Parity of zero (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Parity of zero" in English language version.

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Penner 1999, p. 34: Lemma B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0." Penner, Robert C. (1999), Discrete Mathematics: Proof Techniques and Mathematical Structures, River Edge: World Scientific, ISBN 978-981-02-4088-2
Penner 1999, p. 34. Penner, Robert C. (1999), Discrete Mathematics: Proof Techniques and Mathematical Structures, River Edge: World Scientific, ISBN 978-981-02-4088-2
Salzmann et al. 2007, p. 168 Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (2007), The Classical Fields: Structural Features of the Real and Rational Numbers, Cambridge University Press, ISBN 978-0-521-86516-6
Krantz 2001, p. 4 Krantz, Steven George (2001), Dictionary of algebra, arithmetic, and trigonometry, CRC Press, ISBN 978-1-58488-052-3
Salzmann et al. 2007, p. 224 Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (2007), The Classical Fields: Structural Features of the Real and Rational Numbers, Cambridge University Press, ISBN 978-0-521-86516-6
Keith 2006, pp. 35–68 "There was little disagreement on the idea of zero being an even number. The students convinced the few who were not sure with two arguments. The first argument was that numbers go in a pattern ...odd, even, odd, even, odd, even... and since two is even and one is odd then the number before one, that is not a fraction, would be zero. So zero would need to be even. The second argument was that if a person has zero things and they put them into two equal groups then there would be zero in each group. The two groups would have the same amount, zero" Keith, Annie (2006), "Mathematical Argument in a Second Grade Class: Generating and Justifying Generalized Statements about Odd and Even Numbers", Teachers Engaged in Research: Inquiry in Mathematics Classrooms, Grades Pre-K-2, IAP, ISBN 978-1-59311-495-4
Grimes 1975, p. 156 "...one can pose the following questions to married couples of his acquaintance: (1) Is zero an even number? ... Many couples disagree..." Grimes, Joseph E. (1975), The Thread of Discourse, Walter de Gruyter, ISBN 978-90-279-3164-1
Wilden & Hammer 1987, p. 104 Wilden, Anthony; Hammer, Rhonda (1987), The rules are no game: the strategy of communication, Routledge Kegan & Paul, ISBN 978-0-7100-9868-9
Kaplan Staff 2004, p. 227 Kaplan Staff (2004), Kaplan SAT 2400, 2005 Edition, Simon and Schuster, ISBN 978-0-7432-6035-0
Graduate Management Admission Council 2005, pp. 108, 295–297; Educational Testing Service 2009, p. 1 Graduate Management Admission Council (September 2005), The Official Guide for GMAT Review (11th ed.), McLean, Virginia, USA: Graduate Management Admission Council, ISBN 978-0-9765709-0-5 Educational Testing Service (2009), Mathematical Conventions for the Quantitative Reasoning Measure of the GRE revised General Test (PDF), Educational Testing Service, retrieved 2011-09-06
Arsham 2002; The quote is attributed to the heute broadcast of October 1, 1977. Arsham's account is repeated by Crumpacker (2007, p. 165). Arsham, Hossein (January 2002), "Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives", The Pantaneto Forum, archived from the original on 2007-09-25, retrieved 2007-09-24 Crumpacker, Bunny (2007), Perfect Figures: The Lore of Numbers and How We Learned to Count, Macmillan, ISBN 978-0-312-36005-4
Diagram Group 1983, p. 213 Diagram Group (1983), The Official World Encyclopedia of Sports and Games, Paddington Press, ISBN 978-0-448-22202-8

arxiv.org

Caldwell & Xiong 2012, pp. 5–6. Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C
Gowers 2002, p. 118 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012). Gowers, Timothy (2002), Mathematics: A Very Short Introduction, Oxford University Press, ISBN 978-0-19-285361-5 Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C

books.google.com

Arnold 1919, p. 21 "By the same test zero surpasses all numbers in 'evenness.'"; Wong 1997, p. 479 "Thus, the integer b000⋯000 = 0 is the most 'even.' Arnold, C. L. (January 1919), "The Number Zero", The Ohio Educational Monthly, 68 (1): 21–22, retrieved 2010-04-11 Wong, Samuel Shaw Ming (1997), Computational Methods in Physics and Engineering, World Scientific, ISBN 978-981-02-3043-2
A 1980 Maryland law specifies, "(a) On even numbered calendar dates gasoline shall only be purchased by operators of vehicles bearing personalized registration plates containing no numbers and registration plates with the last digit ending in an even number. This shall not include ham radio operator plates. Zero is an even number; (b) On odd numbered calendar dates ..." Partial quotation taken from Department of Legislative Reference (1974), Laws of the State of Maryland, Volume 2, p. 3236, retrieved 2013-06-02

