Educating Exceptional Mathematics Students: The Role Of Social Capital In Mathematical Eminence

Author(s):

Carmel Diezmann(presenting / submitting)James Watters(presenting)

Conference:

ECER 2017

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 09,

Paper Session

Time:

2017-08-24

13:30-15:00

Room:

K6.04

Chair:

Ken Brown

Contribution

Given the complex social and environmental challenges facing our world, the generation of new knowledge is critical (Nahapiet & Ghoshal, 1998). New mathematical knowledge is of particular importance because it will underpin many of the technological and economic solutions necessary to meet these challenges. According to Wai, Lubinski, Benbow and Steiger (2010) reporting on the Study of Mathematically Gifted Youth [SMPY], career achievement in mathematics and related fields is correlated with the appropriateness of educational opportunities: “Rare accomplishments in STEM [science, technology, engineering, mathematics] appears to emanate from rich talent development opportunities experienced early in life” (p. 870). Hence, there is a need to understand how to educate exceptional students for knowledge generation in mathematics.

The generation of new knowledge has intellectual and social dimensions (Nahapiet & Ghoshal, 1998). There is considerable literature on the intellectual dimension of educating for mathematical eminence (e.g., Phillipson & Callingham, 2009). However, the focus of this paper is on the largely unexplored social dimension. The research question is: What is the role of social capital in education in the attainment of mathematical eminence? This investigation is informed by theories of (1) intellectual capital, social capital and knowledge generation, and (2) extraordinariness.

‘Intellectual capital’ refers “to the knowledge and knowing capability of a social collectivity, such as an organization, intellectual community, or professional practice (Nahapiet & Ghoshal, 1998, p. 245). New intellectual capital is created through the processes of combining and/or exchanging of existing knowledge (Nahapiet & Ghoshal, 1998):

"Knowledge creation involve(s) making new combinations-either incrementally or radically-either by combining elements previously unconnected or by developing novel ways of combining elements previously associated …

Where resources are held by different parties, exchange is a prerequisite for resource combination … Sometimes, this exchange involves the transfer of information within the scientific community or via the Internet. Often, new knowledge creation occurs through social interaction and coactivity". (p. 248)

Thus, there is an interrelationship between knowledge generation and social capital. ‘Social capital’ is “the sum of the actual and potential resources embedded within, available through and derived from the network of relationships possessed by an individual or social unit” (Nahapeit & Goshal, 1998, p. 243). New intellectual capital is created when one of three dimensions of social capital interact with one of four elements in the combination and exchange of intellectual capital (Nahapeit & Goshal, 1998). The social capital dimensions are structural (i.e., a network), cognitive and relational (e.g., trust). Combination and exchange of intellectual capital involve access, anticipation of the value of new knowledge, motivation and the capability of the combination. Thus, educating for mathematical eminence requires attention to developing social capital as well as intellectual capital.

Extraordinary individuals contribute to a field by assuming one of more of four roles, namely The Master, The Maker, The Introspector, and The Influencer (Gardner, 1997). The Master “is an individual who gains complete mastery over one or more domains of accomplishment; his or her innovation occurs within established practice” (Gardner, 1997, p. 11). The Maker “may have mastered existing domains, but he or she devotes energies to the creation of a new domain” (Gardner, 1997, p. 12). The Introspector undertakes “an exploration of his or her inner life: daily experiences, potent needs and fears, the operation of consciousness (both that of the particular individual and that of individuals more generally)” (Gardner, 1997, p. 12). The Influencer “has a primary goal the influencing of other individuals” (Gardner, 1997, p. 12). Hence, educating capable students with the potential to be knowledge generators in mathematics also needs to accommodate the diversity of roles that these extraordinary individuals might assume.

Method

The research question, 'What is the role of social capital in education in the attainment of mathematical eminence'?, was explored using Gardner’s (1997) biographical approach to examining eminence. This approach consisted of explication of the selection of eminent individuals and the employment of biographical methods (Smith, 1994) to create case studies of particular periods in the individual’s life (e.g., schooling). This work builds on previous work on the relationship between education and high achievement of gifted individuals (e.g., Diezmann, Stevenson, & Fox, 2012; Watters, & Diezmann, 2013). Four eminent mathematicians representative of the specific roles assumed by extraordinary individuals (Master, Maker, Introspector, Influencer) were identified to explore the role of social capital in the education of the eminent. The selected mathematicians were Maryam Mirzakhani (1977- ) (Master), Terence Tao (1975- ) (Maker), John Nash (1928-2015) (Introspector) and Paul Erdős (1913-2006) (Influencer). These mathematicians were selected for five reasons. First, the mathematicians were lauded by the international mathematics community for their major contributions and had received significant awards (e.g., Fields Medal). Second, the mathematicians were recognised by authoritative testimony from the international mathematics community as representative of a Master, a Maker, an Introspector or an Influencer. Third, there was adequate documentation of their lives including education whether they were living or deceased. Due to the slowness of judgement within some fields, an individual may die before attaining eminence (Gardner, 1997). Fourth, the mathematicians could be regarded as contemporaries thereby creating the opportunities for interactions among them. Fifth, the mathematicians were selected for diversity including countries of birth, schooling, gender and work roles. Biographical method was used to create chronologies of these individual’s lives (Smith, 1994) and case studies of their education from birth to doctoral completion including opportunities for the development of intellectual and social capital. The data sources were various published documentation (i.e., biographies, articles, websites). Multiple sources were sought to verify reports. The key analytic strategy was pattern matching (Patton, 2002). Evidence of the three dimensions of social capital (i.e., structural, cognitive, relational) in their formal and informal education was particularly sought, as was, how one of more of these dimensions led to the development of new intellectual capital. The intent was to catalogue the breadth of opportunities for the development or leverage of social capital and the extent to which it was integral to each of the four roles of extraordinary mathematicians.

