Journal of Deaf Studies and Deaf Education Advance Access originally published online on June 4, 2007
The Journal of Deaf Studies and Deaf Education 2007 12(4):432-448; doi:10.1093/deafed/enm022
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Visual–Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Students
National Technical Institute for the Deaf, Rochester Institute of Technology
Bowling Green State University
National Technical Institute for the Deaf, Rochester Institute of Technology
University of Rochester
Correspondence should be sent to Gary Blatto-Vallee, Department of Science and Mathematics, National Technical Institute for the Deaf, Rochester Institute of Technology, Rochester, New York 14623-5604 (e-mail: gcvntm{at}rit.edu).
Received October 21, 2006; revised April 3, 2007; accepted April 6, 2007
This research examined the use of visual–spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual–spatial "schematic" representations that encode the spatial relations described in a problem versus visual–spatial "pictorial" representations that encode only the visual appearance of the objects described in a problem. A total of 305 hearing (n = 156) and deaf (n = 149) participants from middle school, high school, and college participated in this study. At all educational levels, the hearing students performed significantly better in solving the mathematical problems compared to their deaf peers. Although the deaf baccalaureate students exhibited the highest performance of all the deaf participants, they only performed as well as the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level. The analysis of the students’ problem representations showed that the hearing participants utilized visual–spatial schematic representation to a greater extent than did the deaf participants. However, the use of visual–spatial schematic representations was a stronger positive predictor of mathematical problem-solving performance for the deaf students. When deaf students’ problem representation focused simply on the visual–spatial pictorial or iconic aspects of the mathematical problems, there was a negative predictive relationship with their problem-solving performance. On two measures of visual–spatial abilities, the hearing students in high school and college performed significantly better than their deaf peers.
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Introduction |
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A generalized disparity in mathematical performance between deaf students and their hearing peers is well documented in the educational literature. Traxler's (2000)
analysis of deaf and hard-of-hearing students’ performance on the Stanford Achievement Test 9th Edition shows they fell largely in the "Below Basic" and "Basic" levels on the "Mathematics Procedures" and "Mathematics Problem Solving" subtests. These levels indicate only partial mastery of the mathematical knowledge and skills in grades 1 through 9 that are fundamental for satisfactory work (p. 343). Deaf students’ mathematical difficulties continue into college. In a recent study, 79% of 248 deaf college freshmen entering associate degree programs scored below the 50th percentile on the ACT Mathematics Subtest, with 32% scoring at chance (Dowaliby, Caccamise, Marschark, Albertini, & Lang, 2000). On average, approximately 50% of the entering deaf students to associate degree college programs at the National Technical Institute for the Deaf (NTID) are required to take prealgebra or introductory algebra courses (V. Danielle, personal communication, January 11, 2007). Similar mathematical deficits for deaf students have been observed in Japan, Norway, and England (Frostad & Ahlberg, 1999; Phelps & Branyon, 1990; Wood, Wood, & Howarth, 1983).
However, hearing loss per se is not a cause of poor mathematical performance but rather more of a risk factor related to the timing, type of instruction, and learning opportunities provided to deaf students (Nunes & Moreno, 1998). Historically, 15% of profoundly deaf individuals have performed at or above average levels for mathematical performance (Wollman, 1965; Wood et al., 1983).
Additional factors potentially affecting deaf students’ mathematical learning have also been identified. Nunes and Moreno (2002) investigated young deaf children's informal mathematical skills learned prior to formal schooling and found that they lacked additive composition, additive reasoning (e.g., two more), multiplicative reasoning (e.g., three children sharing two cookies each), ratio (e.g., 1:1 correspondence), and fractions (e.g., pieces of a whole pizza)—all skills their hearing peers possessed. In another study, Nunes and Moreno (1998) showed that with appropriate instructional intervention, 68.2% of deaf students "out performed" their own predicted score according to the NFER-Nelson Age Appropriate Achievement Test. Although their findings also showed slower reaction times (RTs) for deaf students on basic numerical and arithmetic skills’ tasks, the corresponding RT graphs showed general developmental processes and processing methods similar to their hearing peers. Other recent research has looked at deaf individuals’ automatization of number through the examination of symbolic distance effects in magnitude decisions, the internal number line, subitizing (the rapid, accurate, and confident judgments of number performed for small numbers of items), and the skills involving estimation (Bull, Marschark, & Blatto-Vallee, 2005; Bull, Blatto-Vallee, & Fabich, 2006). Deaf participants’ RTs did not differ statistically from hearing people on these skills, except where research design was a factor (Bull et al., 2006). Zarfaty, Nunes, and Bryant (2004) compared the spatial and temporal numerical skills of deaf and hearing 3- and 4-year-old children and found them to be generally equal, except that the deaf children demonstrated better spatial numerical skills. Bull et al. (2006) showed that deaf learners’ observed mathematical difficulties were not a consequence of absent basic numerical skill and recommended that future investigations of deaf children's mathematical abilities should focus on the more complex relations of cognition and mathematical development.
