Assessing the Effect of Visual Field of Exposure and Mathematical Ability on Acuity of Number Sense - Assignment Example

? Assessing the Effect of Visual Field of Exposure and Mathematical Ability on Acuity of Number Sense The experiment was designed in order to verify the effect of mathematical ability and the visual field in which the stimulus is exposed on the individual’s acuity in discriminating between the numerosity of different objects. Based on theory, it was proposed that higher acuity would be observed for participants with higher mathematical ability and for stimuli presented in the left visual field. Statistical tests revealed no differences between groups, and led to the acceptance of the null hypotheses in each case. Reasons for these results, areas of further research and implications are discussed. Introduction Although most children do not learn to count numbers until well into toddlerhood, even very young children are able to discriminate between more and less numerous groups of objects, and will pay more attention to more numerous objects (Klein & Starkey, 1988). Research into this ability of humans to discriminate between the numerosity of groups of objects yielded fascinating information into Subitizing – an ability of an individual to instantly see the numerosity of a group. Subitizing has been considered as the precursor to counting units; and is believed to be the ability to estimate numbers by focusing on the whole group instead of the units involved (Klein & Starkey, 1988). Although subitizing is rarely confused with counting, evidence for the distinction between subitizing and counting has been found by Pica, Lemer, Izard and Dehaene (2004) who studied the Munduruku speaking tribe. This language does not have exact numbers beyond 5; and although the tribes-persons have difficulty with exact mathematical quantities and calculations beyond 4-5 units, they are capable of identifying and adding approximate numbers that are far greater (Pica et al, 2004). The distinction between language based counting of objects and a non-verbal approximation of quantities is also visible in very young children who can identify the more or less numerous category, but often have trouble counting out smaller sets of objects (Klein & Starkey, 1988). Even though infants and children exhibit the ability the subitize, the ability continues to develop with age, and developmental effects are seen even in early adulthood (Halberda & Feigenson, 2008). Infants can subitize only perceptually and with limited numbers; and as an individual grows, they become able to subitize conceptually and approximate larger and more complex quantities (Halberda & Feigenson, 2008). The ability to cope with patterns and shapes within which objects are placed also developed with age, with rectangular shapes being the easiest which are followed by linear arrangements. Circular arrangements are more difficult to approximate, and scattered arrangements are most difficult for all age groups (Wang, Resnick & Boozer, 1971). Data collected is consistent with Weber’s law at all ages. Response time in subitizing also increases as a function of the number of objects. Most subjects will respond to groups of 1-4 objects within the same amount of time, but as the number of objects increases; the time taken to respond also increases in a linier fashion as explained by Weber’s law (Mandler & Shebo, 1982). This makes it evident that a number of factors affect the ability to subitize, approximate and discriminate groups of stimuli as well as to count. With respect to educational variables, a clear relationship has been found between mathematical ability and these abilities in young children as well as in adults (Libertus, Feigenson & Halberda, 2011). Mathematical achievement at different ages of a child has been found to correlate to their ability to approximate and discriminate groups of objects (Halberda, Mazocco & Feigenson, 2008). More evidence for this relationship has been found when studying children with dyscalculia. Dyscalculia is a learning disability that inhibits mathematical ability; and children with dyscalculia have been found to exhibit reduced ability to subitize and discriminate between groups on the basis of their numerosity (Mazocco, Feigenson & Halberda, 2011b). Another important aspect that could affect approximation of numerical information or number sense is the manner in which the information is obtained (Dehaene, 2011). The centres for spatial ability and number sense typically develop in the right hemisphere of the brain for most persons; though left handed and ambidextrous persons show a greater diffusion of these abilities across hemispheres (Dehaene, 2011). Since information that is collected by each eye is sent to the visual centre in the contra-lateral hemisphere; information about numbers and size that is collected by the left eye are more easily calculated and understood (Palmer, 1999). Each eye picks up information from the left and right visual fields, but as the optic nerves cross at the optic chasm, each visual centre in the brain receives information only from the contra-lateral visual field. When a stimulus is presented briefly in only one visual field, the information is processed by the visual centre of the contra-lateral hemisphere (Palmer, 1999). Thus, if information is available only to one visual centre; this should affect the manner in which will be assessed and used; including the acuity of distinguishing between groups of objects and making judgements about them (Dehaene, 2011). Having understood the various factors that affect the way in which people perceive numbers, it is possible to suggest that mathematical ability could affect the sensitivity of an individual with respect to discriminating between groups of objects. Given that