Journal of Deaf Studies and Deaf Education Advance Access originally published online on March 15, 2008
The Journal of Deaf Studies and Deaf Education 2008 13(4):503-517; doi:10.1093/deafed/enn007
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Family Mediation of Mathematically Based Concepts While Engaged in a Problem-Solving Activity With Their Young Deaf Children
Kent State University
Correspondence should be sent to Karen L. Kritzer, Kent State University, EFSS 405 White Hall, Kent, OH 44242-0001 (e-mail: kkritzer{at}kent.edu).
Received December 16, 2007; revised February 11, 2008; accepted February 12, 2008
This qualitative study examined the relationship between young deaf children's level of mathematics ability ("high" and "low," as defined by test score on the Test of Early Mathematics Ability-3) and opportunities available for the construction of early mathematics knowledge during a problem-solving task implemented by their parents. Findings indicate that the manner in which the mathematically based concepts (number/counting, quantity, time/sequence, and categorization) were incorporated into the activity was more meaningful for children who demonstrated high levels of mathematical ability. In addition, children who demonstrated high levels of mathematical ability experienced a more purposeful use of mediation during activity implementation; however, overall use of mediated learning experience was limited for children from both ability groups.
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Introduction |
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As evidenced by test scores (Traxler, 2000
), and by tasks involving reasoning (Allen, 1995), logical thinking (Marschark & Everhart, 1999), and problem solving (Ansell & Pagliaro, 2006), deaf students currently attain low levels of achievement in mathematics. Limited research is available, however, in documenting precisely when these low achievement levels begin or the basis for them. It is possible that deaf children are beginning formal schooling already performing at levels below their hearing peers, thereby putting them at risk for an ongoing pattern of low achievement.
Although research findings indicate that early school experiences, family interaction, and the home environment are important predictors of later academic success in content areas such as mathematics (Jimerson, Egeland, & Teo, 1999), a dearth of research is available documenting the quality of the early learning experiences to which young deaf children are exposed, and how this impacts their mathematics learning. A critical area in which the early experiences of young deaf children, particularly those with hearing parents, may differ is in the quality of informal mediated learning experiences (MLEs) to which they are exposed while interacting with their parents during learning activities. Suggested by Klein, and based on the theoretical model of Feuerstein and Rand (1997), mediation in early childhood includes five processes: intentionality/reciprocity, transcendence, mediation of meaning, mediation of feelings of competence, and mediation of regulation of behavior.
The focus of this article is the opportunities for the construction of early mathematical knowledge that are available for young deaf children within their home environments. Specifically, it describes the opportunities available for the early learning of mathematics while engaged in a problem-solving activity purposely designed for the development of mathematically based concepts.
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Background |
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Early Mathematics Learning
By the time they start school, young children have already acquired a great deal of information regarding mathematics. Research indicates that young hearing children understand counting principles (Gelman & Gallistel, 1978
), are aware of the quantitative relationships represented by numbers (Sarnecka & Gelman, 2004), make judgments regarding the relationships between size and shape (Sophian, 2002), distinguish shapes and identify their attributes (Clements, Swaminathan, Hannibal, & Sarama, 1999) as well as classify them (Deak, Ray, & Pick, 2002). In addition, by the age of 5 years, young children have been found to be able to demonstrate understanding of quantitative (i.e., part-whole) relationships existent between sets (Sophian & McCorgray, 1994).
Prior to the onset of formal mathematics instruction, preschoolers have been found to demonstrate their early mathematics knowledge informally during play. Ginsburg, Inoue, and Seo (1999) conducted a study in which 4-and-5-year-old children were observed in their preschool classrooms for a total of 469 min during free-play situations. Mathematical activities were found to comprise 209 min (44.6%) of the children's time. The children were observed partaking in activities related to five different types of mathematical awareness: patterns and shapes (36%), exploration of change (22%), relations (18%), classification (13%), and enumeration (11%).
Less research is available regarding the early mathematics skills of young deaf children prior to the onset of formal schooling. In a study investigating the early number understanding of young deaf children, Zarfaty, Nunes, and Bryant (2004) found that young deaf children's abilities to represent number are at least as well developed as their hearing peers before they enter school. When presented with counting tasks, deaf and hearing children were found to perform equally well on temporal counting tasks (i.e., items presented one at a time in a sequence); however, the deaf children performed significantly better on a spatial (i.e., items presented together in a spatial array) counting task. Another study, conducted by Leybaert and Cutsem (2002), examined the development and use of counting by deaf children between the ages of 3 and 6 years. Although findings from this study indicate an age-related lag of approximately 2 years in deaf children's knowledge of the counting string, the children who participated in this study did demonstrate age-appropriate skills in object counting and creating sets of a given cardinality. Both studies examined deaf children's understanding of number. No information is available, however, regarding young deaf children's early construction of mathematics knowledge in other areas such as use of quantitative terminology (i.e., more, less), understanding of time and sequence, or categorization.
Learning that takes place informally, in the home environment, is likely to contribute to the early mathematics knowledge that young children have developed by the time they reach school age. According to Walkerdine (1988), parents purposefully engage in two different types of instructional tasks, "instrumental" and "pedagogic," when interacting with their children to relay mathematical information. In an instrumental task, mathematics is used incidentally. For example, using the domain of number, when asking the child to set the table for dinner, a parent might mention that five plates are needed. Instrumental tasks emerge naturally from daily activities. This is in contrast to pedagogic tasks that are contrived to be instructional in nature. Again using the domain of number, an example of a pedagogic task may be counting blocks as the child plays with them. In this task, counting is the main focus of the activity.
