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Authentic Modelling Problems in Mathematics Education—Examples and Experiences

Authentische Modellierungsprobleme im Mathematikunterricht – Beispiele und Erfahrungen

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Abstract

In this paper, we describe mathematical modelling activities, which deal with authentic problems. These kinds of problems have been tackled in various modelling activities, amongst others in a modelling week. After a description of the theoretical approach used, one of these authentic modelling problems is described in detail showing students’ solutions. Based on the evaluation of a modelling week with several hundred students, it is argued that these kinds of authentic problems are feasible with students from upper secondary level. Furthermore, it became apparent that most students would appreciate these kinds of examples included in school mathematics in order to promote their skills to use mathematics in their real life.

Zusammenfassung

In dem Artikel werden mathematische Modellierungsaktivitäten beschrieben, die von authentischen Problemen ausgehen. Diese Probleme wurden in verschiedenen Modellierungsaktivitäten behandelt, u.a. in Modellierungswochen. Nach einer Beschreibung des zugrundeliegenden theoretischen Ansatzes wird eines dieser authentischen Probleme im Detail mit Lösungsansätzen von Schülerinnen und Schüler beschrieben. Auf der Basis der Evaluation einer Modellierungswoche mit mehreren hundert Schülerinnen und Schüler wird argumentiert, dass diese Art von authentischen Modellierungsproblemen Schülerinnen und Schülern des oberen Sekundarbereichs zugänglich ist. Des Weiteren wird deutlich, dass die meisten Schülerinnen und Schüler der Integration solcher Beispiele in den normalen Mathematikunterricht positiv gegenüber stehen, da mit solchen Beispielen ihre Kompetenzen, Mathematik im wirklichen Leben anzuwenden, gefördert werden.

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Notes

  1. The average number of students attending mathematics at advanced level at the Federal level of Germany is with 29.5% in 2008 remarkably higher than in Hamburg going up to 40% in parts of East Germany, such as Sachsen-Anhalt.

  2. Such a sample solution can be found in Ortlieb (2009).

  3. The used sign “%” was originally used by the group.

  4. When irrigating a quadratic garden (side length a) with only one TVR, there are two equivalent ideal solutions with regard to criterion 1. Both of them have an extent of irrigation of π/4: either the TVR is placed in the middle of the garden and irrigates in a full circle with the radius r=0.5a or the TVR is placed in one corner and irrigates in a quarter circle with the radius r=a according the full side length of the square. The second configuration was probably the origin of the students’ considerations.

  5. A numerical analysis shows that method A is to be preferred when b/a<0.77 is true.

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Acknowledgements

We thank Maren Hoffstall and Anna Orschulik for their extensive work within the evaluation process, Christoph Lederich for editorial support, and Hannah Heinrichs for her translational work.

Author information

Authors and Affiliations

  1. Faculty of Education, University of Hamburg, Von-Melle-Park 8, 20146, Hamburg, Deutschland

    Gabriele Kaiser & Björn Schwarz

Authors
  1. Gabriele Kaiser
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  2. Björn Schwarz
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Corresponding author

Correspondence to Gabriele Kaiser.

Appendices

Appendix A: Questionnaire about the Modelling Week

1.1 A.1 Questions about Mathematics

  1. (1)

    What is mathematics in your opinion? Describe in some sentences.

  2. (2)

    Are you interested in mathematics? If so, why and what are you interested in especially?

    If not, why not and what are you discouraged by?

  3. (3)

    In your opinion, what relevance does mathematics have for this society?

1.2 A.2 Questions about Mathematics Teaching

  1. (1)

    Are you interested in the subject of mathematics in school? If so, why? If not, why not?

  2. (2)

    Describe topics, problems, and activities of mathematic classes which appeal to you. (Max. 3)

  3. (3)

    Describe topics, problems, and activities of mathematic classes which discourage or bore you. (Max. 3)

  4. (4)

    Have you been able to apply any mathematic approaches or methods which you have learned in school to everyday life or other fields of knowledge so far? If so, which ones and where?

