Magnitudes of numbers influence numerical inequality judgments of people. Do symbols representing numbers also affect numerical inequality judgments? To answer the question, I manipulated digit similarity in two-digit number comparison tasks. During the experiment, the participants took part in two comparison tasks – the judging-larger task and the judging-smaller task. Given pairs of two-digit numbers, the participants were required to make numerical inequality judgments (judging larger or judging smaller). To investigate the effect of digit similarity, two kinds of number pairs were used. Two-digit number pairs consisting of same-digits numbers (e.g., 21 – 12) and two-digit number pairs consisting of different-digits numbers (e.g., 21 – 30) were presented at random. The participants needed more time to compare the same-digits number pairs than the different-digits pairs. The result was independent of the findings in number comparison studies such as the numerical-distance effect (Moyer & Landauer, 1967) and the unit-decade compatibility effect (Nuerk, Weger, & Willmes, 2001). The present study poses challenge to the current theories of two-digit number comparison. / Thesis (Master, Psychology) -- Queen's University, 2009-09-02 11:59:06.647

Research study on sixth grade problem-posing instruction: Case of addition, subtraction and number comparison on decimals Abstract The aim of this research project is to investigate the implementation of problem-posing instruction on decimals to one sixth-grade mathematics class. There are four research objectives: 1) design and implement problem-posing instruction on decimals; 2) discuss the status of children¡¦s performance in problem-solving; 3) analyze the type of problems posed by children; and, 4) display categories of misconceptions exhibited when children did problem posing. The stages for instructions were three: 1) children solved the problem given by the instructor; 2) children referred to given problem and posed a problem; and, 3) children solved their own problem. In this study, the type of problem posing chosen for instruction is ¡§similar problem¡¨, which is adapted from Tsubota, a Japan scholar. The researcher collected data by using: own constructed decimal problems question sheet, worksheet on problem solving, worksheet on problem posing, children¡¦s diaries and teachers¡¦ notes on instruction. There are four findings. First, the implementation of sixth grade problem-posing instruction on decimals is feasible. Second, 96.9% of students¡¦ problems are plausible and contain sufficient information for problem solvers. Most students could change the number and content of the question but few revised the structure of the question. There was also multiple development for those problems. Third, children¡¦s performance in posing/solving stage was better than that in problem-solving stage. Finally, the researcher reported that the teacher faced problems such as difficulty in control of time, establishing children¡¦s habit in reporting, and collecting misconceptions of children. Key word : problem solving; problem posing; addition, subtraction and number comparison on decimals

This diploma thesis deals with creation of several random number generators. The data from these prototypes are then compared according to various aspects and statistical methods. The reader is familiar with the basic concepts, the existing random number generators and the technologies used.

Numbers convey different meanings when used in different contexts (Wiese, 2003). In a cardinal context, a number will tell us how many entities are in a set and convey quantity meaning. In an ordinal context, a number will refer to the relative position (or rank) of one element within a sequence; non-numerical ordered series (e.g. the letters of the alphabet) can also be used to provide meaningful order information. Because quantity and order are linked up with each other in the cognitive number domain (the larger the quantity a number refers to, the later it is located in the conventional number sequence), the question of whether they rely on some common or distinct underlying mechanism(s) is theoretically relevant and was addressed in the present thesis. Experimental studies showed evidence of both similarities (similar distance and SNARC effects, recruitment of parietal and frontal regions, and conjoint impairment or preservation after brain damage) and dissociations (different developmental course, dissociation after cerebral lesion, and specific behavioural markers) between quantity and order neuro-functional processes. The aim of the present thesis was to clarify the relationship between numerical quantity and order processing and to test the hypothesis that they rely on (at least partially) dissociated mechanisms. We tested this hypothesis in a single case study, an electrophysiological study and in two behavioural experiments. In the neuropsychological study, we reported the case of patient CO, who showed Gerstmann syndrome after bilateral parietal damage and became unable to process sequence order relations (e.g. he couldn't recite the number sequence backwards, nor decide whether a number, letter, day or month comes before or after a given target in the corresponding sequence, and he was unable to verify the order of items in a pair). Nonetheless, the patient had largely preserved quantity processing abilities (he could compare numbers and dot patterns to find the smaller or larger, and showed a standard distance effect, he could produce a number smaller or larger than a given target, and match dot patterns with Arabic numerals). Overall, CO's pattern of performance was interpreted as reflecting the involvement of different mechanisms when processing quantity or sequence order relations. Our electrophysiological study corroborated this finding since different spatio-temporal patterns of the distance effect were observed when subjects had to process numbers in a quantity comparison task or in an order judgment task. Quantity processing elicited an early distance effect over the P2p component on left parietal sites, whereas the distance effect was slightly delayed and bilaterally distributed in the numerical order judgment task; and this latter task additionally recruited prefrontal regions on a later (P3-counterpart) component. Finally, our behavioural study further emphasized the involvement of different mechanisms underlying the processing of quantity and numerical order and provided some evidence about the nature of these specific mechanisms. In the number comparison (quantity) task, the standard distance effect was proposed to reflect the involvement of a magnitude comparison mechanism; whereas the reverse distance effect observed in the numerical order verification task was taken as evidence for the recruitment of a serial search (recitation) process. Besides, the pair-order effect was also found to specifically affect order but not quantity judgments. Taken together, the data collected in the present thesis lend further support to the hypothesis that quantity and numerical order rely on distinct processing mechanisms that can be damaged selectively after cerebral lesions, that recruit similar brain areas but with a different spatio-temporal course and that show specific behavioural markers.

