# Holistic Teaching and Holistic Learning, Exemplified Through One Example from Linear Algebra

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## Abstract

Here we discuss ways to teach and learn holistically. The holistic method encourages student curiosity and respects student input fully, however qualified. A teacher’s holistic approach leads to open discussions and deep learning in class. Using the students’ innate desire to understand drives the course of such a class. The teacher’s role is to guide and adjust the course as the subject matter, experience, and—in mathematics courses—as the algebraic, geometric and logic rules of mathematics dictate. Holistic Teaching respects and adheres to the ‘necessity principle’ of learning and it applies the ‘holistic management principle’ that is successfully used for many other complex systems to achieve a comprehensive teaching and learning experience for both teachers and students. Teaching holistically is exemplified by one extended in-class study of how to measure angles in \(\mathbb {R}^n\) from first principles of both Geometry and Linear Algebra .

## Keywords

Holistic teaching Necessity principal Holistic management principle Mathematics teaching Linear algebra Geometry Proof Angle Unit vector Linear transformation Matrix Dot product Trigonometry## References

- Fried, M. N. (2014). Mathematics and Mathematics Education: Beginning a Dialogue in an Atmosphere of Increasing Estrangement, chapter 2 in M. N. Fried, T. Dreyfus (eds.),
*Mathematics and Mathematics Education: Searching for Common Ground, Advances in Mathematics Education*. https://doi.org/10.1007/978-94-007-7473-5_2. Springer. - Harel, G. (2013). Intellectual Need. In K.R. Leatham (ed.),
*Vital Directions for Mathematics Education Research*(Chapter 6, p. 119–151), Springer Science and Business Media, New York. https://doi.org/10.1007/978-1-4614-6977-3_6. - Piaget, J. (1960).
*Child’s Conception of Geometry*, (original in French 1948), Basic Books.Google Scholar - Piaget, J. (1985).
*The equilibration of cognitive structures: The central problem of intellectual development*, (first published in 1978), University of Chicago Press.Google Scholar - Piaget, J. (1977).
*Intellectual evolution from adolescence to adulthood*, (original in French, 1970), Cambridge Univ. Press, 1977.Google Scholar - Savory, A. (1998).
*Holistic Management: A New Framework for Decision Making*, 2nd edition, Island Press.Google Scholar - Smuts, J. C. (2010).
*Holism and Evolution*, (Macmillan, 1927), Kessinger Publishing.Google Scholar - Uhlig, F. (2002a)
*Transform Linear Algebra*, Prentice-Hall, ISBN. 0-13-041535-9, 502 + xx p.Google Scholar - Uhlig, F. (2002b). The role of proof in comprehending and teaching elementary Linear Algebra,
*Educational Studies in Mathematics*, vol. 50, 335–346.Google Scholar - Uhlig, F. (2002c)
*Author’s response to comments on “The role of proof in comprehending and teaching elementary linear algebra” in Educational Studies in Mathematics*, 50 (2002), 335–346, Educational Studies in Mathematics, vol. 53 (2003), 271–274.Google Scholar - Uhlig, F. (2003). A new unified, balanced, and conceptual approach to teaching Linear Algebra,
*Lin. Alg. Appl.*, vol. 361, 147–159.Google Scholar - Voisin, A. (1988).
*Grass Productivity*, (original in French, 1957), Island Press.Google Scholar - von Humboldt, A.
*Kosmos, Entwurf einer physischen Weltbeschreibung*, Cotta, Stuttgart; 5 volumes, 1845–1862.Google Scholar - Wendell, B. (2001).
*Life is a Miracle*, Counterpoint, 176 p; ISBN 13: 9781582430584.Google Scholar