Why is linear algebra not taught before calculus

On the justification of Euclidean geometry in academic teaching - Confessions of a mathematical peasant

Summary

To this day, it is a long-cultivated tradition of gaining the foundation of plane Euclidean geometry from abstract points and straight lines which, as objects sui generis, are indivisible and then fulfill numerous axioms of incidence. We are wondering here whether it might not make sense from a practical standpoint to break this tradition and start with a definition of what a Euclidean straight line should be. Building on this and on the Pythagorean theorem, levels and higher-dimensional spaces then arise in a natural and very simple way. In principle, this approach should also be suitable for teaching at grammar schools.

Notes

  1. 1.

    Regarding the dimensions: In the extensive modern Taschenbuch der Mathematik [15] with approx. 2000 pages, about 50 pages are devoted to Euclidean geometry, including elementary areas, volumes, vectorial analytical geometry, conic sections and Hilbert's axioms. According to the preface, the paperback is also aimed at high school students and teachers. How do we want to think about the other 1950 pages? There is e.g. in Chap. 4 at least 36 pages on the topic Fundamentals of Mathematics, i.e. logic and set theory.

  2. 2.

    I'm not sure if there isn't any misunderstanding as to the word Basics of geometry there are religious bureaucrats who might consider it Basic concepts of geometry interpret and then, logically, judge differently. Compare the usual textbooks with titles like Basics of linear algebra, basics of function theory, basics of probability theory or even Fundamentals of financial mathematics.

  3. 3.

    Unfortunately, it is to be feared that this also applies to the “new” axiomatics with 28 axioms, which is proposed in [6] and which aims at an early introduction of the vector space structure.

  4. 4.

    http://www.uni-goettingen.de/de/37363.html.

  5. 5.

    There is no doubt that there is a great deal of beautiful geometry in this book. But as described in the book, the teacher cannot introduce this into the lesson without serious changes.

  6. 6.

    As an example I mention the unhelpful use of the term Linear space in contrast to that in linear algebra [15, 2.3.2], [2, p. 68]. What does the adjective "linear" mean?

  7. 7.

    In the paperback already quoted [15], however, these do not appear to occur. Instead there is a chapter 3.9: Geometries of modern physics.

  8. 8.

    In this context, the author will never forget a biting remark made by Werner Burau (1906–1994), who was around 80 years old, at a geometry conference in Oberwolfach:

    So you know, that's how it works with the basics of geometry: first you chop off a leg, and then you see if you can still walk.

    It should be noted that Mr. Burau, as a classic algebraic geometer, is completely unsuspicious, for example to look down arrogantly at traditional geometry. He had a good sense of the older geometry, including that of the 19th century.

  9. 9.

    http://www.schule-bw.de/unterricht/faecher/mathematik/1bsm/.

  10. 10.

    http://bildungsserver.berlin-brandenburg.de/rahmenlehrplaene.html.

  11. 11.

    http://www.standardsicherung.schulministerium.nrw.de/cms.

  12. 12.

    http://www.hamburg.de/contentblob/2536224/data/mathematik-gy8-sek-i.pdf.

  13. 13.

    The so-to-speak "analytical Pythagoras" would then "only" consist of a formula for triangles with corners (x1,y1),(x2,y2),(x2,y1): How do you calculate the distance between the first two points using the third? What draws the euclidean Distance function in front of others?

  14. 14.

    http://www.standardsicherung.schulministerium.nrw.de/cms.

  15. 15.

    The book [2] on linear algebra should be positively emphasized here, which does not lose sight of the geometric motivation, not even with a topic such as determinants. In this sense, on the other hand, [5] and [13], which almost exclusively place algebraic motivation in the foreground, but still claim to be suitable for the teaching profession, are rather deterrent. Although determinants are dealt with extensively, “areas” or “volumes” appear to be absent. Where should you learn that - geometrically speaking - determinants are nothing other than signed areas or volumes?

  16. 16.

    If this approach seems too straightforward or too clumsy, T2 can be derived from a few other elementary facts, such as in [12], where T2 appears as Theorem 3.

  17. 17.

    There is also the apt remark that the Erlangen program is cited a lot, but rarely put into practice: "The Erlangen program has probably the highest rate\ (\ frac {\ mathit {praised}} {\ mathit {actually} \ \ mathit {used}} \)among mathematical theories ", so V.V. Kisil in Notices AMS 54 (2007), p. 1458.

  18. 18.

    We think of 1-dimensional (real) straight lines. If you really want to have 2-dimensional (complex) straight lines, you would have to modify the following explanations. If you really want that, it would be easier to replace the real coordinates a posteriori with the complex ones and so to one Complexification to pass over. A complex straight line could also be interpreted as a real plane with a certain structure.