deseretnews.com

Sones & Sones 2002 "It follows that zero is even, and that 2/20/2000 nicely cracks the puzzle. Yet it's always surprising how much people are bothered by calling zero even..."; Column 8 readers 2006a "'...according to mathematicians, the number zero, along with negative numbers and fractions, is neither even nor odd,' writes Etan..."; Column 8 readers 2006b "'I agree that zero is even, but is Professor Bunder wise to 'prove' it by stating that 0 = 2 x 0? By that logic (from a PhD in mathematical logic, no less), as 0 = 1 x 0, it's also odd!' The prof will dispute this and, logically, he has a sound basis for doing so, but we may be wearing this topic a little thin ..." Sones, Bill; Sones, Rich (2002-05-08), "To hide your age, button your lips", Deseret News, p. C07, retrieved 2014-06-21 Column 8 readers (2006-03-10), "Column 8", The Sydney Morning Herald (First ed.), p. 18, Factiva SMHH000020060309e23a00049{{citation}}: CS1 maint: numeric names: authors list (link) Column 8 readers (2006-03-16), "Column 8", The Sydney Morning Herald (First ed.), p. 20, Factiva SMHH000020060315e23g0004z{{citation}}: CS1 maint: numeric names: authors list (link)
Sones & Sones 2002 "Penn State mathematician George Andrews, who recalls a time of gas rationing in Australia ... Then someone in the New South Wales parliament asserted this meant plates ending in zero could never get gas, because 'zero is neither odd nor even. So the New South Wales parliament ruled that for purposes of gas rationing, zero is an even number!'" Sones, Bill; Sones, Rich (2002-05-08), "To hide your age, button your lips", Deseret News, p. C07, retrieved 2014-06-21

diva-portal.org

Dickerson & Pitman 2012, p. 191. Dickerson, David S.; Pitman, Damien J. (July 2012), Tai-Yih Tso (ed.), "Advanced college-level students' categorization and use of mathematical definitions" (PDF), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 2: 187–195
Levenson, Tsamir & Tirosh 2007; Dickerson & Pitman 2012 Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero", The Journal of Mathematical Behavior, 26 (2): 83–95, doi:10.1016/j.jmathb.2007.05.004 Dickerson, David S.; Pitman, Damien J. (July 2012), Tai-Yih Tso (ed.), "Advanced college-level students' categorization and use of mathematical definitions" (PDF), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 2: 187–195
Dickerson & Pitman 2012. Dickerson, David S.; Pitman, Damien J. (July 2012), Tai-Yih Tso (ed.), "Advanced college-level students' categorization and use of mathematical definitions" (PDF), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 2: 187–195

