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On the Analysis of Mathematical Practices in Signal Theory Courses | SpringerLink

On the Analysis of Mathematical Practices in Signal Theory Courses

Abstract

The contribution aims at subject-specific analyses of student solutions of an exercise from an electrical engineering signal theory course. The basis for the analyses is provided by praxeological studies (in the sense of the Anthropological Theory of Didactics) and the identification of two institutional mathematical discourses, one related to higher mathematics for engineers and one related to electrical engineering. Regarding the relationship between institutional observations and analyses of students’ solutions, we refer, among others, to Weber’s (1904) concept of ideal types. In the subject-specific analyses of student solutions we address in particular transitions and interrelations within single processing steps that refer to the two mathematical discourses and different forms of embedding of mathematics into the electrical engineering context. Finally, we present a few ideas for teaching.

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Notes

  1. 1.

    The acronyms HM and ET were introduced by Peters and Hochmuth (in press) to denote the two relevant contexts of “Höhere Mathematik” (HM, higher mathematics) and “Elektrotechnik” (ET, electrical engineering) and associated discourses. HM and ET are the standard German actronyms for these contexts. Although the English term electrical engineering requires the acronym EE, for reasons of consistency we stick here to the acronym ET.

  2. 2.

    In the institutional context, we usually skip the institutional and simply speak of subject-specific. This seems justified to us, since discipline-specificity is unthinkable without institutions. With regard to the individual level, we use the term individual subject-specific. The term subject-related, on the other hand, addresses aspects that consider the individual as subject, including societal and psychological moments.

  3. 3.

    For example, compare definitions in the two books by Frey and Bossert (2009) and Fettweis (1996). Both books are recommended as standard literature for the signal theory course we are studying.

  4. 4.

    For a more detailed discussion of such epistemological issues regarding the relationship of mathematics and empirical sciences we refer to Hochmuth und Peters (in press).

  5. 5.

    Circuits are operated with sinusoidal current- and voltage forms in the power supply network as well as in many other important areas.

  6. 6.

    We translated the German term Zeiger with the term phasor, which already refers to electrical engineering concepts. But electrical engineering aspects play no role in the course and Strampp (2012) does not refer to them either. Another possible translation of Zeiger, without the connection to engineering concepts would be pointer. But we decided to use phasor for the following reason: In German, the term Zeiger is used both in electrical engineering and in mathematics courses for engineers, but with different meanings (reference to electrical engineering concepts vs. geometrical object with no further references). By using the term Zeiger instead of vector Strampp (2012) can thus establish a connection to the electrical engineering courses without dropping the inner mathematical conception of complex numbers. This aspect of using the same term, that has different meanings in different course-contexts is in jeopardy of being lost through translation.

  7. 7.

    Within the ATD the term discourse, e.g. in expressions like “reasoning discourse” or “a discourse on praxis” is used in the etymological sense (e.g. Bosch & Gascón, 2014, p. 68).

  8. 8.

    This use of the term discourse goes beyond an etymological understanding (cf. footnote 6). We extend thereby a term which already exists within the ATD. Our extended understanding blends into the already existing concepts (e.g. institutional dependence of knowledge). We do not use the term discourse in the sense of discourse theory. Due to the limited word count, we refrain from further elaboration of possible connections and delimitations.

  9. 9.

    The two mathematical discourses can also be connected to the work of Artaud (2020), where she describes two types of didactical transposition processes: An external didactical transposition process, originating in academic mathematics research institutions. Here one can locate the HM-discourse. And an endogenous didactical transposition process concerning processes within the engineering institution. Here one can locate the ET-discourse.

  10. 10.

    As expressed within ATD as difference between institutional and individual relations to objects of knowledge.

  11. 11.

    This does not contradict the empirical openness discussed above with regard to the two mathematics discourses, since a concrete case could follow a different disciplinary rational, which might be completely independent of the ones we reconstructed.

  12. 12.

    Using observation language means to describe an observation without interpretations (Schwemmer, 1976, p. 165), whereas interpreting means to show actions as rational in purpose or sense (p. 168).

  13. 13.

    The observation correlate of an action is the part of the action that is observed and described in observation language (Schwemmer, 1976, p. 168).

  14. 14.

    Here we also refer to the work of De Oliveira and Nunes (2014) who investigate rotating phasor pathways derived from different standard amplitude modulation systems.

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Appendix: Exercise and sample solution

Appendix: Exercise and sample solution

The exercise under consideration is structured in three items:

  1. 1.

    A message signal s(t) = cos(Ωt) has to be amplitude modulated. The result is x(t) = A[1 + m cos(Ωt)] cos(2πf0t)

  2. 2.

    The result of item 1. Has to be written as the sum of three harmonics. The result is

$$ x(t)=A\cos \left(2\pi {f}_0t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t+\Omega t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t-\Omega t\right) $$
  1. 3.

    The result of item 2. Has then to be displayed graphically in the complex plane as a rotating phasor with varying amplitude.

Our analysis focusses item 3. of the exercise. The exact problem definition of item 3 is:

  1. 3.

    Graphically display x(t) in the complex plane as a rotating phasor with varying amplitude using the relationship \( \cos \left(2\pi ft\right)=\mathfrak{R}\left\{\exp \left(j2\pi ft\right)\right\} \) and the result under item 2.

Sample solution:

One first writes

$$ x(t)=A\cos \left(2\pi {f}_0t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t+\Omega t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t-\Omega t\right)=A\mathfrak{R}\left\{\exp \left(j2\pi {f}_0t\right)\right\}+\frac{Am}{2}\mathfrak{R}\left\{\exp \left(j\left(2\pi {f}_0t+\Omega t\right)\right)\right\}+\frac{Am}{2}\mathfrak{R}\left\{\exp \left(j\left(2\pi {f}_0t-\Omega t\right)\right)\right\}=\mathfrak{R}\left\{\exp \left(j2\pi {f}_0t\right)\underset{A(t)}{\underbrace{\left[A+\frac{Am}{2}\exp \left(j\Omega t\right)+\frac{Am}{2}\exp \left(-j\Omega t\right)\right]}}\right\} $$
(1)

and interprets the expression in the square bracket as a real-valued time-dependent amplitude A(t), which modulates the carrier phasor exp(j2πf0t) rotating at frequency f0 in Fig. 13.

Fig. 13
figure13

Representation of x(t) = A[1 + m cos(Ωt)] cos(2πf0t) as the real part of a rotating phasor A(t) exp(j2πf0t) with ω0 = 2πf0

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Hochmuth, R., Peters, J. On the Analysis of Mathematical Practices in Signal Theory Courses. Int. J. Res. Undergrad. Math. Ed. 7, 235–260 (2021). https://doi.org/10.1007/s40753-021-00138-9

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Keywords

  • Mathematical practices
  • Student solutions
  • Ideal typical discourses
  • Institutions
  • Anthropological theory of the didactic