visual acuity and mathematical ability are correlated; it is possible to suggest that persons with higher mathematical ability will perform better of tasks that require them to discriminate between groups of objects. Based in the understanding of how the visual system functions, it is also possible to suggest that data available to the eye that is contra-lateral to the more developed visual centre would be more effectively evaluated. Based on the description in Dehaene (2011), it may be suggested that stimuli presented in the left visual field (which is processed in the right hemisphere) would be processed with greater sensitivity as compared to stimuli presented in the right visual field. The following research questions were chosen to be studied within this experiment: Do the two cerebral hemispheres have differential acuity for number sense? Does mathematical ability differentially depend on the acuity of number sense in one hemisphere? Based on these research questions, the following hypotheses were developed to be tested: 1. Participants with high mathematical ability will respond correctly to more number sense object discrimination stimuli as compared to participants with low mathematical ability. 2. Participants will respond correctly to more number sense object discrimination stimuli presented in the left visual field as compared to stimuli presented in the right visual field. Method Participants This study was carried out on year two Psychology students at the Department of Psychology. The present sample consisted of 90 female participants and 34 male participants in the age range of 19 to 55 years. Mean age of participants was 30.95 years with a standard deviation of 8.39. Of the participants, 114 were right handed while only 10 were left handed as tested on the Edinburgh Handedness Inventory (Oldfield, 1971). All participants were well versed with the English language, were comfortable with computer administered testing procedures, and were familiar with the use of the E-Prime 2.0 software. Design The design used to conduct this experiment was a mixed factorial design with one randomized independent variable and one repeated independent variable, each of which had two levels. The first independent variable was Mathematical skill on the basis of which participants were divided into two groups having high and low ability using a median split. The median score of the group was 34.5 and participants scoring 34 and below were classified as having low mathematical ability while participants with scores of 35 and above were classified as having high mathematical ability. The second independent variable was field of presentation of stimuli which also had two levels – Right field and Left field. The dependent variable studied was the percentage of correct discriminations among the visual acuity stimuli presented in each visual field. The percentage of correct responses was calculated on the basis of all trials presented to the said visual field. Design: 2 x 2 Mixed factorial design. Field of Presentation (repeated) Left Right Mathematical Ability (randomised) High a b Low c d Materials The study was conducted using the E-Prime 2.0 software which is designed to facilitate Psychological experiments. Stimuli for the Number sense acuity task and the stimuli for the mathematical ability test were both administered using this software. Number Sense Acuity Task: Number sense acuity was gauged using blue and brown circles as stimuli presented against a grey background since both blue and brown were shown to have equal contrast against the grey background. The ratio of blue to brown circles was varied at six levels namely 1.17, 1.25, 1.33, 1.5, 2.0, and 3.0 across the trials to represent different levels of difficulty. All eight practice trials were kept at the 3.0 level. On half the trials, blue circles were more numerous while brown circles were more numerous on the other half trials. In order to ensure that both blue and brown circles had an equal chance of being chosen by chance, total surface area for both was kept equal on 1/3 trials, average size of circles was kept equal on 1/3 trials and 1/3 trials were conducted where these factors varied freely. This was done for stimuli presented to each visual field; and the total correct responses across all trials in each visual field were used to calculate the percentage of correct responses. Four blocks of 72 trials were administered so that a participant responded to a total of 288 trials. Each block had an equal number of each type of trial. Mathematics Task: Mathematical ability was tested by exposing the participants to our blocks of problems which lasted for three minutes. Each block tested for a single ability among addition, subtraction, multiplication, and division and was preceded by three practice problems. The participant was exposed to the problems using the E-Prime 2.0. Software and they responded to each problem using the number pad at the right edge of a standard keyboard. Blocks were presented in a randomised order to each subject. Edinburgh Handedness Inventory: The Edinburgh Handedness Inventory (Oldfield, 1971) is a standard questionnaire which measures handedness along a continuum from +100 (representing fully right-handedness) up till -100 (representing fully left-handedness). A zero score on the inventory represents ambidexterity. For the purpose of this experiment all scores of 1 and above were coded as 1 for right-handed and all scores of -1 and below were coded as 2 for left-handed. Procedure Participants were provided with the instructions for the different tasks before commencing, and were required to register themselves and complete the handedness survey after which they were asked to use the computer to complete the number sense acuity task and the mathematical ability task. Instructions were emphasised again during the tasks. The E-Prime 2.0. software was used and responses were provided using a standard keyboard. The computerised programs were designed so that they were self-administered. For the visual acuity task, the participants responded to stimuli presented on the computer screen placed approximately 40 cm away from their face, and were asked to focus on a small ‘+’ sign at the centre. the Fixation appeared for one second before each trial followed by the array of circles which was offset from the centre by 380 pixels (11.3o away) in either left or right direction for 200 ms. This was followed by a ‘?’ which provided a prompt for the response. Participants were required to respond by pressing the ‘Q’ key to indicate that blue circles were more numerous or the ‘P’ key to indicate that brown circles were more numerous. Participants were advised to focus on the ‘+’ sign as it appeared and to stay focussed on the task through the trials and the succeeding stimuli would appear only briefly. They were also warned that the task may seem difficult and they may guess some answers. For the mathematical ability task, participants responded to the displayed problem using the number pad at the right bottom of a standard keyboard, and pressed enter to confirm their responses. Participants were informed that they should not worry about the task and should simply give their best effort. Results This study attempted to verify if the mathematical ability of an individual and the visual field in which stimuli were presented affected the acuity of number sense displayed. A total of 124 participants were studied of which 62 were classified as having low mathematical ability and 62 were classified as having high mathematical ability using a median split of the scores. All participants in each of these groups responded to stimuli presented at both levels of the second independent variable – field of presentation. There were no missing scores on the dependent variable and Actual hit and miss scores were not used, while only the percentage of correct responses was taken into account. Of all participants, 90 (72.6%) were female and 34 (27.4%) were male. Most participants were right-handed (114; 91.9%) while a few (10; 8.1%) were left handed. Overall, participants responded correctly to an average of 68.39% of trials presented in the left visual field with a standard deviation of the percentage scores of 7.52. participants responded correctly to an average of 68.47% of the stimuli presented in the right visual field (S.D. = 7.018). the means and standard deviations of the scores of participants across the four categories under study are presented in table 1. Table 1: Means and Standard Deviations Field of Presentation (repeated) Left Right Mathematical Ability (randomised) High M = 67.99 S.D. = 7.33 M = 68.55 S.D. = 7.07 Low M = 68.78 S.D. = 7.85 M = 68.40 S.D. = 7.027 Visually, the means and standard deviations in all cells were found to be quite similar. The data was tested using SPSS, and the statistical analysis verified that the cells were not statistically different from each other. A mixed factor two-way ANOVA was calculated to test the hypotheses. The multivariate analyses for each independent factor, the interaction between factors and the test for Sphericity were not significant irrespective of the test used. The effect for the repeated main factor of field of Exposure yielded an F value of 0.033 which was not significant [F (1, 122) = 0.033; ns]. The main effect for the randomised factor of mathematical ability yielded an F value of 0.067 which was also not significant [F (1, 122) = 0.067; ns]. The interaction effect was also not significant and yielded a value of 1.018 [F (1, 122) = 1.018; ns]. These results show that regardless of the mathematical ability of the participants or the visual field in which the stimuli were presented, participants performed in a similar manner on discriminating between circles of different colours. The ANOVA results are presented in table 2. Table 2: ANOVA results SS df Mean Square F Sig. Field of exposure .438 1 .438 .033 .856 Maths Ability 6.318 1 6.318 .067 .796 interaction 13.396 1 13.396 1.018 .315 Error 1604.849 122 13.154 Discussion The experiment conducted wished to explore the effect of field of presentation and mathematical ability on the extent to which an individual was able to identify the more numerous one of two stimuli presented. The results of the experiment showed that percentage of correct responses for participants was not affected by either of the two main factors. The ANOVA results showed no statistical difference either on the basis of mathematical ability of the participant or on the basis of the visual field in which the stimulus is presented. Interaction of the two factors was also not statistically significant. From these results we may conclude that the data collected does not support the hypotheses presented, and accepted the null hypotheses of no difference. The study expected to see differences in the data collected for responses to stimuli presented to the right and the left visual field, since data from each visual field is predominantly delivered to the contra-lateral hemisphere (Dehaene, 2011). This expectation was heightened as most of the sample collected represented right-handed persons for whom the right hemisphere of the brain is mostly responsible for number skills and spatial reasoning. Thus, the dominant effect expected was that the group would show better performance for stimuli presented in the left visual field. This assumption was not supported and the means for the performance of participants with respect to stimuli presented in the left and right visual fields were quite similar. This may be attributed to a number of factors. It is possible