doi.org

Compare Lichtenberg (1972, p. 535) Fig. 1 Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
Lichtenberg 1972, pp. 535–536 "...numbers answer the question How many? for the set of objects ... zero is the number property of the empty set ... If the elements of each set are marked off in groups of two ... then the number of that set is an even number." Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
Lichtenberg 1972, pp. 535–536 "Zero groups of two stars are circled. No stars are left. Therefore, zero is an even number." Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
Lichtenberg 1972, p. 537; compare her Fig. 3. "If the even numbers are identified in some special way ... there is no reason at all to omit zero from the pattern." Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
Lichtenberg 1972, pp. 537–538 "At a more advanced level ... numbers expressed as (2 × ▢) + 0 are even numbers ... zero fits nicely into this pattern." Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
Lorentz 1994, pp. 5–6; Lovas & Pfenning 2008, p. 115; Nipkow, Paulson & Wenzel 2002, p. 127 Lorentz, Richard J. (1994), Recursive Algorithms, Intellect Books, ISBN 978-1-56750-037-0 Lovas, William; Pfenning, Frank (2008-01-22), "A Bidirectional Refinement Type System for LF", Electronic Notes in Theoretical Computer Science, 196: 113–128, doi:10.1016/j.entcs.2007.09.021 Nipkow, Tobias; Paulson, Lawrence C.; Wenzel, Markus (2002), Isabelle/Hol: A Proof Assistant for Higher-Order Logic, Springer, ISBN 978-3-540-43376-7
This is the timeframe in United States, Canada, Great Britain, Australia, and Israel; see Levenson, Tsamir & Tirosh (2007, p. 85). Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero", The Journal of Mathematical Behavior, 26 (2): 83–95, doi:10.1016/j.jmathb.2007.05.004
Levenson, Tsamir & Tirosh 2007, pp. 83–95 Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero", The Journal of Mathematical Behavior, 26 (2): 83–95, doi:10.1016/j.jmathb.2007.05.004
Levenson, Tsamir & Tirosh 2007; Dickerson & Pitman 2012 Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero", The Journal of Mathematical Behavior, 26 (2): 83–95, doi:10.1016/j.jmathb.2007.05.004 Dickerson, David S.; Pitman, Damien J. (July 2012), Tai-Yih Tso (ed.), "Advanced college-level students' categorization and use of mathematical definitions" (PDF), Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 2: 187–195
Hill et al. 2008, pp. 446–447. Hill, Heather C.; Blunk, Merrie L.; Charalambous, Charalambos Y.; Lewis, Jennifer M.; Phelps, Geoffrey C.; Sleep, Laurie; Ball, Deborah Loewenberg (2008), "Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study", Cognition and Instruction, 26 (4): 430–511, doi:10.1080/07370000802177235
Lichtenberg 1972, p. 535 Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number", The Arithmetic Teacher, 19 (7): 535–538, doi:10.5951/AT.19.7.0535
As concluded by Levenson, Tsamir & Tirosh (2007, p. 93), referencing Freudenthal (1983, p. 460) Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero", The Journal of Mathematical Behavior, 26 (2): 83–95, doi:10.1016/j.jmathb.2007.05.004 Freudenthal, Hans (1983), Didactical phenomenology of mathematical structures, Dordrecht, Netherlands: Reidel
Nuerk, Iversen & Willmes (2004, p. 851): "It can also be seen that zero strongly differs from all other numbers regardless of whether it is responded to with the left or the right hand. (See the line that separates zero from the other numbers.)" Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
See data throughout Dehaene, Bossini & Giraux (1993), and summary by Nuerk, Iversen & Willmes (2004, p. 837). Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
Dehaene, Bossini & Giraux 1993, pp. 374–376 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, pp. 376–377 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, p. 376 "In some intuitive sense, the notion of parity is familiar only for numbers larger than 2. Indeed, before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical definition. The evidence, in brief, suggests that instead of being calculated on the fly by using a criterion of divisibility by 2, parity information is retrieved from memory together with a number of other semantic properties ... If a semantic memory is accessed in parity judgments, then interindividual differences should be found depending on the familiarity of the subjects with number concepts." Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Nuerk, Iversen & Willmes 2004, pp. 838, 860–861 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272

ets.org

Graduate Management Admission Council 2005, pp. 108, 295–297; Educational Testing Service 2009, p. 1 Graduate Management Admission Council (September 2005), The Official Guide for GMAT Review (11th ed.), McLean, Virginia, USA: Graduate Management Admission Council, ISBN 978-0-9765709-0-5 Educational Testing Service (2009), Mathematical Conventions for the Quantitative Reasoning Measure of the GRE revised General Test (PDF), Educational Testing Service, retrieved 2011-09-06

handle.net

hdl.handle.net

Ball, Hill & Bass 2005, pp. 14–16 Ball, Deborah Loewenberg; Hill, Heather C.; Bass, Hyman (2005), "Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide?", American Educator, hdl:2027.42/65072

harvard.edu

ui.adsabs.harvard.edu

Caldwell & Xiong 2012, pp. 5–6. Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C
Gowers 2002, p. 118 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012). Gowers, Timothy (2002), Mathematics: A Very Short Introduction, Oxford University Press, ISBN 978-0-19-285361-5 Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C

jewishworldreview.com

Snow 2001; Morgan 2001 Snow, Tony (2001-02-23), "Bubba's fools", Jewish World Review, retrieved 2009-08-22 Morgan, Frank (2001-04-05), "Old Coins", Frank Morgan's Math Chat, The Mathematical Association of America, archived from the original on 2009-01-08, retrieved 2009-08-22