The Importance of Relationships in Mathematics
The need to recognize and utilize relationships in mathematics is vital to one's success in mathematics. This is evident in the research literature as well as in the definitions of mathematics. The American Heritage Dictionary, Fourth Edition (2000), defines mathematics as "the study of the measurement, properties, and relationships of quantities, using numbers and symbols." The Compact Oxford English Dictionary, Second Edition (2000), defines mathematics as "the abstract science which investigates deductively the conclusions implicit in the elementary concepts of spatial and numerical relations" (p. 1048).
To be able to recognize and understand mathematical properties, one needs to be able to look at all the components of a mathematical problem and recognize each item in relation to the other relevant entities. The full meaning of any single mathematical element in a problem task can be understood only by recognizing its relation to the other mathematical elements. Unfortunately, the evidence suggests that deaf adults and children have difficulty in understanding the component relationships in multidimensional complex tasks. Over 25 years ago, research showed that deaf individuals performed significantly less well than their hearing peers on tasks that required considering the relationship between two or more dimensions (Ottem, 1980). In a more recent study that examined the reasoning of deaf and hearing college students involving 4- and 5-term series problems, Marschark and Johnson-Laird (2003) found that for both deaf and hearing students who drew external models showing relationships among the elements of the 4-term series problems, their performances were comparable; but for those students who did not draw external models, the hearing students’ performances were significantly faster. However, with the more complex 5-term series problems, the hearing students’ performances were consistently faster than that of the deaf students regardless whether external models were drawn. Marschark and Johnson-Laird concluded that these results were consistent with other research that indicate deaf individuals are less likely to make use of automatic relational processing in a variety of tasks. In a memory study, Banks, Gray, and Fyfe (1990) showed that deaf children's recall of text tended toward disjointed parts or facts instead of more meaningful whole conceptual units. Marschark's (2003) review of cognitive functioning in deaf adults and children suggests that they focus primarily on the individual words and pieces of text rather than adopting a more holistic, relational approach to abstracting the meaning.
The recent research findings of Kelly and Gaustad (2007) documents the connections between deaf college students’ mathematical skills and their reading and language skills. Their findings show that deaf college students’ language proficiency scores, reading grade level, and morphological knowledge for word segmentation and meaning were all significantly correlated with their scores on both the ACT Mathematics Subtest and the NTID Mathematics Placement Test. On a real-world practical level, the findings of Kelly and Gaustad showed that participating deaf students’ grades in their college mathematics courses were significantly associated with their reading grade level and their knowledge of morphological components of words.
Deaf students’ difficulty in recognizing connections and relationships among elements of mathematical problems occurs whether English or sign language is the presentation mode. Ansell and Pagliaro (2006) examined primary level deaf children's ability to solve mathematical story (word) problems and found that they did not connect the story language to the arithmetic functions necessary for the solution, even when a deaf signer presented it in American Sign Language (ASL). "Most of the children did not appear to view the signing of a story problem as containing any links to its solution. In fact, many children, particularly those in the less successful group, did not seem to attend to the problem at all, focusing primarily on the numbers in the problems. The deaf children ignored or did not recognize any relationship between the story and its solution, thus missing any linguistic markers that could potentially have made for an easier problem." (p. 167).
Visual–Spatial Ability and Mathematics
The long-established positive relationship between spatial ability and achievement in mathematics has been documented as one of the main factors affecting mathematical performance (Battista, 1990; Sherman, 1979). Although the prevalence of visual–spatial instructional methods contained in mathematical textbooks and everyday teacher modeling that occurs in mathematics classrooms daily across the country might be suggestive evidence as to its importance, it is not clear that everyone has a consistent understanding of what visual–spatial ability means, nor of its implications. Furthermore, the anecdotal notion that people who are primarily visual (i.e., pictorial) processors are especially apt at mathematics has long been perpetuated, but research has not substantiated such supremacy. Individuals’ ability to process information visually correlates with neither mathematical performance nor spatial ability tests (Hegarty & Kozhevnikov, 1999). In fact, Lean and Clements (1981) even found a negative correlation between mathematical visuality (i.e., processing mathematical information visually) relative to spatial ability and mathematical performance. This finding with other research evidence led Lean and Clements to conclude that "verbalizers" out perform "visualizers" on mathematical and spatial ability tests.
The negative correlation between students’ visualizing and mathematical success exists in a seemingly paradoxical relationship to what most mathematics educators do and know to be true—that is, being able to illustrate a mathematical concept visually, should aid students’ intuitive understanding of the concepts involved, and deepen overall understanding of mathematics. Both teachers and researchers agree that the use of visual representations is an important part of mathematics education because such representations appear to enhance intuition and understanding in many areas of mathematics (Krutetskii, 1976; Usiskin, 1987). Logically, if students’ intuitive understanding is bolstered, then it follows that their ability to mentally represent that concept should improve. This, in turn, should potentially affect a student's ability to apply previously learned strategies to solutions of new and unique problems. The facilitative quality of the mental representations generated by learners is an important factor in influencing subsequent transfer of acquired problem solving schema to new problems (Chen, 1999). Thus, certain visual images do influence one's understanding of mathematical concepts.