  19. 19.

    This can also be postulated or derived using the usual division method using two straight lines in the Euclidean plane through parallels.

  20. 20.

    Anyone who shies away from treating Pythagoras like an axiom can easily postulate other rules at this point, which - in combination with the axiom of parallels - in turn imply Pythagoras in the sense of the above remarks on Fig. 1.

  21. 21.

    All too often the author of teacher training students has asked about the geometry of the solution set of a linear equation in n≥3 hearing the changeable, that is a straight line. There are obviously deficits here.

  22. 22.

    Compare this with the ideas that are in vogue nowadays about what to buy networked competencies, both in school curricula and in examination regulations for teacher training. The 2010 grammar school teaching examination regulations for BW simply postulate: The graduates have networked skills in specialist science, specialist didactics and school practice, and that at the time of the first state examination. The contents of the teacher training course are also listed in detail. Typical quote from the secondary school curriculum for secondary school in the state of Hamburg from 2011: The concept of competency in mathematics can be structured according to process-related general mathematical competencies and content-related mathematical competencies, arranged according to five main ideas (number, measurement, space and form, functional context, data and chance). In addition to the process and content dimension, there is also the level dimension, which records the cognitive complexity of mathematical activities and tasks. In the state of BW there are also the main ideas of algorithm, variable, networking and modeling. The addition of fractions then belongs to the main idea of ​​"algorithm", not to the main idea of ​​"number". These central ideas and their networking enable an understanding-oriented approach to mathematics. Is that helpful? In spite of or even because of these networked central ideas, the number of high school graduates who do not master fractions but still take STEM courses seems to be increasing.

literature

  1. 1.

    Agricola, I., Friedrich, T .: Elementarge Geometry, 2nd edition Vieweg + Teubner (2009)

  2. 2.

    Artmann, B .: Lineare Algebra, 3rd edition. Birkhauser scripts (1991)

  3. 3.

    Bauer, Th., Partheil, U .: Interface modules in teacher training in mathematics. Math. Semesterber. 56, 85–103 (2009)

    MATHArticle Google Scholar

  4. 4.

    Benz, W .: Level Geometry: Introduction to Theory and Application. University paperback. Spektrum-Verlag (1997)

  5. 5.

    Beutelspacher, A .: Lineare Algebra, 7th edition Vieweg + Teubner (2010)

  6. 6.

    Choquet, G .: New Elementary Geometry, 2nd edition, Vieweg-Verlag (1972)

  7. 7.

    Griesel, H., et al. (Ed.): Elements of Mathematics, Vol. 1–6 (Class 5–10). Schroedel-Verlag, Baden-Württemberg (2004–2008)

  8. 8.

    Hilbert, D .: Fundamentals of Geometry, 13th edition Teubner-Verlag (1987)

  9. 9.

    Karzel, H., Sörensen, K., Windelberg, D .: Introduction to Geometry. Uni-Taschenbücher, Vol. 184. Vandenhoeck & Ruprecht, Göttingen (1973)

    MATH Google Scholar

  10. 10.

    Klein, F .: Lectures on the development of mathematics in the 19th century. Springer, Berlin (1979)

    MATHBook Google Scholar

  11. 11.

    Lötzbeyer, Ph., Gotthardt, G .: Mathematics for the intermediate level, short edition, part II: Geometry and geometric drawing, 8th edition. Verlag Ehlermann, Dresden (1936)

    Google Scholar

  12. 12.

    MacLane, S .: Metric postulates for plane geometry. At the. Math. Mon. 66, 543–555 (1959)

    MathSciNetMATHArticle Google Scholar

  13. 13.

    Wagner, R .: Basics of linear algebra, mathematics for teaching at grammar schools. Teubner-Verlag (1981)

  14. 14.

    Weigand, H.-G., et al .: Didactics of Geometry for Secondary School I. Spektrum-Verlag (2009)

  15. 15.

    Zeidler, E., et al. (Ed.): Teubner-Taschenbuch der Mathematik I, II. Teubner-Verlag (1995/96)

Download references

Author information

Affiliations

  1. Institute for Geometry and Topology, University of Stuttgart, 70550, Stuttgart, Germany

    Wolfgang Kühnel

Corresponding author

Correspondence to Wolfgang Kühnel.

About this article

Cite this article

Kühnel, W. On the Justification of Euclidean Geometry in Academic Lessons - Confessions of a Mathematical Banaus. Math semester60, 105-121 (2013). https://doi.org/10.1007/s00591-012-0111-8

Download citation