Numbers convey different meanings when used in different contexts (Wiese, 2003). In a cardinal context, a number will tell us how many entities are in a set and convey quantity meaning. In an ordinal context, a number will refer to the relative position (or rank) of one element within a sequence; non-numerical ordered series (e.g. the letters of the alphabet) can also be used to provide meaningful order information. Because quantity and order are linked up with each other in the cognitive number domain (the larger the quantity a number refers to, the later it is located in the conventional number sequence), the question of whether they rely on some common or distinct underlying mechanism(s) is theoretically relevant and was addressed in the present thesis. Experimental studies showed evidence of both similarities (similar distance and SNARC effects, recruitment of parietal and frontal regions, and conjoint impairment or preservation after brain damage) and dissociations (different developmental course, dissociation after cerebral lesion, and specific behavioural markers) between quantity and order neuro-functional processes. The aim of the present thesis was to clarify the relationship between numerical quantity and order processing and to test the hypothesis that they rely on (at least partially) dissociated mechanisms. We tested this hypothesis in a single case study, an electrophysiological study and in two behavioural experiments. In the neuropsychological study, we reported the case of patient CO, who showed Gerstmann syndrome after bilateral parietal damage and became unable to process sequence order relations (e.g. he couldn't recite the number sequence backwards, nor decide whether a number, letter, day or month comes before or after a given target in the corresponding sequence, and he was unable to verify the order of items in a pair). Nonetheless, the patient had largely preserved quantity processing abilities (he could compare numbers and dot patterns to find the smaller or larger, and showed a standard distance effect, he could produce a number smaller or larger than a given target, and match dot patterns with Arabic numerals). Overall, CO's pattern of performance was interpreted as reflecting the involvement of different mechanisms when processing quantity or sequence order relations. Our electrophysiological study corroborated this finding since different spatio-temporal patterns of the distance effect were observed when subjects had to process numbers in a quantity comparison task or in an order judgment task. Quantity processing elicited an early distance effect over the P2p component on left parietal sites, whereas the distance effect was slightly delayed and bilaterally distributed in the numerical order judgment task; and this latter task additionally recruited prefrontal regions on a later (P3-counterpart) component. Finally, our behavioural study further emphasized the involvement of different mechanisms underlying the processing of quantity and numerical order and provided some evidence about the nature of these specific mechanisms. In the number comparison (quantity) task, the standard distance effect was proposed to reflect the involvement of a magnitude comparison mechanism; whereas the reverse distance effect observed in the numerical order verification task was taken as evidence for the recruitment of a serial search (recitation) process. Besides, the pair-order effect was also found to specifically affect order but not quantity judgments. Taken together, the data collected in the present thesis lend further support to the hypothesis that quantity and numerical order rely on distinct processing mechanisms that can be damaged selectively after cerebral lesions, that recruit similar brain areas but with a different spatio-temporal course and that show specific behavioural markers.