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Mathematical Maturity for Engineering Students

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Abstract

Facing increased pressure to improve retention and graduation rates, engineering departments are increasingly scrutinizing whether they are getting their desired outcomes from core mathematics coursework. Since mathematics courses are a significant source of attrition and many engineering faculty are unhappy with students’ mathematical abilities, more engineering departments are increasingly looking at drastic options of taking students out of mathematics courses and teaching students mathematics themselves. To mitigate this trend, it may be valuable to better understand what engineering faculty hope students learn from their mathematics coursework. When engineering faculty explain why they require these high-failure prerequisites, many claim that “mathematical maturity”, not calculus skill, is the desired outcome of completing the core math sequence of courses. To better understand what engineering faculty mean by “mathematical maturity”, we conducted a qualitative thematic analysis of how 27 engineering faculty members define “mathematical maturity”. We found that these engineering faculty believed that the mathematically mature student would have strong mathematical modeling skills supported by the ability to extract meaning from symbols and the ability to use computational tools as needed. Faculty frequently lamented that students had underdeveloped epistemic beliefs that undermined their modeling skills, thinking that mathematics is unrelated to the real world and has little practical value. They attributed these dysfunctional epistemic beliefs to their perception that mathematics is too often taught without genuine physical context and realistic examples. We suggest potential avenues for reform that will allow mathematics departments to better serve their client departments in engineering and thus retain control of their courses.

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Notes

  1. The AC T is a standardized exam in the United States. High scores on the math section of the AC T are required for admission to prestigious engineering universities in the United States.

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Acknowledgements

This work was supported by the National Science Foundation under grant EEC 1544388. The opinions, findings, and conclusions do not necessarily reflect the views of the National Science Foundation or the author’s institution.

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Authors and Affiliations

  1. Rose-Hulman Institute of Technology, 5500 Wabash Avenue, Terre Haute, IN, 47083, office C114, USA

    Brian Faulkner

  2. University of Illinois at Urbana Champaign, Champaign, IL, USA

    Katherine Earl

  3. Department of Computer Science, University of Illinois at Urbana Champaign, 201 N Goodwin Ave, Urbana, IL, 61801, office 3213, USA

    Geoffrey Herman

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  1. Brian Faulkner
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  2. Katherine Earl
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Correspondence to Brian Faulkner.

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Appendix: Interview Protocol

Appendix: Interview Protocol

The interview protocol is a semi-structured interview with a set of main questions and samples of follow-up questions depending on how the interviewee responds.

Introductory Questions

  1. 1.

    What engineering courses do you teach?

    1. a.

      How often do you teach these courses?

    2. b.

      Why do you teach these particular courses?

  2. 2.

    Have you been an engineer in industry before your present career in academia?

    1. a.

      What did you do? What mathematics did you use on the job?

  3. 3.

    What are the prerequisite math courses for the courses you teach?

    1. a.

      In what ways does students’ performance in each course you teach depend on their performance in the mathematics prerequisites?

    2. b.

      How do you perceive the content of the courses you teach build on the content of the prerequisite courses?

    3. c.

      In the courses you teach, how would you describe the importance of mathematics?

Primary Questions

  1. 4.

    What is your general perception of the mathematical preparedness of the students rising up to your course?

    1. a.

      How many lectures do you spend on purely mathematical review for your course?

    2. b.

      Does this vary by semester?

    3. c.

      How do you interpret students’ lack of preparation?

    4. d.

      What part do the instructors of the prerequisite courses play?

    5. e.

      What part does the curriculum itself play?

  2. 5.

    What types of attitudes toward mathematics in Engineering do you perceive in your students? How do those attitudes impede students’ learning?

    1. a.

      Do students perceive the “real life” applications of the math they have been taught? Can they connect math to application? The engineering uses?

    2. b.

      What do you do to alter this attitude?

    3. c.

      Do you consider attitudes about learning to be key goals?

  3. 6.

    Is the math students use in your class genuine to the experience of engineering (in your experience in academia/industry)?

    1. a.

      Do you assign problems that have multiple correct solutions? Solutions that aren’t immediately apparent?

    2. b.

      How many minutes do the longest problems take an average students to solve?

    3. c.

      How do your students think about models, critically or hegemonically? Does this improve after taking your course?

Additional Questions

  1. 7.

    What do you think should be the relationship between the engineering curriculum and mathematics curriculum? How well are the current curricula meeting your expectations and needs?

    1. a.

      What do you think about the choice of content and order of content presentation?

    2. b.

      Do students remember/transfer what they have learned in math?

    3. c.

      Are math courses a form of mental exercise for engineers but not directly applicable? Just a form of general education?

  2. 8.

    Do you believe that some mathematics topics or skills are taught better by mathematics faculty and some are taught better by engineering faculty? Why or Why not?

    1. a.

      For example, electrical engineering makes much more extensive use of Laplace transforms, but they are still covered in the standard Differential Equations class

  3. 9.

    What emphasis do you put on derivations in your course? Do you think they are important for future practicing engineers? For future engineering researchers?

  4. 10.

    What specific mathematical knowledge must students have mastered to do well in your course?

    1. a.

      Is it things like integration by parts, or how to define variables, or “what is a scalar?”

  5. 11.

    What mathematical knowledge do you think students should have mastered but is not taught in prerequisite calculus sequence course (dimensional analysis, name that object game, extreme case analysis, etc)?

  6. 12.

    Describe your students’ skills at manipulating notation or working with symbols abstractly.

    1. a.

      Do you encourage graphical/intuitive methods over analytic/formal methods?

    2. b.

      Do you encourage estimation (over calculation) in your class?

    3. c.

      Do they think they can create their own methods or are they beholden to the formula sheet?

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Faulkner, B., Earl, K. & Herman, G. Mathematical Maturity for Engineering Students. Int. J. Res. Undergrad. Math. Ed. 5, 97–128 (2019). https://doi.org/10.1007/s40753-019-00083-8

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