1.3 A.3 Questions about the Modelling Example

  1. (1)

    Please value the modelling example that you dealt with during the modelling week:

    1. (a)

      How interesting was the problem that you dealt with?

      figure a
    2. (b)

      How did you rate the difficulty of the modelling problem?

      figure b
    3. (c)

      How highly was the problem structured?

      figure c
    4. (d)

      How realistic was the modelling example?

      figure d
  2. (2)

    In your opinion, what did you learn when dealing with the subject?

  3. (3)

    Should these examples be increasingly dealt with as part of regular maths classes or would you reject this? Please give reasons for your position

  4. (4)

    Other comments that are relevant to me.

Appendix B: Codes to the Question: “From Your Point of View, What Did You Learn when Dealing with the Modelling Example?”

Application

  • General mathematical application

  • Critical questioning of techniques

  • Everyday-life reference of mathematics

  • Practical reference of mathematics

  • Geometry

Formal mathematics

  • Creating diagrams

  • Using mathematical terms

  • Developing formulae

  • Finding different approaches

  • Setting up equations

  • Stochastic

  • Calculating sectors of a circle

  • Circles in a square

  • Trigonometry

  • Geometry

  • Calculation of areas

  • Iteration

Application of the computer

  • Using mathematical programmes

  • Transforming ideas into algorithms

  • Creating algorithms

Formal work with the computer

  • Programming languages

  • Programming

  • Create graphics

  • Power Point

  • Excel

  • Matlab

Working techniques

  • Problem-solving strategies

  • Working independently

  • Gather information about unknown topics

  • Judging requirements

  • Structuring ideas

  • Recognising the variety of approaches

  • Optimism

  • Goal-oriented work

  • Organising work

  • Stamina

  • Making decisions

  • Structuring

  • Simplifying

Mathematical insights

  • Different approaches

  • Diversity of approach

  • Diversity of aspects in mathematics

  • Complexity

  • Specificity

Social aspects

  • Teamwork

  • Getting to know the fellow students

Appendix C: Codes to the Question: “Should These Examples Be Increasingly Dealt with as Part of Regular Maths Classes or Would You Reject This? Please Give Reasons for Your Position”

3.1 C.1 Yes

Reference to reality

  • Sense of mathematics

  • Recognising connections

  • Reference to reality

  • Reference to own everyday life

  • Clearness

  • Practical relevance

Improvement of working techniques

  • Working independently

  • Stamina

  • One’s own initiative

  • Group work

Variation

  • Diversified

  • Motivation

  • Interest

  • Appealing to the students

  • Fun

Requested change

  • Clearer structure

  • Simplifications

  • More support

  • Structuring of problems

  • Concreteness of problems

Sense

  • Advancement of complex thinking

  • Practise

  • Advancement of mathematical thinking

  • Simplified/“playful” learning

  • Better understanding

  • Memorising of learned things

  • No solution given

  • Puzzling

  • Good preparation/insight into the future tasks of a mathematician

3.2 C.2 No

Time problem

  • Restriction by the curriculum

  • Does not fit into the shortened time until university entrance examination

  • Too complex to be dealt with in class

  • Too extensive to be dealt with in class

Lack of interest

  • Boring

  • Uninteresting

  • Not useful for every student

Meaninglessness

  • Pointlessness

  • No connection to mathematics

  • No mediation of the foundation

  • No relevance concerning university entrance examination

Complexity

  • Complicated

  • Extensive

Inaccuracy

  • Vague formulations

  • No clear tasks

  • Variable level of difficulty

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Kaiser, G., Schwarz, B. Authentic Modelling Problems in Mathematics Education—Examples and Experiences. J Math Didakt 31, 51–76 (2010). https://doi.org/10.1007/s13138-010-0001-3

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Keywords

Mathematics Subject Classification (2000)

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