本研究以一套幾何剛性變換測驗檢視37位學生對於鈍角與銳角三角形心像轉換的能力。測驗要求學生根據給定的三角形，建構對應邊角關係的心像，進行心像旋轉，並評估給定旋轉後的圖形是否正確。研究以E-prime設備收集學生的行為資料，包括答題正確性及反應時間，並利用事件相關電位資料探究學生在進行幾何圖形剛性變換時的腦波。

行為資料分析結果發現，旋轉角度和三角形類型對答對率有顯著效果，角度×類型在答對率有顯著交互作用；旋轉角度對反應時間有顯著效果，而三角形類型對反應時間沒有顯著效果，角度×類型在反應時間上有顯著交互作用。腦波資料分析結果顯示，第一階段三角形的呈現方向、三角形類型對部分頂葉電極平均振幅有顯著效果，90°方向的平均振幅大於0°和180°方向；鈍角三角形的平均振幅大於銳角三角形。第二階段旋轉角度對部分頂葉電極平均振幅和潛伏時間有顯著效果，旋轉0°的平均振幅顯著小於其他旋轉角度，潛伏時間顯著長於其他旋轉角度。第三階段旋轉角度對部分頂葉電極平均振幅有顯著效果，旋轉0°的平均振幅大於旋轉135°。

This study examined the ability of 37 students to transform mental images of obtuse and acute triangles using a geometric rigid transformation test. The test required students to construct mental images based on given triangles, rotate these images according to the relationships between the edges and angles of the triangles, and evaluate the correctness of the results. The study collected students' behavioral data using E-prime, including accuracy in answers and reaction times. It also utilized event-related potentials to investigate students' brain waves during the rigid transformation test of geometric shapes.

Based on the behavioral data, the analysis indicated significant effects of rotation angles and types of triangles on accuracy rates. Additionally, there was a significant interaction between rotation angles and types of triangles. The analysis also indicated significant effects of rotation angles on reaction time, while the types of triangles did not show a significant effect. There was also a significant interaction between rotation angles and types of triangles.

Based on the ERP data, the analysis revealed that in the first stage, there were significant effects of orientations and types of triangles on the mean amplitudes over certain parietal electrodes. The mean amplitudes at 90° were larger than those at 0° and 180°, and the mean amplitudes in response to obtuse triangles were larger than those in response to acute triangles. In the second stage, there were significant effects of rotation angles on the mean amplitudes and latencies over certain parietal electrodes. The mean amplitudes at 0° were smaller, and the latencies were longer compared to other degrees. In the third stage, there were significant effects of rotation angles on the mean amplitudes over certain parietal electrodes. The mean amplitudes at 0° were larger than those at 135°

I.Introduction....................................................1
1.1 Background....................................................1
1.2 Purpose.......................................................3
1.3 Terminology...................................................4
1.4 Limitations...................................................5
II.Literature Review..............................................7
2.1 Mental Imagery................................................7
2.2 Spatial Ability, Mental Rotation, and Mathematics Education...11
2.3 Neurocognitive Science Studies on Mental Rotation.............14
III.Methodology...................................................20
3.1 Research Process..............................................20
3.2 Conceptual Framework..........................................22
3.3 Research Hypotheses...........................................22
3.4 Participants..................................................23
3.5 Research Tools................................................24
3.6 Data Collection...............................................30
3.7 Statistical Analysis..........................................32
IV.Result.........................................................34
4.1 Behavioral Performance........................................34
4.2 Brain Wave Signals............................................39
V.Conclusion......................................................108
Reference.........................................................113