Through the use of interviews with 400 mothers of preschool-age children, Saxe, Guberman, and Gearhart (1987) found that mothers frequently incorporated numbers into their young children's daily routines through play, playing board games and invented number games such as counting steps while walking, or reading numbers on license plates. Nearly the entire sample of mothers reported that their children engaged in self-initiated number activities more than once a week (i.e., counting snacks, toys).
Additional case studies describing young children's developing competencies in mathematics at home indicate that parents tend to engage their children in a variety of pedagogic games and activities related to early mathematics. In a longitudinal case study of two young children from the time they were 18 months old until they were 48 months old, Aubrey, Bottle, and Godfrey (2003) concluded that mothers included mathematics development in the home by frequently engaging their children's interest in counting rhymes, counting books, puzzles, and games. Additionally, these two young children were involved, at least incidentally, in household activities such as setting the table and following a recipe while the mother was cooking.
Findings from a case study by Phillips and Anderson (1993) indicate that a common activity shared between parents and children, reading books, can also be instrumental in relaying mathematical information as it lends itself to mathematical discussions. While reading "The Three Little Pigs" with her 3.5-year-old daughter, a mother was found to effectively relay mathematical information to her daughter by encouraging the child to notice relationships, for example, noticing that each pig's clothing matched the color of his house.
Mediated Learning Experiences
Opportunities to construct early mathematics knowledge may be present not only in the mathematics content that is presented to young children but also in the manner in which the children are exposed to learning opportunities, explicitly or incidentally. Regardless of whether or not a parent views a task as explicitly mathematical in nature, young children are constantly surrounded by instrumental use of mathematical language. Overtime, as findings suggest, children are likely to incorporate the mathematical vocabulary they have overheard into their own language use (Walkerdine, 1988). During a conversation with her mother, for example, a 3.9-year-old child was found to use 10 different relational terms to describe the concept of "big," (i.e., big, bigger, biggest, bit big, very big, as big as, not big enough, not big, bit bigger, too big), indicating an awareness that there are varying degrees to which something can be described as large in size. It is likely that this early mathematical awareness develops through the informal learning experiences to which young children are exposed.
Feuerstein and Rand (1997) describe two main types of informal learning experiences that may relate to young children's early learning of mathematics concepts: "direct exposure" and the MLE. Direct exposure involves direct interaction between the individual and the stimuli; for example, a young child (the individual) playing independently with a small car (the stimulus). The MLE, however, adds a human to the scenario to serve as a bridge between the child and the environment (the stimuli) in order to make an experience more meaningful (Ben-Hur, 1998); for example, given the above situation, the child is playing with the same small car; however, now a mediator is involved. This individual may comment on the size of the car saying, "this car is small, —where is your big car?," thereby exposing the child to the relationship established by size. Effective mediators draw a child's attention to specific stimuli, in this manner teaching the child how to perceive meaning from the environment. Such individuals set up MLEs in ways that teach children how to think (Ben-Hur, 1998).
Research in this study is framed by the perspective that early learning develops through effective exposure to MLEs. According to Klein, and based on the model of Feuerstein and Rand (1997), the dimensions of the MLE that are most critical for young children are as follows: intentionality/reciprocity, transcendence, mediation of meaning, mediation of feelings of competence, and mediation of behavior. Informally, the first three characteristics can also be referred to as the "where/when" (i.e., the environment established for learning), "why" (i.e., the explanations offered to assist children in making sense of the world), and "how" (i.e., the questions asked to encourage use of thinking and problem solving of the learning experience. Additionally, young children require specific encouragement that reinforces their beliefs in their own competence as learners and recognition of their strengths and weaknesses, as well as assistance in developing their inhibitions so that they learn to slow down and think before acting. Because children cannot be directly taught every piece of knowledge they will ever need, they must learn how to learn. This process of learning to self-construct knowledge is the purpose of the MLE and, according to Feuerstein and Rand (1997), is present in every valuable learning exchange between a child and a mediating adult.
Research indicates that when young children are exposed to high-quality meditated learning experiences in the home, they perform at a higher cognitive level (Klein, 1991, 2000). Specifically, Klein (1991, 2000) examined the influence of an intervention program, designed to enhance use of MLE in the home, on the learning of children with a very low birth weight. In this study, mothers were taught how to appropriately use mediation techniques with their 1-year-old children. In a follow-up study when the children were 4 years old, children of mothers in the experimental group demonstrated higher cognitive performance than a control group of children whose mothers had not been formally taught how to use mediating techniques with their children. Similarly, a study conducted by Chiswanda (1999) with mothers of young deaf infants and toddlers found that after implementation of an intervention designed to enhance the use of MLEs, there was a positive change in the mothers’ styles of mediation.
Communication Challenges
It is likely that the communication challenges that some hearing parents experience when interacting with their young deaf child are likely to influence the MLEs that they are able to provide. Hearing parents may find interaction with their deaf child challenging for the following reasons: (a) lack of confidence in their ability to communicate with their child effectively (Spencer, Bodner-Johnson, & Gutfreund, 1992; Spencer & Gutfreund, 1990); (b) communicative passivity or language delays on the part of the child (Spencer et al., 1992; Spencer & Gutfreund, 1990); and/or (c) lack of awareness of how to transition from the auditory–verbal style of communication that they are accustomed to, to an interactive approach that is more accessible to the visual needs of their deaf child (Jamieson, 1994).