Lean and Clements (1981) sought to clarify the different types of visual representational strategies employed by students when solving mathematical problems by separating student-generated imagery into five categories: concrete imagery, pattern imagery, kinesthetic imagery, dynamic imagery, and memory of formulas. Presmeg (1986) subsequently argued that concrete imagery (vivid pictorial images of objects contained in mathematical problems) may actually focus the reasoning on irrelevant details and distract the "solver" from the main elements of the problem. Presmeg determined the most essential role in mathematical problem solving to be pattern imagery—pure relationships depicted in a visual–spatial scheme.
Several research studies have also examined the visuospatial abilities (i.e., the perception of the spatial relationships among objects within the field of vision) of deaf and hard-of-hearing individuals. Persons who use sign language have demonstrated an advantage in several visuospatial domains (Marschark, 2003). Emmorey, Kosslyn, and Bellugi (1993) found that both deaf and hearing users of ASL were faster in generating mental images compared to nonsigners and that mental rotation skills (recognizing and matching objects that were visually rotated to another view angle) were enhanced in both deaf and hearing users of ASL. Talbot and Haude (1993) showed that the level of sign language expertise affected mental rotations of three-dimensional block figures (for further review, see Marschark, 2003). Thus, the enhanced visuospatial ability in users of sign language theoretically has the potential to positively influence deaf students’ mathematical ability.
Hegarty and Kozhevnikov (1999) examined students’ use of two types of visual–spatial representations in mathematical problem solving: (a) "schematic" representations that encoded the spatial relations described in the problem versus and (b) "pictorial" representations that only encoded the visual appearance of the objects described in the problem. Hegarty and Kozhevnikov tested 33 hearing students aged 11.5–13 years from an all-boys school in Dublin, Ireland, with a 15-item test, which they refer to the Mathematical Processing Instrument (MPI), consisting of word problems that were taken from other earlier studies. The 15 mathematical problems were administered to individual students one at a time with a 3-min time limit for each problem. The students were asked to "show all of their work." Following the solution of each problem, students were interviewed with a fixed set of questions designed to help assess the visual–spatial conceptualization of their problem-solving effort. The students’ paperwork for each problem was evaluated and categorized as either a visual–spatial schematic representation that encoded the spatial relations of the problem or as a visual–spatial pictorial representation that only encoded the visual appearance of the objects in the problem. Hegarty and Kozhevnikov found a significant positive correlation between visual–spatial schematic representation and successful mathematical problem-solving performance. Conversely, they found a slightly significant negative correlation for visual–spatial pictorial representation and mathematical problem solving.
A primary focus of this study is whether student generated visual–spatial schematic representations in mathematical problem solving are beneficial to students beyond the middle school level, particularly for deaf students whose learning depends to a large part on visual processing of information.
Research Questions
This study replicates and developmentally extends the research of Hegarty and Kozhevnikov (1999) by examining deaf and hearing students’ ability to see, generate, and use relationships in mathematical problem solving. The research questions of interest were:
- How do deaf and hearing students in middle school, high school, and college perform developmentally on the Hegarty and Kozhevnikov 15-item test of mathematical problems and two visual–spatial ability measures?
- Are there any developmental differences in the visual–spatial schematic and pictorial representations used by deaf and hearing students in middle school, high school, and college while attempting to solve the 15 mathematical problems?
- Is there a predictive relationship between the students’ grade levels and their use of visual–spatial schematic or pictorial representations with their score on the 15 mathematical problems?
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Participants
A total of 305 student participants took part in this study of spatial relational representation in mathematical problem solving. Within this total, there were four groups of deaf students and three groups of hearing students representing middle school, high school, and college students as shown in Table 1.
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The deaf and hearing groups were parallel in educational level with one exception. Two college-level deaf groups participated due to the differences in their entry requirements and programs of study. This grouping recognizes the important distinction that deaf students with different ability levels are enrolled in a variety of college programs. Group D3A comprised deaf students enrolled in programs leading to no higher than an associate's degree, and the entry requirements for D3A students to the associate level programs are an overall 8th grade ability level. Group D3B comprised deaf students enrolled in baccalaureate programs at the same university and are students accepted directly into their baccalaureate programs by meeting the regular university entry requirements. On average, deaf students in Group D3A exhibit lower English language proficiency relative to the deaf baccalaureate-level students in Group D3B (NTID Annual Report, 2005). The hearing college students in Group H3B were all enrolled in traditional 4-year bachelors degree programs and were parallel to the deaf students in Group D3B. No hearing group was parallel to the associate-level deaf Group D3A (i.e., no group designated H3A).