jsonline.com

Steinberg 1999; Siegel 1999; Stingl 2006 Steinberg, Neil (1999-11-30), "Even year, odd facts", Chicago Sun-Times (5XS ed.), p. 50, Factiva chi0000020010826dvbu0119h Siegel, Robert (1999-11-19), "Analysis: Today's date is signified in abbreviations using only odd numbers. 1-1, 1-9, 1-9-9-9. The next time that happens will be more than a thousand years from now.", All Things Considered, National Public Radio Stingl, Jim (2006-04-05), "01:02:03 04/05/06; We can count on some things in life", Milwaukee Journal Sentinel (Final ed.), p. B1, archived from the original on 2006-04-27, retrieved 2014-06-21

maa.org

Snow 2001; Morgan 2001 Snow, Tony (2001-02-23), "Bubba's fools", Jewish World Review, retrieved 2009-08-22 Morgan, Frank (2001-04-05), "Old Coins", Frank Morgan's Math Chat, The Mathematical Association of America, archived from the original on 2009-01-08, retrieved 2009-08-22

nih.gov

pubmed.ncbi.nlm.nih.gov

Nuerk, Iversen & Willmes (2004, p. 851): "It can also be seen that zero strongly differs from all other numbers regardless of whether it is responded to with the left or the right hand. (See the line that separates zero from the other numbers.)" Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
See data throughout Dehaene, Bossini & Giraux (1993), and summary by Nuerk, Iversen & Willmes (2004, p. 837). Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
Nuerk, Iversen & Willmes 2004, pp. 838, 860–861 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272

npr.org

Steinberg 1999; Siegel 1999; Stingl 2006 Steinberg, Neil (1999-11-30), "Even year, odd facts", Chicago Sun-Times (5XS ed.), p. 50, Factiva chi0000020010826dvbu0119h Siegel, Robert (1999-11-19), "Analysis: Today's date is signified in abbreviations using only odd numbers. 1-1, 1-9, 1-9-9-9. The next time that happens will be more than a thousand years from now.", All Things Considered, National Public Radio Stingl, Jim (2006-04-05), "01:02:03 04/05/06; We can count on some things in life", Milwaukee Journal Sentinel (Final ed.), p. B1, archived from the original on 2006-04-27, retrieved 2014-06-21

pantaneto.co.uk

Arsham 2002; The quote is attributed to the heute broadcast of October 1, 1977. Arsham's account is repeated by Crumpacker (2007, p. 165). Arsham, Hossein (January 2002), "Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives", The Pantaneto Forum, archived from the original on 2007-09-25, retrieved 2007-09-24 Crumpacker, Bunny (2007), Perfect Figures: The Lore of Numbers and How We Learned to Count, Macmillan, ISBN 978-0-312-36005-4

semanticscholar.org

api.semanticscholar.org

Nuerk, Iversen & Willmes (2004, p. 851): "It can also be seen that zero strongly differs from all other numbers regardless of whether it is responded to with the left or the right hand. (See the line that separates zero from the other numbers.)" Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
See data throughout Dehaene, Bossini & Giraux (1993), and summary by Nuerk, Iversen & Willmes (2004, p. 837). Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
Nuerk, Iversen & Willmes 2004, pp. 838, 860–861 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272

umich.edu

www-personal.umich.edu

Ball, Lewis & Thames (2008, p. 15) discuss this challenge for the elementary-grades teacher, who wants to give mathematical reasons for mathematical facts, but whose students neither use the same definition, nor would understand it if it were introduced. Ball, Deborah Loewenberg; Lewis, Jennifer; Thames, Mark Hoover (2008), "Making mathematics work in school" (PDF), Journal for Research in Mathematics Education, M14: 13–44 and 195–200, retrieved 2010-03-04
Ball, Lewis & Thames 2008, p. 27, Figure 1.5 "Mathematical claims about zero." Ball, Deborah Loewenberg; Lewis, Jennifer; Thames, Mark Hoover (2008), "Making mathematics work in school" (PDF), Journal for Research in Mathematics Education, M14: 13–44 and 195–200, retrieved 2010-03-04
Ball, Lewis & Thames 2008, p. 16. Ball, Deborah Loewenberg; Lewis, Jennifer; Thames, Mark Hoover (2008), "Making mathematics work in school" (PDF), Journal for Research in Mathematics Education, M14: 13–44 and 195–200, retrieved 2010-03-04
Ball, Lewis & Thames 2008, p. 15. See also Ball's keynote for further discussion of appropriate definitions. Ball, Deborah Loewenberg; Lewis, Jennifer; Thames, Mark Hoover (2008), "Making mathematics work in school" (PDF), Journal for Research in Mathematics Education, M14: 13–44 and 195–200, retrieved 2010-03-04