林福來(民76)。國中生反射、旋轉、平移概念發展研究。行政院國家科學委員會專題研究計畫報告（編號：NSC75-0111-S003-01、NSC76-0111-S003-12）。
Amorim, M. A., Isableu, B., & Jarraya, M. (2006). Embodied spatial transformations:" body analogy" for the mental rotation of objects. Journal of Experimental Psychology: General, 135(3), 327.
Cheng, Y. L., & Mix, K. S. (2014). Spatial training improves children's mathematics ability. Journal of cognition and development, 15(1), 2-11.
Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of experimental psychology: General, 122(3), 371.
Ekstrom, R. B., & Harman, H. H. (1976). Manual for kit of factor-referenced cognitive tests, 1976. Educational testing service.
Farah, M. J. (1988). Is visual imagery really visual? Overlooked evidence from neuropsychology. Psychological review, 95(3), 307.
French, J. W. (1951). The description of aptitude and achievement tests in terms of rotated factors.
French, J. W., Ekstrom, R. B., & Price, L. (1963). Manual for kit of reference tests for cognitive factors . Princeton, NJ: Educational Testing Service.
Guilford, J. P., & Lacey, J. I. (Eds.). (1947). Printed classification tests (No. 5). US Government Printing Office.
Hawes, Z., Moss, J., Caswell, B., & Poliszczuk, D. (2015). Effects of mental rotation training on children’s spatial and mathematics performance: A randomized controlled study. Trends in Neuroscience and Education, 4(3), 60-68.
Hegarty, M., & Waller, D. (2005). Individual differences in spatial abilities. The Cambridge handbook of visuospatial thinking, 121-169.
Heil, M. (2002). The functional significance of ERP effects during mental rotation. Psychophysiology, 39(5), 535-545.
Heil, M., & Rolke, B. (2002). Toward a chronopsychophysiology of mental rotation. Psychophysiology, 39(4), 414-422.
Hilton, C., Raddatz, L., & Gramann, K. (2022). A general spatial transformation process? Assessing the neurophysiological evidence on the similarity of mental rotation and folding. Neuroimage: Reports, 2(2), 100092.
Hsu, H.-Y. (In preparation). Brain activity specific to mental rotations of triangles.
Jansen, P., Render, A., Scheer, C., & Siebertz, M. (2020). Mental rotation with abstract and embodied objects as stimuli: Evidence from event-related potential (ERP). Experimental Brain Research, 238, 525-535.
Kerr, N. H. (1983). The role of vision in" visual imagery" experiments: evidence from the congenitally blind. Journal of Experimental Psychology: General, 112(2), 265.
Kosslyn, S. M. (1975). Information representation in visual images. Cognitive psychology, 7(3), 341-370.
Kosslyn, S. M. (1983). Ghosts in the mind's machine: Creating and using images in the brain. Norton.
Kosslyn, S. M. (1995). Mental imagery. In S. M. Kosslyn & D. N. Osherson (Eds.), Visual cognition: An invitation to cognitive science (2nd ed., pp. 267–296). The MIT Press.
Kosslyn, S. M., Ball, T. M., & Reiser, B. J. (1978). Visual images preserve metric spatial information: evidence from studies of image scanning. Journal of experimental psychology: Human perception and performance, 4(1), 47.
Kozhevnikov, M., Hegarty, M., & Mayer, R. (1999). Students' Use of Imagery in Solving Qualitative Problems in Kinematics.
Krause, D., Richert, B., & Weigelt, M. (2021). Neurophysiology of embodied mental rotation: Event‐related potentials in a mental rotation task with human bodies as compared to alphanumeric stimuli. European Journal of Neuroscience, 54(4), 5384-5403.
Kyttälä, M., & Lehto, J. E. (2008). Some factors underlying mathematical performance: The role of visuospatial working memory and non-verbal intelligence. European Journal of Psychology of Education, 23, 77-94.
Lean, G., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12(3), 267-299.
Leikin, R. (2018). How Can Cognitive Neuroscience Contribute to Mathematics Education? Bridging the Two Research Areas. In Invited Lectures from the 13th International Congress on Mathematical Education (pp. 363-383). Springer International Publishing.