Table 1 also provides the average age and gender breakdown for the seven participant groups. In addition, the pure tone average (PTA) hearing loss for the better ear unaided is provided for the four deaf participant groups. The group averages for PTA hearing loss indicate that the deaf students are in the severe to profound hearing loss categories. On average, the deaf students at each equivalent educational level were about a year older than their hearing peers. No gender differences in performance on the research measures were observed at any level.
All students were compensated for their participation—this consisted of cash payments for the college students and deaf students in the residential/day program and movie tickets for the hearing participants in the three public school programs. The hearing middle and high school students were not compensated with cash due to the policies of the participating schools.
Research Materials
The research instruments for this study were three paper-and-pencil tests. The principal research measure consisted of 15 mathematical problems adapted from the MPI employed by Hegarty and Kozhevnikov (1999) with middle school students. Whereas the 15 mathematical problems contained essentially the same content as the Hegarty and Kozhevnikov word problems, the text of several of them was modified for cultural/language reasons. This was done in order to reflect American English usage rather than the original British English terminology that might be confusing, especially to deaf students (e.g., lorry was changed to truck, tin changed to can). The 15 mathematical problems were presented in a test booklet with one question per page. As shown in Table 2, the 15 mathematical problems ranged in length from one to four sentences, with 22–51 words for an average of 35.2 words per problem. The reading grade level of the 15 mathematical problems ranged from 1.7 to 7.4. Both the readability and mathematical difficulty of the problems were targeted to the middle school participants (same as Hegarty & Kozhevnikov, 1999), which is also an appropriate reading level for use with the deaf college students pursuing associate degrees, who ranged in assessed reading ability from 7.0 to 9.0 grade levels. Prior to beginning the study, the research team met with the teachers of the deaf middle school and high school students. The teachers reviewed the 15 test items and confirmed that the test problems were typical to the type of problems their deaf students were working with in their mathematics courses and therefore within the context of their learning experiences.
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Test booklets with one of four randomly ordered sequences of the 15-word problems were randomly distributed to students. This procedure addressed potential order effects of test item presentation and mitigated against any bias in assigning test versions to participants. Students were tested in a large group setting. They were instructed to explain and show all work in the space provided on the page below each problem. Forty-five minutes were allocated for the test, providing about 3 min for the solution of each mathematical problem. Table 2 presents the exact wording of these 15 mathematical problems.
In order to assess students’ visual–spatial abilities, two other paper-and-pencil tests were also administered. The first was the Primary Mental Abilities Spatial Relations Test (Optometric Extension Program, 1995), a 25-item form completion task. For each item, participants are presented with the line drawing of an incomplete square and then instructed to choose the corresponding missing part from five choices that would complete the square. Per the test guidelines, students were given 6 min to complete the PMA Spatial Relations Test.
The second test was the Revised Minnesota Paper Form Board Test (Likert & Quasha, 1994), a 64-item test designed to test part–whole relationship skills. For this test, participants were instructed to consider the component parts of a figure and then discern the correct form of the whole figure if those parts were pieced together. The format of this visual–spatial test is multiple-choice. Each of the 64 items shows the component parts of a single figure followed by multiple choices of whole figures from which participants must select the correct whole figure that consisted of the component parts. Two A and B versions of this test with the same content items ordered differently were used in this study and were distributed alternately to the students. Per the test guidelines, students were given a maximum of 20 min to complete the Revised Minnesota Paper Form Board Test.
Students were administered all three tests in a single 1.5-hr session. The sequence of test administrations was consistent for every student: First, the 25-item PMA Spatial Relations Test (6 min); second, the 64-item Revised Minnesota Paper Form Board Test (20 min); and third, the 15-item MPI (45 min).
All three tests were scored for the total correct. Additionally, each student's "shown work" for each of the 15 problems on the MPI was evaluated for the type of representation they generated in their solutions per the following four categories:
- Visual–Spatial Schematic Representation (R): If the student's work depicted the relationships between objects and/or parts of an object described in the problem, it was judged to be schematic. If a student's work showed that he/she was approaching a problem utilizing the relationships between several parts of the problem, it was judged to be schematic. The relationships utilized or the answer provided by the student did not need to be the correct ones. Some sort of a visual had to be present in order for a representation to be judged schematic.
- Visual–Spatial Pictorial Representation (P): If the student's work consisted of a picture that depicted the objects described in the problem but depicted no type of relationships between objects or pieces of information contained in the problem, then it was judged to be pictorial.
- Nonvisual Representation (NV): If a student showed either no pictorial information or no work at all but still provided an answer (correct or incorrect), it was judged to be nonvisual.
- No answer or work (.): The student left the question blank.
- Visual–Spatial Pictorial Representation (P): If the student's work consisted of a picture that depicted the objects described in the problem but depicted no type of relationships between objects or pieces of information contained in the problem, then it was judged to be pictorial.
