based on one criterion, color.

Six children organized the shapes into graphic collections. In each case all shapes were visible at all times. No piles were made. Five of these children used the shapes to create pictures or designs.

Relationship between Classification Skills of Young Deaf/Hard-of-Hearing Children and Language Level

It was predicted that language level would be an indicator of performance with children achieving higher scores on the LPP being more likely to organize non-graphic collections. This was not the outcome. The mean LPP score of children who organized non-graphic collections was lower (67.6) than the mean score of children who organized graphic collections (90.7).

However, this finding is complicated by the fact that the two youngest children in the study (ages 5 and 6 years) also received very high scores on the LPP (108), while the three oldest children (ages 9, 10, and 10 years) received very low scores (50, 44 and 81 respectively).

Relationship between Classification Skills of Young Deaf/Hard-of-Hearing Children and Age

It was predicted that age would be an indicator of performance with older children being more likely to organize non-graphic collections. This was the outcome. The mean age of children who organized non-graphic collections was higher (9.6 years) than the mean age of children who organized graphic collections (7.5 years).

Discussion

As this was a pilot study, various limitations were present which prevent firm conclusions from being drawn from the results. The small number of participants and the involvement of children with disabilities extraneous to their hearing loss were two of these limitations. In addition, administration of the language assessment tool used as a measure of language ability was not standardized but rather was administered by individuals who were familiar with each child's individual language use. It is possible that scores between children differed, in part, due to individual administration of the tool. Finally, since hearing children did not participate in this study, it is difficult to compare and generalize the findings. These limitations will be addressed in a future study.

Nevertheless, findings from this pilot study do indicate a possibility that deaf/hard-of-hearing children experience substantial limitations in pre-classification skills as demonstrated through their performance on free-sorting, abstract tasks of the nature used in this study. No child in this study demonstrated an ability to sort items into groups beyond non-graphic stage 3; this means that no child demonstrated an understanding of class-inclusion. While results from other studies with hearing children performing the same tasks are not available, as previously mentioned, research does indicate that an understanding of class inclusion may already be present in hearing children at age 5-6 years (Sophian & MCCorgray, 1994). This is younger than the average age of children who participated in this study.

While children were, in some cases, able to separate groups based on one criterion, they were unable to rearrange the items to sort them in a different way. This reinforces findings from a literature review on cognition in deaf children and adults conducted by Ottem (1980). Findings from this literature review indicate that deaf individuals perform similarly to their hearing peers on uni-dimensional tasks; however, when two dimensions need to be considered, color and shape for example, deaf individuals perform at a lower level. Similar to the findings of Ottem (1980), no child in this study demonstrated an understanding of multi-dimensionality. The high prevalence of graphic collections made by deaf children in this study may also support this finding. Children may have arranged the pictures graphically because they were unable to consider multiple dimensions of the items (e.g., color and shape) simultaneously.

The high prevalence of graphic collections made by deaf children during this study could also indicate that these children experience difficulty in establishing part-whole relationships. If this is true, it would lend credence to the possibility that deaf children, without an adequate language base, may perceive words as properties of objects thereby influencing their ability to establish hierarchical categorical concepts. Further research is needed to investigate this possibility.

Educational Implications

Although not specifically measured, an interesting observation during this study was the deaf children's behavior as they engaged in the three sorting tasks. Episodes of cognitive conflict were observed as children struggled with where to place specific items. For example, one picture card contained a picture of a mouse holding a piece of cheese. Certain children held that card longer and went back and forth between deciding if that card belonged in the food or animal category. Even on the more concrete people sorting task children were observed to hold dolls up to one another, seemingly making comparisons before deciding on group assignment. Most of the children who participated in this study, did not respond impulsively to the tasks. They appeared to enjoy the opportunity to engage in activities that required constructive thinking.

The everyday school experiences of deaf children should incorporate opportunities for this kind of active thinking. In addition, the everyday language that surrounds deaf children must include reference to relationships between items, words, and concepts. Deaf children must be encouraged to see the relationships between items in their environment. They need to see things as belonging together, initially in space and eventually through mental awareness of concepts (Phillips & Phillips, 1996).

For this to happen, teachers and parents must take advantage of opportunities for preclassification in everyday life. Deaf children should be frequently encouraged to find things that are exactly the same and make comparisons between items that are similar. Additionally, deaf children could benefit from practice in organizing things in space and time. This can occur while cleaning up the classroom, or putting away groceries at home. In either case, children should be encouraged to find associations between items (Phillips & Phillips, 1996).

While participating in these daily activities, children should be guided with active questioning. For example, children may be asked to categorize items in a way that makes sense to them and then explain their grouping. In addition, children can be asked if the items could be associated or grouped in a different way.

School age deaf children could benefit from solving problems that require them to use the skills of classification constructively. An example of such a problem would be the challenge of organizing a zoo with a specific group of animals. In such a problem, children are required to consider the feeding habits and habitats of animals in determining those that can and cannot be grouped together.