Ma, H., Li, X., Liu, M., Ma, H., & Zhang, D. (2018). Mental rotation effect on adult immigrants with long-term exposure to high altitude in Tibet: an ERP study. Neuroscience, 386, 339-350.
McGee, M. G. (1979). Human spatial abilities: psychometric studies and environmental, genetic, hormonal, and neurological influences. Psychological bulletin, 86(5), 889.
Ministry of Education. (2014). Curriculum guidelines of 12-year basic education.
Muthukumaraswamy, S. D., Johnson, B. W., & Hamm, J. P. (2003). A high density ERP comparison of mental rotation and mental size transformation. Brain and Cognition, 52(2), 271-280.
Núñez-Peña, M. I., & Aznar-Casanova, J. A. (2009). Mental rotation of mirrored letters: evidence from event-related brain potentials. Brain and Cognition, 69(1), 180-187.
Pearson, J., & Kosslyn, S. M. (2015). The heterogeneity of mental representation: Ending the imagery debate. Proceedings of the national academy of sciences, 112(33), 10089-10092.
Pellkofer, J., Jansen, P., & Heil, M. (2012). Lateralization of event-related potential effects during mental rotation of polygons. NeuroReport, 23(10), 585-589.
Peronnet, F., & Farah, M. J. (1989). Mental rotation: An event-related potential study with a validated mental rotation task. Brain and cognition, 9(2), 279-288.
Provost, A., Johnson, B., Karayanidis, F., Brown, S. D., & Heathcote, A. (2013). Two routes to expertise in mental rotation. Cognitive science, 37(7), 1321-1342.
Pylyshyn, Z. W. (1973). What the mind's eye tells the mind's brain: A critique of mental imagery. Psychological bulletin, 80(1), 1.
Ramful, A., Lowrie, T., & Logan, T. (2017). Measurement of spatial ability: Construction and validation of the spatial reasoning instrument for middle school students. Journal of Psychoeducational Assessment, 35(7), 709-727.
Schendan, H. E., & Lucia, L. C. (2009). Visual object cognition precedes but also temporally overlaps mental rotation. Brain Research, 1294, 91-105.
Shepard, R. N., & Metzler, J. (1971). Mental rotation of three-dimensional objects. Science, 171(3972), 701-703.
Stuss, D. T., Sarazin, F. F., Leech, E. E., & Picton, T. W. (1983). Event-related potentials during naming and mental rotation. Electroencephalography and clinical Neurophysiology, 56(2), 133-146.
Tao, W., Liu, Q., Huang, X., Tao, X., Yan, J., Teeter, C. J., ... & Sun, H. J. (2009). Effect of degree and direction of rotation in egocentric mental rotation of hand: an event-related potential study. NeuroReport, 20(2), 180-185.
ter Horst, A. C., Jongsma, M. L., Janssen, L. K., Van Lier, R., & Steenbergen, B. (2012). Different mental rotation strategies reflected in the rotation related negativity. Psychophysiology, 49(4), 566-573.
VandenBos, G. R. (2007). APA dictionary of psychology. American Psychological Association.
Varriale, V., van der Molen, M. W., & De Pascalis, V. (2018). Mental rotation and fluid intelligence: A brain potential analysis. Intelligence, 69, 146-157.
Vastano, R., & Widerstrom-Noga, E. (2023). Event-related potentials during mental rotation of body-related stimuli in spinal cord injury population. Neuropsychologia, 179, 108447.
Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of educational Psychology, 101(4), 817.
Weckbacher, L. M., & Okamoto, Y. (2014). Mental rotation ability in relation to self-perceptions of high school geometry. Learning and Individual Differences, 30, 58-63.
Wijers, A. A., Otten, L. J., Feenstra, S., Mulder, G., & Mulder, L. J. (1989). Brain potentials during selective attention, memory search, and mental rotation. Psychophysiology, 26(4), 452-467.
Yanik, H. B., & Flores, A. (2009). Understanding rigid geometric transformations: Jeff's learning path for translation. The Journal of Mathematical Behavior, 28(1), 41-57.
Zacks, J. M., Ollinger, J. M., Sheridan, M. A., & Tversky, B. (2002). A parametric study of mental spatial transformations of bodies. Neuroimage, 16(4), 857-872.
Zhao, B., Della Sala, S., Zeman, A., & Gherri, E. (2022). Spatial transformation in mental rotation tasks in aphantasia. Psychonomic Bulletin & Review, 29(6), 2096-2107.