Research Report 2019: Grade- Distinction PDF

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THE PERPLEXITY OF ARITHMETIC

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The Perplexity of Arithmetic: Examination of Grouping Perception Arithmetic is a field in mathematics that consists of small numbers and a few operations commonly used in daily life tasks (Rivera & Garrigan, 2016). The inferior temporal gyrus processes the visuals of numbers, while the prefrontal cortex influences cognitive thinking in mathematics (Shum et al., 2013; Amalric & Dehaene, 2016). Thus, these neural regions are heavily involved in developing a spatial representation and visual grouping (Reiner, De Volder, & Rauschecker, 2014). It is imperative to understand students’ thought process when calculating equations, as multiple factors can contribute to students’ perplexity in comprehending arithmetic. The present study will analyse these issues regarding one’s cognitive skills in mathematics. A research conducted by Marghetis et al (2016) investigated participants’ capabilities on perceptual discriminability when examining coloured numbers between and within subexpressions. Participants were assessed if they could identify the correct order of operation without being confused by coloured operands. Syntax knowers had better accuracy in algebra task than non-syntax knowers, when assessing object-based attention for mathematical subexpressions. Algebraic performance was enhanced when a participant has more object based attention. This study also supports the signal detection theory (Green & Swets, 1966) where several participants experienced uncertainty when discriminating coloured operands and became more certain when solving non-coloured equations. Nevertheless, this research did contained limitations, with no affirmation of a randomly allocated sample. Also, the participants were referred to as adults with no specific age range, so mathematical skills may drastically vary. The present study targets to encompass a randomly allocated sample and particularly surveying on university undergraduates.

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The current study aims to replicate Marghetis et al. (2016) findings but with a more rigorous measure on syntax knowledge. A study conducted by Landy and Goldstone (2010) investigated participants’ abilities of analysing spacing conditions within an expression, since variation of sizes and perceptual groupings of an equation can cause confusion. (Landy & Goldstone, 2007). Therefore, spacing conditions will be experimented in the current study when scrutinizing participants’ abilities to recognise colour changes, and thus extends our measure from Marghetis et al.’s (2016) research. Additionally, if one has knowledge of mathematical syntax, they perform better in detecting sub-expression groupings within equations. However, visual cues, such as inconsistent spacing conditions, can confuse one’s perception. Landy and Goldstone (2010) deduced that the participants’ performance in consistent and inconsistent spacing conditions were significantly distinct, and thus majorly affects participants’ mathematical skill. The current study will analyse if visual cues can disrupt participants’ recognition in colour changes within and between algebraic expressions. This study aims to investigate respondents’ detection on colour changes within (x) arithmetic expressions, and between (+) sub-expression groupings. This would be conducted through providing participants with equations containing different types of spacing conditions and coloured operands, similar to the experimental design from Weinauer et al. (2017). The study’s hypotheses were (1) mathematical syntax knowers, but not non-knowers, will show an advantage detecting colour changes occurring within (x), compared to between (+) subexpression groupings in algebraic expressions; (2) this effect of mathematical syntax knowledge will remain when these sub-expression groupings are supported by syntax consistent spacing; and (3) this effect of mathematical syntax knowledge will be eliminated when these sub-expression groupings are disrupted by syntax inconsistent spacing. Method

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Participants The participants were selected from PSYC105/PSYX105 students who volunteered. Participants were randomly allocated into each of the three spacing conditions (neutral, consistent and inconsistent). The survey initially had 642 respondents, but ultimately finalised into 450 to have an equal amount of participants with 75 syntax knowers and 75 non-syntax knowers in each condition. The age range was between 17 to 62 years (M =20.7, SD=5.71). Female participants consisted of 74% of the total sample while male participants consisted of 22.44%. Measures Mathematical syntax knowledge among participants were determined from a measure with 25 maths questions, with the experiment being programmed through the PsyToolkit software. This measure is a modified version to Marghetis et al.’s (2016) experiment, based on the two questions of their research. Each math question contained four response choices in a random order, with a limited time of 10 seconds to answer. If the time runs out, a ‘Too Slow” message will appear. Participants receive a score of one for each correct response, with 25 being the highest score. If they score 85% or over (≥22/25), they were assigned into the ‘Syntax Knowers’ group. If the score was under 85%, they were assigned into the ‘Syntax Non-Knowers’ group. The dependent variable is the syntax grouping index, which provides a measure based on the participants’ performance on the colour verification task. This was based on subtracting the participants’ accuracy in calculating coloured letters between mathematical syntax groupings, from accuracy on coloured letters within mathematical syntax groupings. It was also referred to as ‘Accuracy within –Accuracy between’. Procedure

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Participants were required to sign an informed consent and answer demographic questions, such as age and gender, before the experiment started. After the participants were randomly assigned into each of the three spacing conditions, they were instructed to complete a computer-based task containing four blocks of trials, with a total of 352 trials. Every condition included the same set of instructions and tasks. Each trial was initiated when black text appeared on the computer screen for 3000ms, before the participant could start on the colour judgement task or the algebraic equivalence task. Each block eventually lead into a pause, which ended if one presses ‘space bar’ to continue. In the colour judgement task, the participants’ response time and accuracy were tested when they were required to distinguish between two coloured letters. Participants’ response time and accuracy were also tested in the algebraic equivalence task, when they were provided two equations on either side of ‘=’ and determine if the equations were equivalent. In both tasks, participants were given 10 seconds to respond to the mathematical expression and were either given ‘Correct’, ‘Wrong’ or ‘Too Slow’ as feedback for 500ms before the next question. Results Descriptives The syntax grouping index ranged from -2 to 5. Syntax-knowers tend to score above 0 while non-syntax knowers tend to score below 0. The results of syntax knowers’ accuracy exceeded over 0 in all spacing conditions, except for inconsistent spacing conditions. The results of non-syntax knowers’ accuracy scored below 0 in all spacing conditions, except for neutral spacing conditions. Hypothesis testing