unicog.org

See data throughout Dehaene, Bossini & Giraux (1993), and summary by Nuerk, Iversen & Willmes (2004, p. 837). Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
Dehaene, Bossini & Giraux 1993, pp. 374–376 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, pp. 376–377 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, p. 376 "In some intuitive sense, the notion of parity is familiar only for numbers larger than 2. Indeed, before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical definition. The evidence, in brief, suggests that instead of being calculated on the fly by using a criterion of divisibility by 2, parity information is retrieved from memory together with a number of other semantic properties ... If a semantic memory is accessed in parity judgments, then interindividual differences should be found depending on the familiarity of the subjects with number concepts." Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13

uwaterloo.ca

cs.uwaterloo.ca

Caldwell & Xiong 2012, pp. 5–6. Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C
Gowers 2002, p. 118 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012). Gowers, Timothy (2002), Mathematics: A Very Short Introduction, Oxford University Press, ISBN 978-0-19-285361-5 Caldwell, Chris K.; Xiong, Yeng (2012-12-27), "What is the Smallest Prime?", Journal of Integer Sequences, 15 (9), arXiv:1209.2007, Bibcode:2012arXiv1209.2007C

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See data throughout Dehaene, Bossini & Giraux (1993), and summary by Nuerk, Iversen & Willmes (2004, p. 837). Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13 Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect", The Quarterly Journal of Experimental Psychology A, 57 (5): 835–863, doi:10.1080/02724980343000512, PMID 15204120, S2CID 10672272
Dehaene, Bossini & Giraux 1993, pp. 374–376 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, pp. 376–377 Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Dehaene, Bossini & Giraux 1993, p. 376 "In some intuitive sense, the notion of parity is familiar only for numbers larger than 2. Indeed, before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical definition. The evidence, in brief, suggests that instead of being calculated on the fly by using a criterion of divisibility by 2, parity information is retrieved from memory together with a number of other semantic properties ... If a semantic memory is accessed in parity judgments, then interindividual differences should be found depending on the familiarity of the subjects with number concepts." Dehaene, Stanislas; Bossini, Serge; Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude" (PDF), Journal of Experimental Psychology: General, 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371, archived from the original (PDF) on 2011-07-19, retrieved 2007-09-13
Snow 2001; Morgan 2001 Snow, Tony (2001-02-23), "Bubba's fools", Jewish World Review, retrieved 2009-08-22 Morgan, Frank (2001-04-05), "Old Coins", Frank Morgan's Math Chat, The Mathematical Association of America, archived from the original on 2009-01-08, retrieved 2009-08-22
Steinberg 1999; Siegel 1999; Stingl 2006 Steinberg, Neil (1999-11-30), "Even year, odd facts", Chicago Sun-Times (5XS ed.), p. 50, Factiva chi0000020010826dvbu0119h Siegel, Robert (1999-11-19), "Analysis: Today's date is signified in abbreviations using only odd numbers. 1-1, 1-9, 1-9-9-9. The next time that happens will be more than a thousand years from now.", All Things Considered, National Public Radio Stingl, Jim (2006-04-05), "01:02:03 04/05/06; We can count on some things in life", Milwaukee Journal Sentinel (Final ed.), p. B1, archived from the original on 2006-04-27, retrieved 2014-06-21
Arsham 2002; The quote is attributed to the heute broadcast of October 1, 1977. Arsham's account is repeated by Crumpacker (2007, p. 165). Arsham, Hossein (January 2002), "Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives", The Pantaneto Forum, archived from the original on 2007-09-25, retrieved 2007-09-24 Crumpacker, Bunny (2007), Perfect Figures: The Lore of Numbers and How We Learned to Count, Macmillan, ISBN 978-0-312-36005-4