When a communication barrier is present in the home, some hearing parents might adapt an approach to communication with their deaf child, which is limited in its ability to access high levels of thinking. For example, the predominant role of conversation with the deaf child might be to obtain answers to direct questions or to gain specific information (Charlson, Bird, & Strong, 1999). Although it is through asking and answering questions that children learn to derive meaning from their environment (Feurstein, 1997), deaf children may not frequently be asked cognitively challenging questions; rather, they may be requested to recall labels for items in their environment or directed to perform specific tasks. As described by Finn (1995):
It is natural for hearing children and deaf children of deaf parents to seek causes and meanings; they constantly ask, "Why?", "How?", "What if?" But deaf children of hearing parents do not ask these questions. Rather, their mothers or teachers use meaningless monologues, asking "What's this?" and "Do that" as if the child understood English (p.10).
There is evidence of use of this simpler form of communication in the findings from a study by Moeller and Schick (2006). When observed while conducting a problem-solving task (playing with building toys) with their young deaf children, mothers with limited sign communication skills tended to use language with their children, which was immediate and concrete, such as discussing the colors of the toys; in contrast, mothers with better sign language skills were more likely to use the toys as tools to initiate complicated fictional scenarios that were more abstract in nature.
Use of simple concrete language will not encourage the development of higher level thinking skills such as comprehension, comparison, and evaluation that are developed during MLEs and necessary for informal learning (Feuerstein & Rand, 1997). This premise provides a starting point from which to examine the early opportunities for construction of mathematics knowledge that young deaf children experience in their home environments. It is possible that young deaf children experience a lack of early mathematically based learning opportunities within the home environment and that this ultimately has an influence on the cognitive readiness skills with which they begin their formal education.
The study presented in this article adds to the limited body of research currently available regarding young deaf children's early mathematics learning in the home, by examining the opportunities for early construction of mathematics knowledge available for young deaf children during an activity purposefully designed to stimulate awareness of mathematically based concepts.
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Background
Twenty-nine deaf children from seven schools for the deaf across the United States participated in the larger study of which the study discussed in this article is a part. The children were between the ages of 4 and 6 years, but not yet in first grade, with no additional disabilities, and from homes in which either American Sign Language (ASL) or spoken English was the primary language spoken. This age level was chosen in order to examine deaf children's early learning experiences prior to the onset of formal instruction.
A background questionnaire was completed by an adult family member of each participant, and the Test of Early Mathematics Ability (TEMA-3) was administered to all 29 participants in the original study individually. The TEMA-3 (Ginsburg & Baroody, 2003) is a test that utilizes both informal and formal tasks to measure the mathematics ability of young children between the ages of 3.0 and 8.11 years.
The scores of the TEMA-3 were used to select participants for the portion of the study described in this article. Participants were selected as follows: scores within 1 standard deviation of the group mean formed the "average" group; scores 2 standard deviations above and below the mean formed the "high" ability and "low" ability groups, respectively. One child, who received a score on the TEMA-3 that was 2 standard deviations higher than the next highest score, was deemed an outlier. All information regarding this child was removed from data considered for further analysis. Three participants from each group (i.e., high ability and low ability) were randomly selected and invited to participate in the portion of the study described in this article. It is critical to note that the terms high and low are relative and refer only to the children who participated in this study. When compared with the normed scores established by the TEMA-3 (Ginsburg & Baroody, 2003), disregarding the outlier, no child who participated in this study scored above average.
Participants
Of the six children classified as having high mathematics ability, one child had hearing parents; the other five children had at least one deaf parent. The three children who were randomly selected to participate in the second level of the study all had at least one deaf parent. As noted in Table 1, the three children in the high ability group (i.e., Andrew, Emma, and Sara) ranged in age from 4 years, 8 months to 5 years, 10 months; all three had at least one sibling and were the oldest children in their families. For each child, ASL was the primary language used in the home. All three children lived in rural to semirural areas.
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Of the seven children classified as having low mathematics ability, five had hearing parents; two children had at least one deaf parent. The three children randomly selected to participate in the second level of the study all had hearing parents. As noted in Table 1, children in the low ability group (i.e., Rosa, Malcom, and Joey) ranged in age from 5 years 9 months to 5 years 11 months; one child had two younger siblings, and the other two were the only children in the family. Spoken English, with some sign support, was the primary language used in the home. One child had a cochlear implant, and the other two used hearing aids. In terms of geography, one child lived on a farm, another lived in a semirural area, and the third lived in an apartment in an urban area.
All six children had been attending school for at least 3 years at the time of the study, with the exception of the 4-year-old, who had only been in school for 2 years. Five of the six children were currently enrolled in Kindergarten classrooms at the schools they attended; the 4-year-old was in a classroom classified as preschool. Although specific data regarding socioeconomic status was not collected during this study, observationally all six families appeared to be at a similar status, at least materialistically: all families had at least one television in the home (at least two televisions were noticed in the home of each child with low mathematics ability), and two families in each group had a computer in the home. Parents’ level of education was also similar, although slightly higher for parents of children with lower mathematics ability (at least one parent of each child with low mathematics ability had a degree beyond a high school diploma).
