Visual-Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Students

Mr. Gary Blatto-Vallee/Lecturer. Science & Mathematics - National Technical Institute for the Deaf WAS available from 4/28/08 until 5/18/08 to answer questions and share ideas concerning his research and its implications for parents of children who are deaf/hard of hearing, their teachers and other professionals who work with them.

You are encouraged to read the research summary below and review the attached discussion.

Abstract: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 5 156) and deaf (n 5 149) 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. Paragraph One:If you are reading this, you probably already know about the disparity in mathematics performance between deaf and hearing peers. On average, deaf children lag their hearing peers by roughly three years. This separation stays constant through high school and post secondary education. This phenomenon is not caused by deafness (in an audiological sense), and it is not isolated to the United States (Frostad & Ahlberg, 1999; Phelps & Branyon, 1990). The good news is that intervention strategies work. Nunes and Moreno (2002) designed an intervention focused primarily on developing the informal mathematical knowledge that is assumed all children have prior to entering their first year of school. Skills taught were additive composition, additive reasoning, and multiplicative reasoning. After using this intervention, the deaf students involved in the study were able to outperform their own previously established (by assessment) projected mathematics developmental milestones. Paragraph Two:The ability to recognize and use relationships is paramount to success in mathematics. The importance of this is inherent in the very definition of mathematics. 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). The ability to see the explicit and implicit connections in a mathematics problem and to other related problems that may aid in finding a solution is critical in developing mathematical knowledge. In terms of deaf learners there has been research indicating that this very fundamental skill is underdeveloped, and that this cuts across several domains of knowledge (See Marschark, 2003 for a review).

Paragraph Three:Visual-Spatial ability is a main factor affecting mathematical performance (Battista, 1990; Sherman, 1979). In the past there has been some confusion as to why some visual learners perform well in mathematics while others do not. Leans & Clements (1981) actually concluded that “verbalizers” (people who process thing verbally) outperformed “visualizers” (people who process things visually) on mathematical and spatial ability skills tests. For most people this will contradict the belief that people who are good at mathematics and science tend to be visual people, and that a well designed visual aid greatly enhances the “understandability” of a math problem. In fact, I tell my students that without a diagram some problems are almost impossible to understand. Several researchers sought to clarify this apparent contradiction (Leans & Clements, 1981; Presmeg, 1986,). What emerged is two different groups of “visualizers”. One group tends to focus on the concrete aspects of visual imagery, the pictorial details. Things are seen in terms of the objects. The other group sees the relationships between objects, the connections that are explicit and implicit in the problem. When these two groups are separated out from each other you see a high degree of success on mathematics problems and spatial ability tests associated with this second group (Hegarty & Kozhevnikov, 1999). The first group (the object visualizers) tend not to do well on spatial and mathematical ability tests. In terms of deafness and sign language there has been several studies documenting the enhanced visual-spatial skills of deaf and hearing signers. People who sign have been shown to be faster at generating mental images and in completing mental spatial rotational tasks than non-signing peers (Emmory, Kosslyn, and Bellugi, 1993; Talbot & Haude, 1993). Both of these skills have been taken to indicate an enhancement of spatial ability in deaf and hearing signers. Paragraph Four:Our study was a replication study of Hegarty and Kozhevnikov (1999), where they sought to document and classify visual-spatial strategies used by students when solving mathematics problems. They were able to distinguish two main types of visual-spatial strategies, ones that were primarily iconic (focused on the concrete images contained in a problem) and others that were primarily spatial-relational in nature (focusing more on the relationships between objects). What they found was that the students who saw and used the relationships within a problem were more accurate in their solutions, and students who focused primarily on the concrete images were less accurate. They also gave several tests of spatial ability and found that the students who used spatial-relational strategies tended to perform better on these tests as well when compared to the concrete visualizers. Paragraph Five:As mentioned previously we replicated this study with deaf and hearing students across several educational levels, from middle school through college. We found that on the whole, hearing students had higher scores on our mathematics test than their deaf peers. Although important, this was not the focus of our study. We wanted to know what differences there were in the way that deaf and hearing students approached these problems. We (like Hegarty & Kozhevnikov (1999)) classified students recorded responses as being more concrete or more relational. Our analysis of the students’ work indicated that the hearing students were using relationships to a greater degree than deaf students over all. Both deaf and hearing students who used relationships scored higher than those who did not (the ones who focused primarily on the concrete images in the problem). These results are in line with previous research underlining the importance of relationships in mathematical problem solving. Paragraph Six:Our research has shown that deaf students do not necessarily see or use the relationships in mathematics problems, and that this negatively affects their problem solving success. Teachers (myself included) need to emphasize the explicit and implicit relationships in problems. We cannot assume that because we see the relationships that our students do as well. Parents need to point out the relationships that we take for granted in everyday life. Draw connections to previously seen or learned information. When you make a decision (which brownie pan to use... very important in my house) explain why you decided what you decided, what previous experience lead you to the better brownie pan (again high stakes stuff). “Talk through” your thought processes to your students and children while you are working on a problem or seeking to understand something. Talk to your students and children about how you decided which approach was better. Explain your evaluation process. Talk about what your thoughts are and how you are drawing from your previous experience when you approach something new. References: Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. Journal of Research in Mathematics Education, 21, 47–60. Emmorey, K., Kosslyn, S. M., & Bellugi, U. (1993). Visual imagery and visual-spatial language: Enhanced imagery abilities in deaf and hearing ASL signers. Cognition, 46(2), 139–181. Frostad, P., & Ahlberg, A. (1999). Solving story-based arithmetic problems: Achievement of children with hearing impairment and their interpreting of meaning. Journal of Deaf Studies and Deaf Education, 4(4), 283–293. Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representation and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689 Lean, C., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12, 267–299. Marschark, M. (2003). Cognitive functioning in deaf adults and children. In M. Marschark & P. E. Spencer (Eds.), Oxford handbook of deaf studies, language, and education (pp. 264–277). New York: Oxford University Press. Nunes, T., & Moreno, C. (2002). An intervention program for mathematics. Journal of Deaf Studies and Deaf Education, 7, 120–133. Oxford University Press. (2000). The Compact Oxford English Dictionary (2nd ed.). New York, NY: Oxford University Press Inc. Phelps, L., & Branyon, B. J. (1990). Academic and nonverbal intelligence in public school hearing-impaired children. Psychology in the Schools, 27(3), 210–217. Presmeg, N. C. (1986). Visualization in high school mathematics. For Learning of Mathematics, 63, 42–46. Sherman, J. A. (1979). Predicting mathematical performance in high school girls and boys. Journal of Educational Psychology, 71, 242–249. Talbot, K. F., & Haude, R. H. (1993). The relationship between sign language skill and spatial visualizations ability: Mental rotation of three-dimensional objects. Perceptual and Motor Skills, 77, 1387–1391.

