# Computational Triangulation in Mathematics Teacher Education

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Triangulation within a Method

- Brain Teaser: Five identical plates are lined up and each one is filled with candies. How many ways can one put candies on the first two plates so that when each plate beginning from the third has as many candies as the previous two plates combined, the fifth plate has 13 candies?
- Discussion: As a method of solving this brain teaser, consider the following algebraic approach and computational triangulation within the method. Let x and y represent the number of candies on the first and second plates, respectively. Then the sequence x, y, x + y, x + 2y, 2x + 3y can be developed. Therefore, one has to solve the equation

## 4. Triangulation between Methods

_{3}and F

_{4}) as coefficients in the unknown number of candies on the first and second plates, respectively. So, in the case of n plates, the numbers ${F}_{n-2}$ and ${F}_{n-1}$ would be the coefficients in x and y, respectively, and a more general equation (to be used in spreadsheet programming) would be as follows:

^{18}in Figure 5). This shows how computational triangulation not only brings rigor and confidence in problem solving but, through modification of conditions of existing problems, may contribute to problem posing.

## 5. Computational Triangulation and Internal Rival Factors

^{18}and formulate a problem similar to the Brain Teaser. The coefficient 4 in z

^{18}points at four ways to put 18 candies on the first two plates. However, using the Graphing Calculator, one can locate inside the segment connecting the coordinate axes only two points with integer coordinates. As was mentioned above, the difference in the number of solutions provided by different digital instruments can be explained conceptually in terms of the conditions of the Brain Teaser—there should be no empty plates. If we allow for one of the first two plates to be empty, then putting nine candies on the first plate and keeping the second plate empty, one will end up with 18 candies on the fifth plate. Likewise, keeping the first plate empty and putting six candies on the second plate, the fifth plate would have 18 candies. This is exactly what one can see on the graph of Figure 8—the segment connects the points (9, 0) and (0, 6) residing on the x and y axes, respectively. Furthermore, the equation 2x + 3y = 18 (unlike the equation 2x + 3y = 14) shows that both x = 0 and y = 0 provide a solution to this equation as both 2 and 3 divide 18 (whereas only 2 divides 14).