Qualitative Data Collection and Analysis
This study made use of qualitative methodology to examine the opportunities available for the young deaf children who participated in this study to construct early mathematics knowledge. Because qualitative research is typically used in research projects that seek to examine a problem in detail and in its natural setting (Creswell, 1998), and is particularly applicable when there is high variability in the population (Mertens, 1998), it was found to be particularly well suited to the study described in this article. Case study research, in particular, has been used previously in research to examine related areas such as the informal learning of mathematics in the home (Aubrey et al., 2003; Benson & Baroody, 2002; Phillips & Anderson, 1993) and the family/communication experiences of deaf children (Blackburn, 1999; Evans, 1994, 1995, 1998) and was, therefore, the approach chosen for this study. This study was designed to be exploratory in nature.
Instrumentation.
To examine the opportunities available for early mathematics learning for each child under similar conditions, data for the portion of the study described in this article were collected through use of a planned problem-solving activity, "This is In, This is Not," from the book "Family Math for Young Children" (Coates & Stenmark, 1997), that parents were asked to conduct with their deaf children. The mathematical purpose of the activity was to observe, describe, and sort into categories. The written description of the activity suggests that parents collect a variety of items (i.e., toys, kitchen utensils, food products) to be used during the activity and then designate a sorting space. The activity takes place as parent and child take turns sorting items into the sorting space. The parent might begin, for example, by picking up a blue car and saying, "This is in (the sorting space) because it is blue." According to activity directions, all items are sorted dichotomously as either belonging to a designated category or not. This activity was chosen due to its focus on informal mathematics, the use of categorization in particular. As designed, the activity lends itself to discussions regarding making comparisons and finding relationships. For each participating family, this activity was videotaped and later transcribed.
Each family received the written activity description at least 1 day prior to implementation. In order not to influence the manner in which parents implemented the activity, the researcher did not discuss the activity with parents beyond what was written on the instructions that all families received. When the researcher arrived at the home to videotape the activity, however, she asked if there were any questions. No parent expressed confusion regarding the activity. The researcher set up the camera and then in order to maintain an interaction environment that was as natural as possible, the researcher left the room while the activity was conducted. The researcher was called back into the room when the activity was completed, at which time the camera was turned off. The duration of videotaped interaction ranged from 7 minutes to 32 minutes, with an average duration of 20 minutes per family.
Unfortunately, when the videotapes were reviewed, it was discovered that Andrew's parent did not conduct the activity described. During the time the videotape was running, the child appeared to be engaged in a free-play situation with his mother. Because this interaction was not comparable to the other five families, data from Andrew's videotape for this task were excluded from analysis.
Data coding and analysis.
The entire set of videotaped data collected during the problem-solving activity was transcribed. Then, during the process of coding, tags or labels were used to assign meaning to data regarding opportunities for early mathematical learning to which the children were exposed. Codes were assigned to data marking mathematical content in the areas of: number/counting, quantity, time/sequence, and categorization. Mathematically based codes were derived from Early Childhood Mathematics: Promoting Good Beginnings (Retrieved, 2006), a joint position statement of the National Association for the Education of Young Children and the National Council of Teachers of Mathematics. Codes were assigned to data marking evidence of MLEs in the areas of: intentionality/reciprocity, transcendence, meaning, mediation of feelings of competence, and mediation of regulation of behavior. Codes marking aspects of mediation were derived from the Criteria for Observation of Mediated Learning Experience in Infancy and Early Childhood (Feuerstein and Rand, 1997).
To establish credibility (Guba & Lincoln, 1989) for data collected during the portion of the study described in this article, a variety of "checks" were in place. These checks took the form of member checks and code checking.
Member checks (Mertens, 1998) were incorporated into the study in order to ensure that the perspectives and viewpoints of participants were accurately recorded. When the videotapes from the planned activity had been transcribed, the transcript was shared with parents of participants to verify that the information recorded reflected a typical portrayal of their interaction with their child and accurately captured the child's knowledge and behavior. Four families signed and returned the transcripts by mail indicating their agreement. The remaining two families were contacted by phone. They also indicated their agreement with the information recorded.
Reliability (Wiersma, 2000) was measured in two ways during this study. First, because the language used in the majority of videotaped data was ASL, 25% of each videotape was reviewed to ensure that transcriptions were accurately recorded into English. The reviewer was a deaf faculty member and instructor of ASL in the department of linguistics at a university. All transcripts were found to be accurately recorded.
Second, an expert in the field of deaf education was solicited to perform a reliability check (Miles & Huberman, 1994) on data that were coded for demonstrating reference to mathematics content and use of MLEs. Of the five transcripts that provided the data for the portion of the study described here, a stratified random sample of two transcripts (i.e., one transcript from the high ability group and one transcript from the low ability group) were selected for review. After the expert coded the transcripts, differences were discussed until a mutual agreement was reached; adjustments were made to the coding as necessary.
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During data analysis, it became evident that the incorporation of mathematically based concepts and provision of MLEs were both critical to the opportunities available for the young deaf children who participated in this study to construct early mathematics knowledge. Each of these areas will be discussed in turn below.
Incorporation of Mathematically Based Concepts
Transcripts were coded for evidence of: "reference to numbers/counting," "reference to quantity," "reference to time/sequence," and "reference to categorization." As the data presented in Table 2 demonstrate, these mathematically based concepts were incorporated to varying degrees by each of the five families. The definition of each of the four mathematically based concept areas, and the manner through which it was addressed by each family, is discussed below:
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Reference to numbers/counting.