Reference: Blatto-Vallee, G., Kelly, R.R, Gaustad, M.G., Porter, J., & Fonzi, J. (2007). Visual-Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Students. Journal of Deaf Studies and Deaf Education, 12(4), 432 – 448.

Visual-Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Studentsto answer questions and share ideas concerning his research and its implications for parents of children who are deaf/hard of hearing, their teachers and other professionals who work with them.WASavailable from 4/28/08 until 5/18/08Abstract: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 5 156) and deaf (n 5 149) 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.Paragraph One:If you are reading this, you probably already know about the disparity in mathematics performance between deaf and hearing peers. On average, deaf children lag their hearing peers by roughly three years. This separation stays constant through high school and post secondary education. This phenomenon is not caused by deafness (in an audiological sense), and it is not isolated to the United States (Frostad & Ahlberg, 1999; Phelps & Branyon, 1990). The good news is that intervention strategies work. Nunes and Moreno (2002) designed an intervention focused primarily on developing the informal mathematical knowledge that is assumed all children have prior to entering their first year of school. Skills taught were additive composition, additive reasoning, and multiplicative reasoning. After using this intervention, the deaf students involved in the study were able to outperform their own previously established (by assessment) projected mathematics developmental milestones.Paragraph Two:The ability to recognize and use relationships is paramount to success in mathematics. The importance of this is inherent in the very definition of mathematics.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). The ability to see the explicit and implicit connections in a mathematics problem and to other related problems that may aid in finding a solution is critical in developing mathematical knowledge. In terms of deaf learners there has been research indicating that this very fundamental skill is underdeveloped, and that this cuts across several domains of knowledge (See Marschark, 2003 for a review).Paragraph Three:Visual-Spatial ability is a main factor affecting mathematical performance (Battista, 1990; Sherman, 1979). In the past there has been some confusion as to why some visual learners perform well in mathematics while others do not. Leans & Clements (1981) actually concluded that “verbalizers” (people who process thing verbally) outperformed “visualizers” (people who process things visually) on mathematical and spatial ability skills tests. For most people this will contradict the belief that people who are good at mathematics and science tend to be visual people, and that a well designed visual aid greatly enhances the “understandability” of a math problem. In fact, I tell my students that without a diagram some problems are almost impossible to understand. Several researchers sought to clarify this apparent contradiction (Leans & Clements, 1981; Presmeg, 1986,). What emerged is two different groups of “visualizers”. One group tends to focus on the concrete aspects of visual imagery, the pictorial details. Things are seen in terms of the objects. The other group sees the relationships between objects, the connections that are explicit and implicit in the problem. When these two groups are separated out from each other you see a high degree of success on mathematics problems and spatial ability tests associated with this second group (Hegarty & Kozhevnikov, 1999). The first group (the object visualizers) tend not to do well on spatial and mathematical ability tests. In terms of deafness and sign language there has been several studies documenting the enhanced visual-spatial skills of deaf and hearing signers. People who sign have been shown to be faster at generating mental images and in completing mental spatial rotational tasks than non-signing peers (Emmory, Kosslyn, and Bellugi, 1993; Talbot & Haude, 1993). Both of these skills have been taken to indicate an enhancement of spatial ability in deaf and hearing signers.Paragraph Four:Our study was a replication study of Hegarty and Kozhevnikov (1999), where they sought to document and classify visual-spatial strategies used by students when solving mathematics problems. They were able to distinguish two main types of visual-spatial strategies, ones that were primarily iconic (focused on the concrete images contained in a problem) and others that were primarily