To assure reliability in the judgment categorization of the students’ problem representations, three people were trained to assess the students’ work per these four categories. All three raters assessed the same sample of 50 students’ shown representation work for solving the 15 mathematical problems. Interrater reliability of .96 and .97 for the second and third raters, respectively, with the primary rater was calculated using a measure of correlation as the procedure (Borg & Gall, 1983). Such high interrater reliability suggests that the criteria for categorizing the students’ problem representations were clear and defined more objective than subjective aspects. The few differences in scoring were discussed and resolved. The primary rater assessed all remaining tests per the agreed-upon criteria. When students’ representations of the mathematical problems appeared ambiguous to the primary rater, they were set aside and reevaluated by a second rater to assure consensus in the assessment of every student's representations.
The number of judged representations for each of the four categories above (sum total = 15) was then entered into the data files for every student, so the manner in which they represented each problem in trying to solve it could be analyzed relative to their total score on the 15 mathematical problems. Thus, in addition to each student's total score for solving the 15 problems, they potentially could have four representation scores for (a) visual–spatial schematic, (b) visual–spatial pictorial, (c) nonvisual, and (d) no-work. All results that follow are reported as group mean performance data, not as individual performances.
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Results |
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Mathematical Problem Solving, Visual–Spatial Tasks, and Student Representations
"How did the deaf and hearing students in middle school, high school, and college perform on the 15-item mathematical problems and two visual–spatial ability tasks?" Table 3 presents the means and standard deviations of the participating students’ test scores by educational grade levels for the 15 mathematical problems, the 25-item PMA Spatial Relations Test, and the 64-item Revised Minnesota Paper Form Board Test. For the 15 mathematical problems, no ceiling effect occurred for any of the participating student groups. Table 3 also shows the group means and standard deviations for the student generated visual–spatial schematic and pictorial representations in solving the mathematical problems.
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As presented at the bottom of Table 3, all the one-way analyses of variance (ANOVAs) that show significant overall differences among the seven participating student groups on the mathematical problem-solving task, the two visual–spatial measures, and the students'-generated use of visual–spatial schematic and pictorial representations. Post hoc pair-wise comparisons show that the hearing students at all grade levels performed consistently and significantly higher than their deaf counterparts on the mathematical problem solving, the two visual–spatial tasks, and the use of visual–spatial schematic representations. In contrast, the deaf students generated and used significantly more visual–spatial pictorial representations consistently across all grade levels with one exception. The deaf college bachelor's degree level students (D3B) averaged the same use of pictorial representations as their hearing bachelor's degree level peers (H3B). The results presented in Table 3 indicate that hearing students appear to have a consistent and significant edge over their deaf peers on the two visual–spatial ability measures and the use of visual–spatial schematic representations in mathematical problem solving, even though according to the research previously cited, sign language users have demonstrated advantages in several visuospatial domains. Also at the bottom of Table 3 are the results of a two-factor ANOVA showing a significant interaction for the main effects of hearing status and educational level for the number of schematic representations generated by the students in solving the 15 mathematical problems.
Table 4 provides the correlations among the students' scores on the 15-item mathematical test, PMA test, Revised Minnesota Paper Form test, and the students’ use of visual–spatial schematic and pictorial representations in solving the mathematical problems.
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Note that the students use of visual–spatial pictorial representations correlates negatively with their mathematics test score—that is, as students’ use of pictorial representations increased, their test score on the mathematical problems decreased.
These measured variables provide a framework for the primary focus of this study: (a) to determine whether the deaf and hearing students performed similarly or differently on the tasks due to their hearing status and grade level and (b) how their assessed visual–spatial ability and student-generated use of visual–spatial schematic and pictorial representations predict their performance on the 15 mathematical problems. Because the PMA and Revised Minnesota Paper Form Board tests similarly measure visual–spatial ability, these measures are combined into one score to simplify the subsequent analyses and interpretations.
Predicting Students’ Mathematical Performance From Their Participant Groupings
To determine how hearing status and educational level influenced the students’ performance on the 15-item mathematical problem-solving task, a multiple regression analysis was conducted using a continuous ordinal predictor variable (educational level with four categories: middle school, high school, college associate, and baccalaureate programs) and two dummy predictor variables (hearing status and the interaction between educational level and hearing status). As shown in Table 5, both educational level and hearing status were significant predictors for their total score on the 15 mathematical problems, whereas the interaction predictor was also significant, indicating that the deaf and hearing groups performed in a different pattern across the educational levels for solving the mathematical problems. Because of the concern that the associate degree category for deaf students with no parallel category for hearing students might be the potential cause of the interaction, this regression analysis was also run without the associate degree category and the interaction effect was still significant (coefficients for intercept = 6.791, interaction = 1.283, t = 4.004, p < .0001). Thus, the interaction appears not to be caused by the missing associate category for the hearing college students.