Any aspect of parent–child interaction during the problem-solving activity that included reference to a specific number, or required the child to count, for example, by answering the question "how many," was incorporated under the theme of reference to numbers/counting. As demonstrated in Table 2, Sara's family used almost twice as many references to number as any other family. This is largely due to the manner in which her parents framed the activity, through the creation of informal addition problems that required Sara to combine groups and determine the total. For example, Sara was asked to, "Pick two pink things," then to get, "Two red cans," and finally to determine, "How many altogether?"
Although with less frequency, Emma's family also used the question, "how many?" to encourage their daughter to determine group size after putting various groups of items together. For example, Emma was asked, "How many are there?" after putting all of the "big" things together. Emma's parents were also observed to prompt their daughter with comments such as, "There's one more," if an item that belonged in a group was left out.
Malcom experienced references to numbers/counting with the same frequency as Emma, although these references served a different purpose. Malcom was asked to retrieve a specific quantity of items, for example, "I want 2 horses," rather than being asked to determine how many of a specific item was present. With less frequency, the same was true for Rosa and Joey's experience with the activity. The children were directed to, "Give me 6" and, "Two ... red," respectively, but were never requested to determine how many items were in a group.
Reference to quantity.
Any reference to a quantity greater than one, without the use of numbers, was counted under the theme of reference to quantity. As demonstrated in Table 2, Emma's family used language making reference to quantities with considerably greater frequency than any other family. Quantities were referenced in a variety of ways. For example, Emma was asked, "Is there more, or is that it?" when she finished adding items to a group. As Emma was working putting items together, her parents were observed to make comments such as, "Many, many." They also guided the activity by requesting that Emma find "everything" that belonged in a certain group, or find "all" the items that belonged, rather than stating a specific number.
Although with less frequency, Sara's family also used language referencing quantity. During the activity, Sara was requested to make a "group" of red things. Her parents were also observed to reference, "Everything here" while directing Sara's attention to the next task.
As Table 2 demonstrates, Rosa and Malcom did not experience use of quantitative language during the activity. Joey experienced one reference to a quantitative term when his father asked him if there were any "more" red items present.
Reference to time/sequence.
Any aspect of parent–child interaction during the problem-solving task that included a reference to placement of an event in time or sequence was counted under the theme of reference to time/sequence. As demonstrated in Table 2, Malcom's mother made reference to time/sequence with the greatest frequency. She did this largely through comments that were designed to establish a system of turn taking, such as, "I'm going to go first," "My turn," and, "Your turn." In addition, Malcom, Rosa, and Joey were each told more than once to, "Wait," "Wait a minute," or, "You need to wait." Rosa and Joey's parent also used the term, "Now" to direct their children's attention, saying, for example, "Now look."
Emma's parents also used the word/sign "now," although for a different purpose. For Emma, the word/sign appeared to be used to structure the sequence of the task or to transition into a new category, for example, "Okay good, ok, all of these are big, now how many are small?," or, "Now, which has a funny feel to it?" In addition, Emma's parents used language referencing time/sequence to encourage the child to recall prior learning. For example, when Emma was having difficulty recalling the meaning of a particular concept, her parents prompted her with, "Before we talked about ..."
Sara's family made use of the word/sign "now" to signal a transition saying, for example, "Okay, those are hard, now give me one soft." They were also observed to reference change over time asking, "You haven't learned that yet?" in response to their daughter's obvious discomfort when a subtraction problem was attempted.
Reference to categorization.
The problem-solving activity was designed to explicitly facilitate the use of categorization skills. Nevertheless, as demonstrated in Table 2, children were asked to use categorical grouping with varying degrees of frequency. Emma's family used the greatest quantity of categorical groupings, creating 15 different categories throughout the activity. Specifically, the items were sorted by the following attributes sequentially: big, small, "funny feel," smooth, sticky, light or bright in color, dark in color, short, long, noisy, red, things you can eat, toys, heavy, and things that roll. No category was repeated throughout the activity and each (i.e., with the exception of things that were sticky and things that you eat) included more than one item. Because Emma's parents set up the activity to include a variety of materials, such as toys (cars, stuffed animals, etc.) and tools (hammer, wrench, screwdriver, etc.), the same materials were sorted and resorted in a variety of ways throughout the activity. For example, a stuffed animal that was classified as big was also classified as a "toy." Emma's parents appeared to take advantage of the activity to teach their daughter categorical labels for concepts that were new to her, the attribute of "bright" or "light" in color being one of these concepts. In addition to creating categories, Emma was also requested to explain her placement for items within a particular category.
As demonstrated in Table 2, the other four children explored considerably fewer categories during the activity. Sara was asked to explore four categories: pink, red, hard, and soft. Rosa explored two categories, dolls with long hair and dolls with short hair. Although other attribute labels were mentioned, they were used to examine the items individually rather than categorically. For example, Rosa's mother asked, "Is this yellow, or not yellow?" while holding a specific doll. Malcom also explored two categories during the activity, hard and soft. Similar to Rosa, other attribute labels were mentioned but under a specific direction, for example, "Find something blue." Although this line of questioning required Malcom to examine items individually, he had little opportunity to practice grouping items categorically. Joey explored one category during the activity, red things.