spatial-relational in nature (focusing more on the relationships between objects). What they found was that the students who saw and used the relationships within a problem were more accurate in their solutions, and students who focused primarily on the concrete images were less accurate. They also gave several tests of spatial ability and found that the students who used spatial-relational strategies tended to perform better on these tests as well when compared to the concrete visualizers.Paragraph Five:As mentioned previously we replicated this study with deaf and hearing students across several educational levels, from middle school through college. We found that on the whole, hearing students had higher scores on our mathematics test than their deaf peers. Although important, this was not the focus of our study. We wanted to know what differences there were in the way that deaf and hearing students approached these problems. We (like Hegarty & Kozhevnikov (1999)) classified students recorded responses as being more concrete or more relational. Our analysis of the students’ work indicated that the hearing students were using relationships to a greater degree than deaf students over all. Both deaf and hearing students who used relationships scored higher than those who did not (the ones who focused primarily on the concrete images in the problem). These results are in line with previous research underlining the importance of relationships in mathematical problem solving.Paragraph Six:Our research has shown that deaf students do not necessarily see or use the relationships in mathematics problems, and that this negatively affects their problem solving success. Teachers (myself included) need to emphasize the explicit and implicit relationships in problems. We cannot assume that because we see the relationships that our students do as well. Parents need to point out the relationships that we take for granted in everyday life. Draw connections to previously seen or learned information. When you make a decision (which brownie pan to use... very important in my house) explain why you decided what you decided, what previous experience lead you to the better brownie pan (again high stakes stuff). “Talk through” your thought processes to your students and children while you are working on a problem or seeking to understand something. Talk to your students and children about how you decided which approach was better. Explain your evaluation process. Talk about what your thoughts are and how you are drawing from your previous experience when you approach something new.References:Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. Journal of Research in Mathematics Education, 21, 47–60.

Emmorey, K., Kosslyn, S. M., & Bellugi, U. (1993). Visual imagery and visual-spatial language: Enhanced imagery abilities in deaf and hearing ASL signers. Cognition,

46(2), 139–181.

Frostad, P., & Ahlberg, A. (1999). Solving story-based arithmetic problems: Achievement of children with hearing impairment and their interpreting of meaning. Journal of Deaf Studies and Deaf Education, 4(4), 283–293.

Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representation and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689

Lean, C., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12, 267–299.

Marschark, M. (2003). Cognitive functioning in deaf adults and children. In M. Marschark & P. E. Spencer (Eds.), Oxford handbook of deaf studies, language, and education (pp. 264–277). New York: Oxford University Press.

Nunes, T., & Moreno, C. (2002). An intervention program for mathematics. Journal of Deaf Studies and Deaf Education, 7, 120–133.

Oxford University Press. (2000). The Compact Oxford English Dictionary (2nd ed.). New York, NY: Oxford University Press Inc.

Phelps, L., & Branyon, B. J. (1990). Academic and nonverbal intelligence in public school hearing-impaired children. Psychology in the Schools, 27(3), 210–217.

Presmeg, N. C. (1986). Visualization in high school mathematics. For Learning of Mathematics, 63, 42–46.

Sherman, J. A. (1979). Predicting mathematical performance in high school girls and boys. Journal of Educational Psychology, 71, 242–249.

Talbot, K. F., & Haude, R. H. (1993). The relationship between sign language skill and spatial visualizations ability: Mental rotation of three-dimensional objects. Perceptual and Motor Skills, 77, 1387–1391.

Reference:Blatto-Vallee, G., Kelly, R.R, Gaustad, M.G., Porter, J., & Fonzi, J. (2007). Visual-Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Students. Journal of Deaf Studies and Deaf Education, 12(4), 432 – 448.