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All hypotheses were tested in independent sample t-tests with equal variances. The results supported Hypothesis 1, with the syntax grouping index being significantly higher among syntax knowers (M= 3.03, SD= 1.95) than non-syntax knowers (M= 0.36, SD= 1.99); t(148)= 8.2976, p < .001. Hypothesis 2 was also supported, as syntax grouping was significantly higher among syntax knowers (M= 4.47, SD= 1.81) than non-syntax knowers (M= -0.28, SD= 2.66); t(148)= 12.76, p < .001. However, the results did not support Hypothesis 3, with syntax knowers (M= -0.35, SD= 2.13) scoring non-significantly more than non-syntax knowers (M= -0.55, SD= 2.2) in syntax grouping, t(148)= 0.56, p= 0.57. Discussion This current study examined two types of conditions in the tested equations; spacing conditions and colour verification. Hypothesis 1 and 2 were supported since syntax knowers have more advantage in detecting colour changes than non-syntax knowers in neutral and consistent spacing conditions, but Hypothesis 3 was not supported as syntax knowers still performed better than non-syntax knowers in inconsistent spacing conditions. The results are in line and extended upon previous research that demonstrated the same conclusion (e.g, Landy & Goldstone, 2010; Rivera and Garrigan, 2016). Strengths of the study include a large sample size being randomly allocated into 3 conditions. By ensuring a large sample size can help increasing the validity of our research, whilst a randomly allocated sample provides an unbiased and more variation of the data. However, limitations of the study includes the lack of information on participants’ true capabilities of syntax knowledge. Since the experiment is voluntary, some individuals with poor arithmetic skills may likely choose to ignore participating, thus the ‘non-syntax’ knowers may possibly be more capable in arithmetic than initially perceived. Furthermore, the mark required to classify as a syntax knower may potentially be too high. The measure of the 85

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percent cut off can be quite arbitrary and does not precisely represent a syntax knower’s capabilities; thus possibly underestimating the actual number of algebraically skilled participants. Future research could incorporate further ideas into engaging students to partake in the research so the results can be unbiased, while also experimenting a shorter cut off to more accurately measure the participants’ arithmetic ability. These results can apply to real-world tasks such as mathematics in schools. Colour perception and spatial layout can potentially affect student’s abilities when solving equations. For example, if students are undergoing a math examination without properly laid out equation, their performance may potentially be reduced due to confusion. Also, future studies can investigate how coloured symbols and spatial layouts can affect non-mathematical areas, such as advertisement. In conclusion, this study is in line and extended upon previous research that spacing conditions and colour perception affect one’s ability to determine the correct order of operation. Future research can inspect ways to improve students’ mathematical skill on identifying the correct order of operation, regardless of problematic conditions.

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References Amalric, M., Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 49094917. doi: 10.1073/1603205113 Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, ER., Landy, D. (2017). Adapting Perception, Action, and Technology for Mathematical Reasoning. Current Directions in Psychological Science, 26, 434-441. doi: 10.1177/0963721417704888 Green, D. M., Swets, J. A. (1966). Signal detection theory and psychophysics. New York: Wiley. Landy, D., Goldstone, R. L. (2007). How abstract is symbolic thought? J Exp Psychol Learn Mem Cogn, 33, 720-733. doi: 10.1037/0278-7393.33.4.720 Landy, D., Goldstone, R. L. (2010). Proximity and precedence in arithmetic. The Quarterly Journal of Experimental Psychology, 63(10), 1953-1968. doi: 10.1080/17470211003787619 Marghetis, T., Goldstone, R. L., Landy, D. (2016). Mastering algebra retrains the visual system to perceive hierarchical structure in equations. Cognitive Research: Principles and Implications volume, 1, 25-35. doi: 10.1186/s41235-016-0020-9 Reiner, L., De Volder, A. G., Rauschecker, J. P. (2014). Cortical plasticity and preserved function in early blindness. Neuroscience and Biobehavioral Reviews, 41, 53–63. doi: 10.1016/j.neubiorev.2013.01.025 Rivera, J., Garrigan, P. (2016). Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions. Memory and Cognition, 44, 750-761. doi: 10.3758/s13421016-0593-z

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Shum, J., Hermes, D., Foster, B. L., Dastjerdi, M., Rangarajan, V., Winawer, J., . . . Parvizi, J. (2013). A brain area for visual numerals. Journal of Neuroscience, 33(16), 6709–6715. doi: 10.1523/JNEUROSCI.4558-12.2013 Stoet, G. (2010). PsyToolkit - A software package for programming psychological experiments using Linux. Behavior Research Methods, 42(4), 1096-1104 Stoet, G. (2017). PsyToolkit: A novel web-based method for running online questionnaires and reaction-time experiments. Teaching of Psychology, 44(1), 24-31...