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Figure 1 illustrates the significant interaction effect shown in Table 5. The plot for each group was generated from the regression line equation with two X terms and interaction term as follows: Y = mX1 + mX2 + mX1 x X2 + intercept. Based on these calculations, the slope of the interaction lines for the hearing students’ predicted performance increased 1.2 points from middle school to high school and 2.4 points from high school to college. In contrast, the deaf students'-predicted performance only increased a consistent .47 of a point between each of the levels from middle school to the college baccalaureate program level. Clearly, the slope of the hearing students increased at a greater rate than that of the deaf students whose developmental progress was relatively flat from middle school to college level.
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Visual–Spatial Ability and Use of Schematic and Pictorial Representations as Predictors
Since the previous analysis step showed a significant interaction between the students’ hearing status and educational level indicating a different pattern of problem-solving performance, this section examines the deaf and hearing students’ performances separately by educational level. A multiple regression model was applied to examine how much the students’ visual–spatial ability and their self-generated use of visual–spatial schematic and pictorial representations in solving the 15 mathematical problems predicted their total correct score. The multiple regression analysis included three independent predictor variables: (a) measured visual–spatial ability (each students’ combined score from the PMA Test and Revised Minnesota Paper Form Board Test); (b) Visual–Spatial Schematic Representation Score (each students’ total number of schematic representations used in solving the 15 mathematical problems), and (c) Visual–Spatial Pictorial Representations (each students’ total number of pictorial representations generated in solving the 15 mathematical problems). Tables 6 (deaf students) and 7 (hearing students) present the results of the multiple regression analyses per educational level.
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Tables 6 and 7 present four models of regression analysis for the deaf and hearing participants, respectively, per each educational level: Model A shows the results of the Combined PMA/Minnesota Score only for predicting the students performance score on the 15 mathematical items; Model B shows the results of the students’ use of Visual–Spatial Schematic Representation only to predict their performance score on the mathematical problems; Model C shows the students’ use of Visual–Spatial Pictorial Representation only to predict their performance score on the mathematical problems; and Model D is the multiple regression analysis that includes all three predictor variables (Combined PMA/Minnesota, Schematic, and Pictorial Representations) to predict the students’ mathematical problem-solving score.
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Table 6 shows that the deaf students’ self-generated use of visual–spatial schematic representation (Model B) is the single best predictor for their mathematical problem solving score for all educational levels (D1 R2 = .67, D2 R2 = .54, D3A R2 = .49, and D3B R2 = .40). R2 is the proportion of variance predicted or accounted for in the dependent measure; in this case, the students’ total correct score for solving the 15 mathematical problems. Additionally, R2 increment testing was conducted to determine the predictive contributions of other predictors to the total variance accounted for in the multiple regression Model D for each of the educational levels—that is, the students’ measured visual–spatial ability (Combined PMA/Minnesota Score). Because the students’ use of pictorial representation was generally not a reliable predictor and contributed only minimally to the variance explained in any of the multiple regression analyses (Model D) at the various grade levels, it was not included in the increment analyses. Procedurally, the increment testing began with the total variance accounted for in the multiple regression analysis (R2 for Model D) and subtracted the variance explained by the use of schematic representation (R2 for Model B) to obtain the R2 difference or incremental contribution of the predictor Combined PMA/Minnesota predictor (Model A). The results of the R2 increment testing for the deaf students in each educational level are shown at the bottom of Table 6. The incremental prediction contributions of the students’ measured visual–spatial ability (Combined PMA/Minnesota) was significant for the middle school deaf students (D1 R2 increment = .19, p < .01), associate program deaf students (D3A R2 increment = .06, p < .01), and the baccalaureate program deaf students (D3B R2 increment = .09, p < .05). Thus, for the deaf participants, both their visual–spatial ability and use of visual–spatial schematic representation in solving problems generally appear to be significant predictors of their mathematical problem-solving performance from middle school through college level.
In contrast, only the findings for the hearing middle school students show that their measured visual–spatial ability (Combined PMA/Minnesota) and their use of self-generated visual–spatial schematic representation in problem solving are significant predictors of their total score on the 15 mathematical problems (see Table 7). For both the hearing high school and baccalaureate program students, the predicted R2 is relatively insignificant for both their visual–spatial ability and their use of visual–spatial schematic representations (12% or less explained variance of their total correct score predicted in solving the 15 mathematical problems).
When one looks at each educational level for both the deaf and hearing participants, it is clear that both visual–spatial ability and the use of visual–spatial schematic representation by deaf students are very important factors for their successful mathematical problem solving. For the hearing participants, only the middle school students’ visual–spatial ability and their use of visual–spatial schematic representation appear to be important predictors of their mathematical problem solving. Although both the hearing high school and college students scored relatively higher than their deaf peers on the visual–spatial measurements and clearly used significantly more visual–spatial schematic representation in solving the 15 mathematical problems (see Table 3), they seem to have gone beyond their dependence on schematic representations, and other experiential factors appear to be contributing to their successful problem-solving skills.