## 6. Computational Triangulation of the Second Order

## 7. A Misprint in Computational Triangulation as a Window on New Phenomenon

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Campbell, D.T. A Study of Leadership among Submarine Officers; Ohio State University Research Foundation: Columbus, OH, USA, 1953. [Google Scholar]
- Campbell, D.T. From Description to Experimentation: Interpreting Trends as Quasi-Experiments. In Problems in Measuring Change; Harris, C.W., Ed.; University of Chicago Press: Madison, WI, USA, 1963; pp. 212–242. [Google Scholar]
- Webb, E.J.; Campbell, D.T.; Schwartz, R.D.; Sechrest, L. Unobtrusive Measures; Rand McNally: Chicago, IL, USA, 1966. [Google Scholar]
- Boaler, J.; Ball, D.L.; Even, R. Preparing Mathematics Education Researchers for Disciplined Inquiry: Learning from, in, and for Practice. In Second International Handbook of Mathematics Education; Bishop, A.J., Clements, M.A., Keitel, C., Kilpatrick, J., Leung, F.K.S., Eds.; Springer International Handbooks of Education, Springer: Dordrecht, The Netherlands, 2003; Volume 10, pp. 491–521. [Google Scholar]
- Sharma, S. Qualitative approaches in mathematics education research: Challenges and possible solutions. Educ. J.
**2013**, 2, 50–57. [Google Scholar] [CrossRef] - Löwe, B.; van Kerkhove, B. Methodological Triangulation in Empirical Philosophy (of Mathematics). In Advances in Experimental Philosophy of Logic and Mathematics; Aberdein, A., Inglis, M., Eds.; Bloomsbury Publishing: London, UK, 2019; pp. 15–37. [Google Scholar]
- Abramovich, S. Advancing the concept of triangulation from social sciences research to mathematics education. Adv. Educ. Res. Eval.
**2022**, 3, 201–217. [Google Scholar] [CrossRef] - Wing, J.M. Computational thinking. Comm. ACM
**2006**, 49, 33–35. [Google Scholar] [CrossRef] - Freudenthal, H. Weeding and Sowing; Kluwer: Dordrecht, The Netherlands, 1978. [Google Scholar]
- Harrison, J. Formal proof—Theory and practice. Not. Am. Math. Soc.
**2008**, 55, 1395–1406. [Google Scholar] - McFee, G. Triangulation in research: Two confusions. Educ. Res.
**1992**, 34, 215–219. [Google Scholar] [CrossRef] - Denzin, N.K. The Research Act in Sociology: The Theoretical Introduction to Sociological Methods; Butterworth: London, UK, 1970. [Google Scholar]
- National Council of Teachers of Mathematics. Principles to Actions: Ensuring Mathematical Success for All; National Council of Teachers of Mathematics: Reston, VA, USA, 2014. [Google Scholar]
- Char, B.W.; Geddes, K.O.; Gonnet, G.H.; Leong, B.L.; Monagan, M.B.; Watt, S.M. Maple V Language Reference Manual; Springer: New York, NY, USA, 1995. [Google Scholar]
- Power, D.J. A Brief History of Spreadsheets, DSSResources.COM. 2000. Available online: http://dssresources.com/history/sshistory.html (accessed on 7 February 2023).
- Avitzur, R. Graphing Calculator [Version 4.0]; Pacific Tech: Berkley, CA, USA, 2011. [Google Scholar]
- Denzin, N.K. Triangulation. The Blackwell Encyclopedia of Sociology; Ritzer, G., Ed.; Blackwell Publishing: Malden, MA, USA, 2007; Volume 10, pp. 5075–5080. [Google Scholar]
- Saukko, P. Doing Research in Cultural Studies: An Introduction to Classical and New Methodological Approaches; Sage: London, UK, 2003. [Google Scholar]
- National Curriculum Board. National Mathematics Curriculum: Framing Paper. Author: Australia. 2008. Available online: http://www.acara.edu.au/verve/_resources/National_Mathematics_Curriculum_-_Framing_Paper.pdf (accessed on 7 February 2023).
- Ontario Ministry of Education. The Ontario Curriculum, Grades 1–8, Mathematics (2020). 2020. Available online: http://www.edu.gov.on.ca (accessed on 7 February 2023).
- Western and Northern Canadian Protocol. The Common Curriculum Framework for Grades 10–12 Mathematics. 2008. Available online: http://www.bced.gov.bc.ca/irp/pdfs/mathematics/WNCPmath1012/2008math1012wncp_ccf.pdf (accessed on 7 February 2023).
- Felmer, P.; Lewin, R.; Martínez, S.; Reyes, C.; Varas, L.; Chandía, E.; Dartnell, P.; López, A.; Martínez, C.; Mena, A.; et al. Primary Mathematics Standards for Pre-Service Teachers in Chile; World Scientific: Singapore, 2014. [Google Scholar]
- Advisory Committee on Mathematics Education. Mathematical Needs of 14–19 Pathways; The Royal Society: London, UK, 2007. [Google Scholar]
- Ministry of Education Singapore. Mathematics Syllabuses, Secondary One to Four; The Author: Curriculum Planning and Development Division. 2020. Available online: https://www.moe.gov.sg/-/media/files/secondary/syllabuses/maths/2020-express_na-maths_syllabuses.pdf?la=en&hash=95B771908EE3D777F87C5D6560EBE6DDAF31D7EF (accessed on 7 February 2023).
- Department of Basic Education. Mathematics Teaching and Learning Framework for South Africa: Teaching Mathematics for Understanding; Private Bag: Pretoria, South Africa, 2018.
- Common Core State Standards. Common Core Standards Initiative: Preparing America’s Students for College and Career. 2010. Available online: http://www.corestandards.org (accessed on 7 February 2023).