Mediated Learning Experiences
In addition to the content incorporated, it is likely that the manner in which parents facilitated the problem-solving activity contributed to the mathematics knowledge that their young deaf children were able to take away from it. In the following sections, the manner in which parents facilitated the activity will be discussed in relation to the MLEs that their children were exposed to during the activity. MLEs will be discussed in terms of the following five criteria: intentionality and reciprocity, transcendence, mediation of meaning, mediation of feelings of competence, and mediation of regulation of behavior.
Intentionality and reciprocity.
During their implementation of the problem-solving activity, parents were found to incorporate the intentionality and reciprocity dimension in a variety of ways and to varying degrees. Use of this dimension was evident in the materials that parents used to organize the activity, as well as the manner in which they invited responses from their child, and adapted the activity to address the child's needs and interests. Perhaps most critically, successful use of intentionality and reciprocity was evident when children demonstrated recognition of the goal for the activity by responding accordingly to their parents’ attempts at mediation.
In terms of materials, Emma's parents set up the activity to include toys (cars, stuffed animals, etc.) and tools (hammer, wrench, screwdriver, etc.). The wide variety of materials that were used facilitated the type of categorical arrangements that the family was able to organize during the activity. The task appeared to be set up based upon the goal of categorization as the same materials were sorted and resorted in a variety of ways throughout the activity (i.e., a hammer was included in three different sorts: hard, long, and, "not a toy"). Although they organized the activity differently, Sarah's parents also used a wide variety of materials including toys (i.e., cars, stuffed animals) and soda cans. Grouping was still required; however, the goal of the activity appeared to be different as Sara was asked to engage in concrete problem solving involving primarily addition and counting concepts.
Parents of the other three children organized the activity using materials that were more homogenous in nature. Rosa's mother, for example, chose to use her daughter's favorite doll collection (Disney Princess dolls). Although these materials were engaging for the child, the homogeneity of the toys restricted the ways that the materials could be grouped. Although the pair began with a sort that was categorical in nature, separating the dolls with long and short hair, the parent's goal for the activity appeared to change later in the task. Eventually the activity transitioned to the labeling of attributes. Rosa's mother would pick up individual items and ask her daughter dichotomous questions about each: small or not small,? yellow or not yellow,? noisy or not noisy,? and so on.
Although there was more variety in the materials used by Malcom's parent (i.e., a variety of small toys including sponge letters and a toy dinosaur), the items were still relatively homogenous as all toys were used. As with Rosa, the parent's goal appeared to change during the activity. She began with a sort that was categorical in nature, sorting hard things, then soft things; then the activity changed. Malcom and his mother began taking turns telling each other one thing to put in the sorting space (e.g., "blue"). As with Rosa, the activity became centered on the labeling of attributes (e.g., yellow, big, blue square toy).
A variety of materials were used by Joey's parents during the task, including small toys and household items; however, the items were all similar in the one categorical attribute that they chose to sort, "red." Among the items used for the activity, everything that was red was similar in shade. In contrast, when Emma was asked to sort red things, the task was more complicated as the items she had to choose from varied in color and shade. This was evident in the following conversational exchange that occurred when Emma picked up a wrench and showed it to her mother:
Mother: Is this red? (shows the wrench to Emma's father) I think this is more brown. What do you think?During this exchange, Emma had the opportunity to observe her parents as they engaged in a brief problem-solving discussion. Humor was also used to make the learning experience fun and entertaining (handing the child the Mr. Potato Head glasses to assist her in "looking carefully").Father: Let me see (takes the wrench). No, that's brown (puts it to the side). Look carefully. (hands his daughter Mr. Potato Head glasses)
While implementing the activity, parents used different techniques to invite responses from their children. In addition to humor as described in the exchange above, Emma's parents used a variety of question forms to keep their daughter interested and involved. For example, when sorting large then small items, Emma was asked, "Where's big?" then, "How many are small?" Later on, when Emma's attention appeared to be waning a bit, her mother posed the question, "I'm curious, can you eat any of these things?" The question, to which Emma responded, "Of course not!" appeared to be absurd to the child and served its purpose of rearousing her interest.
Similarly, Sara's parents also used questions to involve the child in the task. For example, Sara was asked, "Can you make a group of red things?" then to figure out, "Altogether, how many?" Throughout the task, Sara was asked to combine and count different groups of items, then later, to take a group apart. Sara was attentive to her parents’ requests throughout the activity and responded accordingly.
Malcom, Rosa, and Joey's parents appeared to rely on imperative statements such as, "Pay attention," "Look," "Come on," "Sit down," and, "Stop," to direct and maintain their children's attention. Imperative statements were also used to involve the children in the task by directing their attention toward what they were expected to do next. For example, Malcom was told to, "Find something hard" and to, "Get the horse." Similarly, Joey was told, "Want red" more than once. Rosa's mother used an interrogative format to invite her daughter to participate; however, the child often appeared confused by the request and would respond by repeating back part of her mother's statement. For example, when Rosa's mother asked her, "Why inside? Why is it inside?" after putting a doll with long hair inside the sorting space, Rosa repeated, "Inside?"
Transcendence.
During their implementation of the problem-solving activity, only one family was found to take the activity beyond the "here and now" and incorporate the dimension of transcendence. Use of this dimension was evident in the explanations of concepts that Emma's parents incorporated into the activity, as well as the manner in which they encouraged their daughter to use critical thinking skills, to make comparisons, use past experience, and explain the reasoning behind her actions.