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Discussion |
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TOPIntroductionMethodologyResultsDiscussionSummaryReferences |
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This research compared the performance of deaf and hearing peers on a mathematical problem-solving task adapted from the MPI used by Hegarty and Kozhevnikov (1999)
. The MPI was designed to distinguish students’ use of visual–spatial schematic representations that primarily focus on the spatial relations described in a problem versus visual–spatial pictorial representations that focus on a problem's iconic nature.
Students’ Use of Visual–Spatial Schematic and Pictorial Representations
Hearing students across the board generally utilized visual–spatial schematic representations to a greater degree than the deaf students. Although the deaf and hearing baccalaureate students were statistically similar in their use of visual–spatial schematic representation, the hearing students’ performance in solving the 15 mathematical problems was significantly better. These results are consistent with the earlier findings of Kelly and Mousley (2001), which showed that although deaf college subjects were able to generate an appropriate diagram that demonstrated understanding of the component parts of a multidimensional mathematical word problem, they were unable ultimately to achieve the same degree of success as hearing peers in calculating the correct answer to that problem.
With respect to visual–spatial pictorial representation, deaf students used this iconic-only representation strategy to a greater degree than hearing subjects, suggesting that many of the deaf students may be focusing on a problem's surface structure and potentially irrelevant information while solving mathematical problems (Hegarty & Kozhevnikov, 1999; Lean & Clements, 1981; Presmeg 1986). Lewis (1989) found that when students were taught to translate the statements of a compare word problem (thereby focusing only on a problem's surface structure and not the deeper meaning of the problem), overall problem-solving skills deteriorated.
The ability to see and utilize relationships and its effect on mathematical problem-solving efficacy was of principal concern for this study. As reported in Tables 6 and 7, deaf students’ performance variance on the mathematical problem-solving task was primarily explained by their use of visual–spatial schematic representation to a much greater degree than it was for their hearing peers. The prediction strength of utilizing visual–spatial schematic representation use decreases sharply for the hearing subjects beyond middle school, but not so for the deaf students. In terms of problem-solving performance, the deaf baccalaureate students scored similarly to hearing middle school students. There are several potentially plausible explanations for this. Developmentally, other factors may become more operatively available and important in solving those problems. For example, the hearing high school and college students may have already automatized the formulaic calculation regimen and beyond middle school are not so dependent on the visual–spatial schematic representations to arrive at successful problem solutions. Similar to Bebco's (1998) discussion of language automatization and working memory constraints for deaf learners, maybe the older deaf participants have not fully automatized fundamental analytical skills for organizing relationships and thus have less working memory and cognitive capacity available to focus on the problem solution. Lee, Swee-Fong, Ee-Lynn, and Zee-Ying (2004) found effects of working memory on algebraic problem solving to be mediated by literacy skills. Marschark (2006) in discussing the strategic and content differences in memory and cognition observes "the fact that deaf individuals both have a greater reliance on visual information than hearing peers and have to deal with visual and verbal (also via the visual modality) information consecutively rather than simultaneously ... clearly will result in their having different perceptual and cognitive strategies than those who can draw on both visual and auditory input" (p. 84). It is also possible that hearing high school and baccalaureate students were not sufficiently challenged. Therefore, it must be considered that the participating hearing students at the secondary and postsecondary educational levels might have shown a greater dependence on visual–spatial schematic representations had they been sufficiently challenged. Although there was no apparent ceiling effect on the mathematical problem-solving task performance, there may be an implicit ceiling effect. It could be that the relative "ease" of these problems for the more advanced hearing students had a negative impact on student effort that may have contributed to careless errors. Such factors may have played a greater role in affecting the hearing college students’ problem-solving performance on the mathematical task than actual methodology choice. As conducted, this study provides a quasi-developmental picture in mathematics skill acquisition. If students are appropriately challenged in mathematics (i.e., the middle school hearing subjects and all the deaf subjects), then visual–spatial schematic representations are highly predictive of success in solving mathematical problems. However, as students move from novice to more proficient in mathematics, factors other than visual–spatial schematic representation appear to become larger determinants of success.
Additional Observations and Findings
The current study extended the research of Hegarty and Kozhevnikov (1999) and provided a wider developmental scope by making comparisons between deaf and hearing populations at three educational levels for hearing subjects (middle school, high school, and baccalaureate degree program) and four levels for deaf subjects (middle school, high school, associates degree, and baccalaureate degree programs). Also, a group-testing rather than individual administration procedure was used. This change was made due to both the larger sample size involved as well as concerns about the original design that could possibly influence subjects’ subsequent responses if questions were asked immediately following the solution of each problem. Although a group-testing procedure guards against inadvertent influences on participant responses, it does sacrifice the additional qualitative information about problem-solving strategies. Given that this was not a goal of this study, it did not adversely affect resulting data but might be worth pursuing in future research because of the greater effect of visual–spatial schematic representation outcomes for deaf participants. However, the group design procedures of this study provided a more conservative measure of participant performance by not inadvertently providing leading questions for subsequent problems.