Expected Outcomes

Erdős and Nash represent the antithesis of social capital in mathematics. Erdős published with over 500 coauthors and was regarded as “a weaver of social networks and thus a builder of social capital” (Krebs, 2009). Nash lacked social skills (American Experience, 2001), shunned classes at Princeton (O’Connor & Robertson, 2015), and led a relatively solitary life (Nasar, 2011). However, Nash’s contributions were outstanding winning the 1994 Nobel Prize for a paper written 45 years previously (Kuhn, 1995) and the Abel prize shortly before his death (O’Connor & Robertson, 2015). Hence, while social capital might facilitate the generation of intellectual capital (Nahapeit & Goshal, 1998), it does not preclude it. Tao and Mirzakhani differed substantially in early opportunities to develop social capital. Tao worked with intellectual peers from a young age (Muratori et al., 2006), was mentored by mathematics professors (Muratori et al., 2006), and at age 10, discussed mathematics with Erdős (Klarreich, 2014b). Mirzakhani’s mathematical capabilities were not apparent until middle school (Klarreich, 2014a) but she successfully advocated for girls’ inclusion in preparatory classes for the International Mathematical Olympiad (Klarreich, 2014a). Both Mirzakhani (Zorich, 2014) and Tao won gold medals at the International Mathematical Olympiad (O’Connor, & Robertson, 2011); were awarded PhDs from prestigious universities (Mirzakhani, Harvard; Tao, Princeton); and won the prestigious Fields Medal (American Mathematical Society, 2014; O’Connor & Robertson, 2011). Mirzakhani’s achievements at and post the International Mathematical Olympiad, suggest that it afforded her particular opportunities to develop the social capital dimensions for knowledge generation (i.e., social structures, cognition, relationships). The value of social capital in knowledge generation was integral to the Influencer, important to the Master/Maker, and peripheral to the Introspector. Education should acknowledge these differences but support the development of social capital recognising that it can be developed rapidly through engagement in strategic events.

References

American Experience. (2001). People & events: John Nash (1928 - ). PBS. Retrieved from http://www.pbs.org/wgbh/amex/nash/peopleevents/p_jnash.html Diezmann, C. M., Stevenson, M. K., & Fox, J. L. (2012). Supporting exceptional students to thrive mathematically. In B. Perry. T. Lowrie, T. Logan, A. MacDonald & J. Greenlees (Eds.). Research in Mathematics Education in Australasia: 2008-2011 (pp. 89-109). Rotterdam: Sense Publishers. Gardner, H. (1997). Extraordinary minds: Portraits of exceptional individuals and an examination of our extraordinariness. New York: Basic Books. Klarreich, E. (2014a, August 12). A tenacious explorer of abstract surfaces. Quanta Magazine. Retrieved from: https://www.quantamagazine.org/20140812-a-tenacious-explorer-of-abstract-surfaces/ Klarreich, E. (2014b, December 22). Mathematicians make a major discovery about prime numbers. Quanta Magazine. Retrieved from https://www.wired.com/2014/12/mathematicians-make-major-discovery-prime-numbers/ Krebs, V. (2009). The social graph of a famous mathematician. Retrieved from http://www.orgnet.com/Erdos.html Muratori, M.C., Stanley, J.C., Ng, L., Ng, J., Gross, M.U.M., Tao, T., & Tao, B. (2006). Insights From SMPY’s Greatest Former Child Prodigies: Drs. Terence (‘Terry’) Tao and Lenhard (‘Lenny’) Ng Reflect on Their Talent Development. Gifted Child Quarterly, 50(4), 307-324. Nahapiet, J. & Ghoshal, S. (1998). Social capital, intellectual capital, and the organizational advantage. Academy of Management. The Academy of Management Review, 23(2), 242-266. Nasar, S. (2011). A beautiful mind: A biography of John Forbes Nash, Jr., winner of the Nobel Prize in economics, 1994. New York: Simon & Schuster. O’Connor, J. J., & Robertson, E. F. (2011). Terence Chi-Shen Tao. Retrieved from http://www-history.mcs.st-and.ac.uk/Biographies/Tao.html O’Connor, J. J., & Robertson, E. F. (2015). John Forbes Nash. Retrieved from: http://www-history.mcs.st-and.ac.uk/Biographies/Nash.html Patton, M.Q. (2002). Qualitative research and evaluation methods (3rd ed.). Thousand Oaks: Sage. Phillipson, S. N., & Callingham, R. (2009). Understanding mathematical giftedness: Integrating self, action repertoires and the environment. In L. V. Shavinina (Ed.), International Handbook of Giftedness (pp. 671-698). Dordrecht, The Netherlands: Springer. Smith, L. M. (1994). Biographical method. In N. K. Denzin, & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 286-305). Thousand Oaks, CA: Sage. Wai, J., Lubinski, D., Benbow, C. P., & Steiger, J. H. (2010). Accomplishment in science, technology, engineering, and mathematics (STEM) and its relation to STEM educational dose: A 25-year longitudinal study. Journal of Educational Psychology, 102, 860-871. doi: 10.1037/a0019454 Watters, J. J. & Diezmann, C. M. (2013). Starting small: A staged approach to professional development in gifted education. Australasian Journal of Gifted Education, 22(2), 5-17. Zorich, A. (2014). The magic wand theorem of A. Eskin and M. Mirzakhani. Gazette des mathematiciens, 142(2014), 39-54. Retrieved from: http://arxiv.org/pdf/1502.05654v1.pdf

Author Information

Carmel Diezmann (presenting / submitting)

Australian Catholic University

Faculty of Education and Arts

Brisbane

James Watters (presenting)

Queensland University of Technology, Australia

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