A Proposed Exercise to Reinforce Abstract Thinking for Upper-Division Computer and Electrical Engineering Students: Modeling a High-Speed Inverter Using Cognitive Representations and Abstract Algebra
Abstract: In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.
Cite this paper: Melendy, R. (2014) A Proposed Exercise to Reinforce Abstract Thinking for Upper-Division Computer and Electrical Engineering Students: Modeling a High-Speed Inverter Using Cognitive Representations and Abstract Algebra. World Journal of Engineering and Technology, 2, 298-304. doi: 10.4236/wjet.2014.24031.
References

[1]   Razavi, B. (2008) Fundamentals of Microelectronics. John Wiley & Sons, New York.

[2]   Neamen, D. (2008) Microelectronic Circuit Analysis and Design. McGraw-Hill, New York.

[3]   Jaeger, R. and Blalock, T. (2010) Microelectronic Circuit Design. McGraw-Hill, New York.

[4]   Sedra, A. and Smith, K. (2010) Microelectronic Circuits. Oxford University Press, New York, Oxford.

[5]   IEEE Standard 1076-2008 (2009) VHDL Language Reference Manual.
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4772740

[6]   Martin, A. (2007) The Representation of Object Concepts in the Brain. Annual Review of Psychology, 58, 25-45. http://dx.doi.org/10.1146/annurev.psych.57.102904.190143

[7]   Damasio, R. (1989) Time-Locked Multiregional Retro-Activation: A Systems Level Proposal for the Neural Substrates of Recall and Recognition. Cognition, 33, 25-62.
http://dx.doi.org/10.1016/0010-0277(89)90005-X

[8]   Simmons, K. and Barsalou, W. (2003) The Similarity-in-Topography Principle: Reconciling Theories of Conceptual Deficits and the Organization of Conceptual Knowledge in the Brain: Neuropsychological and Neuroimaging Perspectives. Cognitive Neuropsychology, 20, 451-486.
http://dx.doi.org/10.1080/02643290342000032

[9]   Barsalou, L. (1999) Perceptual Symbol Systems. Behavioral and Brain Sciences, 22, 577-609.

[10]   Simon, A., Tzur, R., Heinz, K. and Kinzel, M. (2004) Explicating a Mechanism for Conceptual Learning: Elaborating the Construct of Reflective Abstraction. Journal for Research in Mathematics Education, 35, 305-329. http://dx.doi.org/10.2307/30034818

[11]   Georgia Tech (2014) In 2014-2015 School of Electrical and Computer Engineering Projected Schedule of Graduate Courses. ECE Course Information.

[12]   Oregon State University (2014) ECE Graduate Courses. In 2014-2015 General Catalog and Schedule of Classes.

[14]   Purdue (2014) ECE Graduate Level Courses. In 2014-2015 Course Information: School of Electrical and Computer Engineering.

[15]   Stanford University (2014) Explore Courses. In 2014-2015 Stanford Bulletin.

[16]   Texas A&M University (2014) In 2014-2015 Course Descriptions—Graduate Courses, Electrical & Computer Engineering.

[17]   University of Washington (2014) Course Descriptions. In 2014-2015 University of Washington Electrical Engineering. http://www.ee.washington.edu/academics/course_descriptions.html

[18]   Lidl, R. and Pilz, G. (1998) Applied Abstract Algebra. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-2941-2

[19]   Durbin, J. (2005) Modern Algebra: An Introduction. John Wiley & Sons, Hoboken.

[20]   Roberts, F. and Tesman, B. (2008) Applied Combinatorics. CRC Press, Boca Raton.

[21]   Even, G. and Medina, M. (2012) Digital Logic Design: A Rigorous Approach. Cambridge University Press, Cambridge, New York, Melbourne. http://dx.doi.org/10.1017/CBO9781139226455

[22]   Judson, T. (2013) Abstract Algebra: Theory and Applications. Orthogonal Publishing, Ann Arbor.

[23]   Rao, V. (1989) Switch-Level Timing Simulation of MOS VLSI Circuits. Kluwer Academic Publishers, Norwell.

[24]   Fabricius, E. (1992) Modern Digital Design and Switching Theory. CRC Press, Boca Raton.

[25]   Epstein, G. (1993) Multiple-Valued Logic Design. IOP Publishing, England.

[26]   Kohavi, Z. and Jha, N. (2010) Switching and Finite Automata Theory. Cambridge University Press, Cambridge, New York, Melbourne.

[27]   Hill, F. and Peterson, G. (1981) Introduction to Switching Theory & Logical. John Wiley & Sons, New York.

[28]   Sasao, T. (1999) Switching Theory for Logic Synthesis. Kluwer Academic Publishers, Norwell. http://dx.doi.org/10.1007/978-1-4615-5139-3

[29]   Astola, J. and Stankovic, R. (2006) Fundamentals of Switching Theory and Logic Design: A Hand on Approach. Springer, Berlin.

[30]   Ojose, B. (2008) Applying Piaget’s Theory of Cognitive Development to Mathematics Instruction. The Mathematics Educator, 18, 26-30.

[31]   Hershkowitz, R., Schwarz, B. and Dreyfus, T. (2001) Abstraction in Context: Epistemic Actions. Journal for Research in Mathematics Education, 32, 195-222. http://dx.doi.org/10.2307/749673

[32]   Tervaniemi, M., Maury, S. and Näätänen, R. (1994) Neural Representations of Abstract Stimulus Features in the Human Brain as Reflected by the Mismatch Negativity. Neuroreport, 5, 844-846.
http://dx.doi.org/10.1097/00001756-199403000-00027

[33]   McGuire, P., Paulesu, E., Frackowiak, R. and Frith, C.D. (1996) Brain Activity during Stimulus Independent Thought. Neuroreport, 7, 2095-2099.

[34]   Kandel, E., Schwartz, J. and Jessell, T. (2000) Principles of Neural Science. McGraw-Hill Health Professions Division, Upper Saddle River, New York, San Francisco, London.

[35]   NXP Semiconductor (2014) 74LVC1G10 Single 3-Input NAND Gate Product Data Sheet.
http://www.nxp.com/documents/data_sheet/74LVC1G10.pdf

[36]   NXP Semiconductor (2014) 74LVC1G332 Single 3-Input OR Gate Product Data Sheet.
http://www.nxp.com/documents/data_sheet/74LVC1G10.pdf

[37]   Wechsler, D. (1958) The Measurement and Appraisal of Adult Intelligence. The Williams & Wilkins Company, Baltimore. http://dx.doi.org/10.1037/11167-000

[38]   Wechsler, D. (2008) Wechsler Adult Intelligence Scale, Fourth Edition (WAIS-IV). Pearson PsychCorp., San Antonio.

[39]   ACER (2014) Test of Abstract Reasoning Manual.
https://www.psychometrics.com/docs/tar-manual.pdf

[40]   Australian Council for Educational Research (ACER) (2014) Clinical Assessments.
http://www.acer.edu.au/assessment/all-clinical-assessments

Top