that given the limited size of the computer screen and the distance at which the participants typically sat; the stimuli presented did not fall fully into one single visual field. People are also liable to slightly move their heads to favour better judgement (Dehaene, 2011), and thus, the information of all trials may have been received in the same visual field inadvertently. It is also possible that as psychology students, the participants were not naive to the experiment. Practice can improve participant scores, and exposure to similar material or knowledge about the stimuli may have affected the participant’s ability to respond. The study also expected that mathematical ability would be associated with the participants’ ability to respond to the number acuity task (Libertus, Feigenson & Halberda, 2011) such that participants with high mathematical ability would have a higher proportion of correct responses as compared to those with low mathematical ability. The data collected did not support this assumption, and the means for percentage of correct responses on the number acuity task for participants with high and low mathematical ability were very similar (68.59 and 68.25 respectively). It is possible that the familiarity participants had with similar stimuli and practice with similar stimuli may have affected the results. Some participants may have reason to use mathematical computations on a regular basis, and thus would be able to solve more problems simply as a function of practice as against actual mathematical ability. Another possible concern is that high and low mathematical ability groups were distinguished using a median split of scores rather than any standardised means of defining ability. If a large number of scores were concentrated around the median score, it is possible that the two groups are not very different in terms of actual mathematical ability, and the differences seen are merely a matter of chance. It would be interesting to see if the results of the experiment would be different if these concerns were addressed. Using a wider area for stimulus presentation, restricting head movement in subjects, using naive subjects and differentiating groups on the basis of standardised tests of mathematical ability are some alterations that may reduce the effects of confounding variables that may have affected this particular experiment. Further search may be conducted to address these concerns. It is also important to address the effect of demographic variables like handedness, age and gender in providing a response, which was not studied in this experiment. The sample studied had very few left handed participants as it did male participants; while the age range of participants was quite high. Data collected on participants who are balanced for gender and handedness may help in identifying if these factors did influence the present data. Restricting the age group of participants to smaller cohorts would help in identifying if age of respondent did affect the data. It may also be noted that the response time of participants was not recorded during this experiment; and thus, it is not possible to examine the relationship between response time and correctness of the participants’ responses. If participants took more time to process the information, they could have provided a correct response regardless of the visual field in which the stimulus was presented. During further research, it would be of interest to note the response time of participants, or to restrict the time frame in which a response may be given. The interaction of response time with correctness would help in assessing if the participants were more concerned about giving the right answer. An analysis of participant experiences and strategies used would also provide valuable insight into the results of the present study. A final concern is that the data collected is not representative, given that all participants are part of a single group of students of psychology. Using a more representative sample could provide data that are more representative of the population and thus are more valid compared to the present data. In conclusion, it may be stated that the data collected from the present study did not support the hypotheses tested. Both, Field of exposure of stimulus and Mathematical ability had no effect on the percentage of correct responses on the number acuity task. The possible reasons for this have been explored and possible concerns have been discussed. it is important to verify these results using more stringent methods so that confounding may be reduced. References Dehaene, S. (2011). The Number Sense: How the Mind Creates Mathematic. New York: Oxford University Press. Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the “number sense”: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44, 1457-1465. Halberda, J., Mazocco, M. M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665-668. Klein, A., & Starkey, P. (1988). Universals in the Development of Early Arithmetic Cognition. In Saxe, G.B. & Gearhart, M.(eds.). Children's Mathematics, 27-54. San Francisco: Jossey-Bass. Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14, 1292-1300. Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111, 1-22. Mazocco, M. M. M., Feigenson, L., & Halberda, J. (2011b). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Development, 82, 1224-1237. Oldfield, R. C. (1971). The assessment and analysis of handedness: The Edinburgh inventory. Neuropsychologia, 9, 97-113. Palmer, S. E. (1999). Vision science: From photons to phenomenology. Cambridge, MA: MIT Press. Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499-503. Wang, M., Resnick, L. & Boozer, R.F. (1971). The Sequence of Development of Some Early Mathematics Behaviors. Child Development 42, 1767-78. Read More