The hearing middle school subjects in this study showed similar performance to that of the subjects in the Hegarty and Kozhevnikov (1999) study. Although no direct statistical comparisons can be made in comparing the group mean scores for the mathematical problem-solving task, our middle school subjects achieved slightly higher scores than the subjects in the original study, and the mean use of visual–spatial schematic representation was much higher by the middle school students in our study as well (this may be related to curricular innovations having been introduced in the area public schools over the past several years by a National Science Foundation-funded curricular project for mathematics education at the Warner Graduate School of Education and Human Development at the University of Rochester). Further, similar moderate correlation coefficients were observed across both studies between students’ mathematical problem-solving performance and their utilization of visual–spatial schematic representation. Similarly, in both studies, a low negative correlation was observed between hearing students’ use of visual–spatial pictorial representation and their accuracy on the mathematical problem-solving task. Both the spatial–visual assessments (Revised Minnesota Paper Form Board Test and the PMA Spatial Test) were positively correlated to overall performance on the mathematical problem-solving task but showed only minimum correlation with students’ use of schematic relational and pictorial representations similar to the Hegarty and Kozhevnikov findings. The results of this study are consistent with earlier findings and thus provide validation and confidence in our examination of hearing and deaf subjects’ performance across several educational levels.
Comparison of the visual–spatial processing skills of the deaf and hearing subjects in our study reveals that hearing subjects performed consistently better than deaf subjects as evidenced by scores on the Revised Minnesota Paper Form Board Test and the PMA Spatial Test. Since no tests were conducted for nonverbal reasoning, the effects of overall intelligence on these results cannot be ruled out. These results are in seeming contradiction to the enhanced visuospatial skills documented with deaf and hearing signers reported in earlier studies. If users of sign language are faster at generating and rotating 2- and 3-dimensional figures, one could hypothesize that sign language users should potentially out perform their nonsigning peers on other tests of visual–spatial skills. Future testing in this area needs to be done with controls of general intelligence and sign language skill included in the research design. Such results could be a product of the interaction between visual–spatial skills and relational processing, whereby deaf individuals have been shown to be stronger in visuospatial skills but have demonstrated a weakness where the latter is concerned.
As noted earlier, the hearing subjects’ performance on the mathematical problem-solving task was consistently higher than their deaf peers across all educational grade levels. The deaf baccalaureate students exhibited the highest performance of all the deaf participants but only performed at the level of the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level and then showed a 41% increase in performance from the associate degree level to the baccalaureate degree level. In contrast, the hearing subjects evidenced a 29% in performance increase between middle and high school and an 11% increase between the high school and baccalaureate levels. Hearing baccalaureate students scored 44% higher than their deaf baccalaureate peers, this was consistent with previous research. However, an issue that was surprising and one that should raise concern among educators was the finding that the deaf students showed minimal increase in performance from middle school to the college associate degree level.
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Summary |
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TOPIntroductionMethodologyResultsDiscussionSummaryReferences |
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This study demonstrates the importance of students’ understanding and use of visual–spatial schematic representations in solving mathematical problems. It also highlights critical differences in this area between deaf and hearing peers as well as the detrimental effects of these differences on mathematics problem-solving performance when not using visual–spatial schematic representations. The minimal increase in scores of deaf students on the mathematical problem-solving task from middle school to college associates degree level is of considerable concern. The fact that the deaf baccalaureate students performed similarly to the hearing middle school students is also cause for concern. Although this study cannot provide any definitive answers as to the causality of these findings, it does clearly demonstrate that when deaf students generate and use visual–spatial schematic representation to show the spatial relationships contained in mathematics problems they will realize greater problem-solving success. Students who primarily focus only on the pictorial iconicity in representing a problem will not be as successful in mathematical problem solving.
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Acknowledgments |
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This research was supported by a grant (SBE-0350277) from the National Science Foundation to the National Technical Institute for the Deaf, Rochester Institute of Technology, Rochester, NY, with collaborative subcontracts to Gallaudet University, Washington, DC, and Bowling Green State University, Bowling Green, OH. We gratefully acknowledge the participation in this study of students attending the Rochester Institute of Technology, National Technical Institute for the Deaf at RIT, Rochester School for the Deaf, Rochester, NY, Byron-Bergen Middle and High Schools, Brighton Middle School, Webster High School, and the Board of Cooperative Educational Services (BOCES) #1 and #2, all in Western New York. We are also indebted to Stan Barringer, Cynthia Callard, John Callard, Shari Dressler, Heather Mooney, Marty Nelson Nasca, and Patti Wink for their invaluable assistance in facilitating the gathering of the student data analyzed in this study. We are especially grateful to Sujay Narayanswamy for his capable management and preparation of data submitted to statistical analysis. No conflicts of interest were reported.
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