Research evidence indicates that deaf children benefit from classroom exposure to activities that require them to explore relationships and establish categories. A study by Boyd and George (1973) examined the abstract classification skills of deaf children between the ages of 10 and 13 years in the domain of science. Findings indicated that the deaf children benefited from participation in an experimental program that encouraged the development of inquiry skills through the physical manipulation and categorization of objects. For example, in the first of thirty sessions, children were given 20 objects that represented the following categories: living and non-living. The child was instructed to sort these items into two groups. Once the child demonstrated the ability to accurately create the two groups, these categories were then labeled with the appropriate vocabulary terms. A similar procedure was followed to encourage children's development of concepts such as size, shape, texture, state, material, weight, odor, taste, transparency, etc. Children in the experimental group of this study learned the skill of categorization in addition to science concepts. Results indicated a developmental gain on the part of the students who participated in the experimental group. The authors of this study indicated the need for research on the categorization skills of younger deaf children. It is possible that being taught thinking skills such as categorization at a younger age could improve academic performance.

Conclusion

There is a need for further research to investigate the pre-classification skills of hearing and deaf children in relationship to background variables such as age, language and experience. While it has already been established that the mathematics achievement levels of deaf students are well below that of their hearing peers, it is necessary to examine if this difference is also evident in the thinking skills demonstrated by both populations. If it is known precisely when and in what capacity differences in cognitive performance between deaf and hearing children occur, then steps can be take to accommodate for these differences before they develop into delays.

References

Ansell, E. & Pagliaro, C.M. (2006). The relative difficulty of signed arithmetic story problems for primary level deaf and hard-of-hearing students. Journal of Deaf Studies and Deaf Education, 11(2). Bebko, J.M., Calderon, R., & Treder, R. (2003). The language proficiency profile-2: Assessment of the global communication skills of deaf children across languages and modalities of expression. Journal of Deaf Studies and Deaf Education, 8(4).

Bebko, J.M., & McKinnon, E.E. (1998). Assessing pragmatic language skills in deaf children: The Language Proficiency Profile. In Psychological perspectives on deafness, Vol. 2. (Eds. Mark Marschark & M. Diane Clark). Lawrence Erlbaum Assoc. Publishers: NJ.

Best, B.J., & Roberts, G.C. (1975). Cognitive development in young deaf children. (Research Report No. 92) Minnesota University: Minnesota. ED 116362

Best, B.J. (1972). The development of classification skills in deaf children with and without early manual communication. Unpublished doctoral dissertation. University of California, Berkley.

Boyd, E., & George, K.D. (1973). The effect of science inquiry on the abstract categorization behavior of deaf children. Journal of Research in Science Teaching, 10(1), 91-99.

Furth, H.G. (1961). Influence of language on the development of concept formation in deaf children. Journal of Abnormal and Social Psychology, 63, 386-389.

Inhelder, B., & Piaget, J. (1970). The early growth of logic in the child. Routledge and Kegan Paul: London.

Luckner, J.L., & McNeil, J.H. (1994). Performance of a group of deaf and hard-of-hearing students and a comparison group of hearing students on a series of problem solving tasks. American Annuals of the Deaf 193(3).

Lundy, J.E.B. (2002). Age and language skills of deaf children in relation to theory of mind development. Journal of Deaf Studies and Deaf Education, 7(1).

Lunzer, E.A. (1964). Translator's Introduction to Inhelder, B., & Piaget, J. The early growth of logic in the child. W.W. Norton & Company, Inc: NY.

Mervis, C.B., Johnson, K.E., & Mervis, C.A. (1994). Acquisition of subordinate categories by 3 year olds. Cognitive Development, 9.

Mix, K.S., Huttenlocher, J., & Levine, S.C. (2002). Quantitative development in infancy and early childhood. Oxford University Press: NY.

Ottem, E. (1980). An analysis of cognitive studies with deaf subjects. American Annuals of the Deaf, 125(5), 564-575.

Phillips, D.G., & Phillips, D.R. (1996). Structures of thinking: Concrete operations. Iowa: Kendall/Hunt Publishing.

Piaget, J. (1962). Play, dreams and imitation in childhood. NY: W.W. Norton & Company.

Sophian, C., & McCorgray, P. (1994). Part-whole knowledge and early arithmetic problem solving. Cognition and Instruction, 12(1), 3-33.

Traxler, C.B. (2000). The Stanford Achievement Test, 9th Edition: National norming and performance standards for deaf and hard-of-hearing students. Journal of Deaf Studies and Deaf Education 5(4).

Vygotsky, L.S. (1962). Thought and language. Cambridge, Massachusetts. Massachusetts Institute of Technology Press.

Wood, D., Wood, H., Griffiths, A., & Howarth, I. (1986). Teaching and talking with deaf children. Chichester: John Wiley & Sons.

Karen L. Kritzer

University of Pittsburgh