During the activity, Emma's parents incorporated reference to concepts that their daughter had not yet mastered. While categorizing items by these new conceptual attributes, opportunities for explanations emerged. One such opportunity arouse when Emma was asked to sort things that were light or bright in color. When Emma placed a dark-colored item into the group, her mother explained that, "We want really light in color, like you know yellow, or light blue, maybe orange." Later, when her daughter again placed a dark-colored item in the sorting space, her mother explained again, "Blue, red, brown, black, they're all dark. We want light, bright colors. That's really not bright." Throughout this exchange, Emma's mother guided her to the understanding of a new concept that had the potential of transcending the current learning experience. This new concept was referenced again later in the activity when Emma was asked to sort dark items. When she put a light-colored item into the sorting space, she was reminded of what she previously learned regarding dark and light colors, "Before we talked about bright and dark ..."
To develop her understanding of another new concept, "smooth," Emma was encouraged to make comparisons. Pointing to two objects, Emma's father told her, "... feel this one, then feel that one." Comparative language was used again later when Emma was asked to, "Look for short things, not long, short." Through use of comparative language and behavior, Emma was not only exposed to a new concept but also a new strategy for learning.
Mediation of meaning.
During their implementation of the problem-solving activity, only one family was found to make use of the dimension of mediation of meaning.
Although the problem-solving activity required only sorting, Emma was encouraged to make use of critical thinking skills to explain why she sorted items the way she did. For example, after sorting "noisy" items, Emma was asked, "Why is that noisy?" After sorting "toys" and "not toys," Emma was asked, "Why aren't those toys?" In being asked these questions, Emma was learning that acquisition of meaning goes beyond the concrete information that one acquires through their senses.
Mediation of feelings of competence.
During their implementation of the problem-solving activity, all families were observed to use praise, such as, "Good" and "Very good" in response to their child's correct response to a question or request. In addition to this type of praise, one family also used language that served the purpose of mediating their child's feelings of competence.
While creating a group of big things, Emma initially looked to her mother for approval after each item she picked up. In response, Emma's mother said, "Decide for yourself, which?" Later on, while creating a group of smooth things, Emma again looked for approval. This time she was told, "It's your decision." Finally, while grouping dark-colored items, Emma again looked for her mother's agreement. Again she was told, "Why do you keep looking at me? You know." Critical is that the later two concepts, "smooth" and "dark," were new concepts for Emma. Yet even in these unfamiliar areas, Emma's mother communicated that she had faith in her daughter's ability to make decisions, thereby mediating her child's feelings of competence.
Mediation of regulation of behavior.
During their implementation of the problem-solving task, two families were found to engage in processes to mediate the regulation of behavior. Both families were found to emphasize a need for precision and orient their children's attention to specific information required to finish the task completely and accurately. In addition, use of inhibitions was encouraged as thinking before acting was emphasized.
When Emma thought she had found all the members of a particular group, her parents oriented her attention toward items she had missed using pointing gestures and comments such as, "There's one more", "Look over there," and, "I see something else over here." Similarly, Sara's parents pointed out when she missed a member of the group she was making saying, "You missed one" and pointing in the general direction of the missing item. When Sara struggled to solve a problem related to subtraction concepts, rather than giving her the answer or giving up on her daughter's readiness, her mother broke it down, directing Sara's attention to the pile of items left when part of the group had been removed. In this manner, both sets of parents were mediating the need for precision and accuracy. Precision was again emphasized as Emma's mother told her, "You have to pay attention. Look at them all carefully, how many,?" smiling and nodding along with a, "Be careful" after Emma self-corrected a sorting error and pointing out that, "The top and bottom are the same" when her daughter accidentally counted a tool twice.
While Malcom, Rosa, and Joey all experienced language use that stressed control of impulsivity (i.e., each was told more than once to, "Wait," "Wait a minute," or "You need to wait"), an emphasis on thinking before acting was not included with these requests. When Emma was encouraged to control her inhibitions, there was an emphasis on slowing down and thinking. This was most apparent as Emma surveyed the items available for her to sort to find those that qualified as light or bright. Rather than telling her answers, Emma's mother gave her time to think, occasionally prompting her with, "There's more." Emma silently reviewed the items, pointing to each in turn, shaking her head "no," until she got to a toy that was light yellow. She paused, then picked up the yellow toy to place it in the group of light/bright items.
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The opportunities available for the young deaf children who participated in this study to construct early knowledge of mathematics varied in both quantity and quality.
This variation was evident both through the mathematically based concepts that the children were exposed to as well as the MLEs that were used to guide their learning. Given the small number of participants in this study, however, these findings should be accepted with caution.
Although, as seen in Table 2, the mathematically based concept area of reference to time/sequence was used more frequently by parents of children with low mathematical ability (Rosa: 11 instances; Malcom: 15 instances) than by parents of children with high mathematical ability (Emma: 10 instances, Sara: 6 instances), overall the four mathematically based concepts (reference to numbers/counting, reference to quantity, reference to time/sequence, and reference to categorization) were used more frequently by the parents of the two children with high mathematical ability (Emma: 47 instances; Sara: 30 instances) than the parents of the children who demonstrated lower mathematical ability (Rosa: 16 instances; Malcom: 26 instances; Joey: 7 instances). More critical, however, may be the quality of the manner in which these concepts were used. Across all four areas, Emma and Sara were exposed to mathematically based concepts in a style that was more purposeful and meaningful than that experienced by the three children with low mathematics ability.