- Conference Board of the Mathematical Sciences. The Mathematical Education of Teachers II; The Mathematical Association of America: Washington, DC, USA, 2012. [Google Scholar]
- Association of Mathematics Teacher Educators. Standards for Preparing Teachers of Mathematics. 2017. Available online: amte.net/standards (accessed on 7 February 2023).
- Wing, J.M. Research Notebook: Computational Thinking—What and Why. Link Mag.
**2011**, 6, 20–23. Available online: https://people.cs.vt.edu/~kafura/CS6604/Papers/CT-What-And-Why.pdf (accessed on 7 February 2023). - Arnheim, R. Visual Thinking; University of California Press: Berkeley, CA, USA; Los Angeles, CA, USA, 1969. [Google Scholar]
- Vygotsky, L.S. Mind in Society; Harvard University Press: Cambridge, MA, USA, 1978. [Google Scholar]
- Gershenfeld, N. Fab: The Coming Revolution on Your Desktop—From Personal Computers to Personal Fabrication; Basic Books: New York, NY, USA, 2005. [Google Scholar]
- Beghetto, R.A.; Kaufman, J.C.; Baer, J. Teaching for Creativity in the Common Core Classroom; Teachers College Press: New York, NY, USA, 2015. [Google Scholar]
- Koshy, T. Fibonacci and Lucas Numbers with Applications; Wiley: New York, NY, USA, 2001. [Google Scholar]
- Catalan, E. Notes sur la Théorie des Fractions Continues et sur Certaines Séries. In Mémoires de L’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, Tome XLV; Hayez, F.F., Ed.; Imprimeur de L’Académie Royale: Bruxelles, Belgique, 1884. [Google Scholar]
- Knuth, D.E. The Art of Computer Programming, Fundamental Algorithms; Addison Wesley: Reading, MA, USA, 1977; Volume 1. [Google Scholar]
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers; Penguin Books: Middlesex, UK, 1986. [Google Scholar]
- Weisstein, E.W. CRC Concise Encyclopedia of Mathematics; Chapman & Hall/CRC: Boca Raton, FL, USA, 1999. [Google Scholar]
- Apostol, T.M. Calculus: One-variable Calculus, with Introduction to Linear Algebra; Wiley: Hoboken, NJ, USA, 1967; Volume 1. [Google Scholar]
- McCallum, W. Mathematicians and Educators: Divided by a Common Language. In Mathematics in Undergraduate Study Programs: Challenges for Research and for the Dialogue between Mathematics and Didactics of Mathematics; Report No. 56/2014; Mathematicsches Forschungsinstitut Oberwolfach: Oberwolfach, Germany, 2015; pp. 48–49. [Google Scholar] [CrossRef]
- Kilpatrick, J. Reformulating: Approaching Mathematical Problem Solving as Inquiry. In Posing and Solving Mathematical Problems: Advances and New Perspectives; Felmer, P., Pehkonen, E., Kilpatrick, J., Eds.; Springer: New York, NY, USA, 2016; pp. 69–81. [Google Scholar]
- Béguin, P.; Rabardel, P. Designing instrument-mediated activity. Scand. J. Inform. Syst.
**2000**, 12, 173–190. [Google Scholar] - Abramovich, S.; Leonov, G.A. Revisiting Fibonacci Numbers Through a Computational Experiment; Nova Science Publishers: New York, NY, USA, 2019. [Google Scholar]
- Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences. Available online: http://www.research.att.com/~njas/sequences/ (accessed on 7 February 2023).
- Matijasevic, J.V. Diophantine representation of enumerable predicates. Math. USSR—Izvestiya
**1971**, 5, 1–28. [Google Scholar] [CrossRef] - Hilbert, D. Mathematical problems (Lecture delivered before the International Congress of Mathematicians at Paris in 1900). Bull. Am. Math. Soc.
**1902**, 8, 437–479. [Google Scholar] [CrossRef] - Langtangen, H.P.; Tveito, A. How Should We Prepare the Students of Science and Technology for a Life in the Computer Age? In Mathematics Unlimited—2001 and Beyond; Engquist, B., Schmid, W., Eds.; Springer: New York, NY, USA, 2001; pp. 809–825. [Google Scholar]
- Kaput, J.J. Technology and Mathematics Education. In Handbook of Research on Mathematics Teaching and Learning; Grouws, D.A., Ed.; Macmillan: New York, NY, USA, 1992; pp. 515–556. [Google Scholar]
- Pierpont, J. Mathematical rigor, past and present. Bull. Amer. Math. Soc.
**1928**, 34, 23–53. [Google Scholar] [CrossRef] [Green Version] - Pólya, G. Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving; Wiley: New York, NY, USA, 1965; Volume 2. [Google Scholar]
- Engquist, B.; Schmid, W. (Eds.) Mathematics Unlimited—2001 and Beyond; Springer: New York, NY, USA, 2001. [Google Scholar]

**Figure 17.**The staircase diagram demonstration of the convergence of $\frac{{f}_{n+1}}{{f}_{n}}$ to $\frac{-1-\sqrt{5}}{2}$.

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Abramovich, S.
Computational Triangulation in Mathematics Teacher Education. *Computation* **2023**, *11*, 31.
https://doi.org/10.3390/computation11020031

**AMA Style**

Abramovich S.
Computational Triangulation in Mathematics Teacher Education. *Computation*. 2023; 11(2):31.
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**Chicago/Turabian Style**

Abramovich, Sergei.
2023. "Computational Triangulation in Mathematics Teacher Education" *Computation* 11, no. 2: 31.
https://doi.org/10.3390/computation11020031