Emma and Sara were both required to use critical thinking skills while being exposed to mathematically based concepts during the problem-solving activity. In the area of number/counting, for example, Emma and Sara were both required to determine how many of a certain item was present in a group. They were also exposed to language referencing quantity, requiring cognitive understanding of abstract terms such as, "everything," "all," and "group." In addition, they were exposed to linguistic terms used to transition between activities (such as the purposeful use of "now") and reference prior learning (such as the use of "yet" and "before") requiring them to determine how to cognitively sequence events in time. For Emma in particular, the categorization aspect of the activity provided multiple opportunities for the child to engage in high levels of thinking, as she was required to consider the various attributes of the items she was presented with, and consider each attribute individually to determine an item's eligibility for membership in a group.
In contrast, while the three children with lower mathematical ability were also exposed to mathematically based concepts, this exposure occurred at a lower cognitive level. Malcom, Rosa, and Joey were requested to provide a prescribed quantity of items, but never required to determine how many items were in a group. These three children experienced almost no exposure to quantitative language, leaving them no opportunity to develop awareness that the concept of quantity can be expressed without the use of number. In terms of the concept area of "time/sequence," while, especially for Malcom, linguistic terms were used with relative frequency, these terms were used concretely, to signal turn taking, for example, rather than placement in time or to transition between activities. This is in comparison to the two children with higher mathematical ability, for whom the use of these terms was more purposeful. Most critical, however, is that children with lower mathematical ability experienced few opportunities to engage in categorization, even though this was the explicit purpose of the activity. Instead, for each child, the activity became more of a practice in labeling, a skill on a much lower cognitive level as it requires use of memorized knowledge rather than higher-order thinking skills such as comparison or analysis.
Differences in the manner in which the parents incorporated mathematically based concepts into the problem-solving activity could be a function of the parents’ ability to communicate easily with their children. The hearing parents may not have known the appropriate signs to use to reference quantity or categories or how to reference time purposefully using a visual language and therefore found it easier to limit the use of these concepts altogether as they implemented the activity.
Lack of access to easy communication could also be, in part, responsible for the limited use of MLEs incorporated by parents of children who demonstrated lower levels of mathematical ability. Parents of children in the high group made sure that their children understood the task they were doing. They reoriented their child's attention as needed and varied the questions asked to keep their children interested and involved. Emma and Sara both appeared to understand the goals established for the activity as they responded appropriately to their parents’ requests. This reciprocity, however, appeared to be lacking in the interactive exchange between parents and children who demonstrated lower levels of mathematical ability. Children in the lower group did not always appear to clearly understand the goals for the task established by their parents. This was most clearly evidenced by Rosa, who repeated her mother's question rather than answering it, and Joey, who frequently walked away from his parents during the activity. Although parents of children who demonstrated lower levels of mathematical ability may have recognized that their children did not understand the task the way it was presented, communication challenges may have prevented them from rewording the request differently, thereby leaving them with the strategy of reorienting the task by reducing the level of cognitive challenge, as Rosa and Malcom's parents did when they turned the categorization activity into a labeling task.
However, it is necessary to note that the challenges associated with providing MLEs to children in this study went beyond language. Although more MLEs were used by families of children who demonstrated high mathematics ability, one of these two families only included two of the five critical dimensions of mediation. With limited mediation, less opportunity was available for the learning of mathematically based concepts and overall construction of early mathematics knowledge. This is especially critical considering that the references to high and low mathematical ability, as used in this study, were relative and applicable only to the study participants. Out of all the young deaf children who participated in the larger study upon which the study described in this article is based, none demonstrated levels of mathematical ability that were higher than average based on norms established for hearing children on scores from the TEMA-3. Although this could, in part, be due to the test not being developed for, or normed on, deaf children, it could also demonstrate a need for mathematics intervention even before young deaf children begin formal schooling.
In general, the findings from this study indicate a possibility that young deaf children may be lacking opportunities within their home environments to engage in high levels of thinking. This being the case, it is likely that when these children encounter learning situations in school that require them to use more than concrete information, they will experience difficulty. Not having learned how to process information beyond that which can be acquired through their senses, they may be unable to effectively engage in formal learning tasks that require them to think more critically. However, it is important to remember that this study was designed to be exploratory in nature. Although the results of this study cannot be generalized beyond these five participants during one specific activity, they do indicate a need for more research in this area, with a larger number of participants.
Need for Future Research
In particular, research is needed to examine the influence of parent education on young deaf children's early mathematics learning and the long-term effects of this influence on mathematics achievement once the children reach school age. Specifically, a study is needed that tracks the impact of parent education in the following areas: (a) incorporating MLEs into everyday learning and (b) bringing activities that incorporate mathematically based concepts into young deaf children's early learning experiences.
It is possible that young deaf children are beginning formal schooling already performing at a disadvantage in terms of their understanding of foundational mathematics concepts and early mathematical thinking. As findings from this study suggest, families play a substantial role in mediating early mathematics learning for young deaf children and may ultimately be responsible for building the foundation that is necessary for later achievement.
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Department of Education, University of Pittsburgh.
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The author wishes to thank Dr. Claudia M. Pagliaro (Michigan State University) for her feedback on previous versions of this manuscript. In addition, the author wishes to thank the families of the six children who participated in the study upon which this article is based. No conflicts of interest were reported.
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