Relevant knowledge concerning the derivative concept for students of economics - A normative point of view and students' perspectives

International audienceThe concept of the derivative plays a major role in economics. A proper understanding of the concept and its application in economics is therefore important for students of economics. In this paper two perspectives on the relevant knowledge concerning the derivative for students of economics are presented: a normative point of view based on literature and the students' perspectives identified in an empirical study. It can be seen that the students' perspectives differed from the normative point of view. An interesting result is that, although emphasized in the course, the students of economics seemed to consider the economic interpretation of the derivative and the corresponding mathematical background knowledge to be much less important than pure mathematical procedures like differentiation rules.</p

Using a lecture-oriented flipped classroom in a proof-oriented advanced mathematics course

In traditional advanced mathematics lectures the instructor usually provides definitions, theorems, and proofs on the board rather quickly. The students often cannot make sense of these during the lecture as they are busy writing. In order to gain an understanding of the content, an intensive post-class processing on the basis of their notes would therefore be necessary for many students. However, students often do not post-class process lectures carefully. Furthermore, they often do not record the lecturer’s oral explanations that are essential to make sense of the formal lecture content. One approach to address these problems is a flipped classroom, in which students come into contact with new content before class, while the in-class time is used for activities that help students make sense of it. We implemented an adaptation of this approach in a proof-oriented analysis course, in which the instructor used the in-class time to illustrate how experts in mathematics make sense of new definitions, theorems, and proofs, while keeping the lecture as the mode of teaching. In this article we describe this adaptation, its possible benefits for addressing students’ problems in learning from traditional mathematics lectures, and its actual effects, which we investigated in an empirical study.Peer Reviewe

Changes in Students’ Mathematical Competencies at the Beginning of Higher Education Within the Last Decade at a German University

Mathematics plays a significant role in many study programs. However, several studies show deficiencies and a decline in beginning undergraduates’ skills in mathematics in many content domains. However, it remains unclear whether they have improved in so-called process competencies like modeling, mathematical reasoning, or using different representations instead, because there has been a shift towards the acquisition of such in many recent curricula. We investigated this issue at a university in Germany based on data from a (non-standardized) mathematics entry test taken by 3076 economics students divided into different cohorts from 2012 to 2019. Using regression analyses, we found that, on the one hand, students’ ability to carry out symbolic calculations decreased. On the other hand, their performance increased in some test questions focusing on other process competencies like reasoning, mathematizing, or using different representations, which have become common tasks at school due to a stronger emphasis on these process competencies after a curriculum reform. Our data indicate that this reform might have had the desired effect.Peer Reviewe

Students’ Understanding of the Economic Interpretation of the Derivative in the Context of Marginal Cost

Many disciplines make use of mathematical concepts. However, there are often discrepancies between the way mathematical concepts are understood and taught in mathematics and the way they are used in other disciplines. The literature suggests that such discrepancies might make it hard for students in mathematics service courses to make a connection between the mathematical concepts taught and the way they are used in the students’ major disciplines. We investigated this hypothesis for one specific example in mathematics for economics students – the derivative and its interpretation commonly used in economics as the amount of change when increasing the production by one unit. We conducted an interview study investigating to what extent economics students can make a connection between the mathematical concept of the derivative and this common economic interpretation of the derivative. This study provides empirical evidence that it is actually difficult for economics students to make this connection, even though it was covered in their calculus course. In particular, the study reveals difficulties students have when trying to make this connection, which could be addressed in teaching.Volkswagen Foundation http://dx.doi.org/10.13039/501100001663Peer Reviewe

Can Guided Notes Support Students’ Note-taking in Mathematics Lectures?

In traditional mathematics lectures the instructor normally writes the definitions, theorems, and proofs covered on the board, and gives informal oral explanations that help to make sense of them. The students have to take notes. However, there are serious problems concerning students’ note-taking in traditional mathematics lectures. Students often cannot think about the information presented during the lecture as they are busy writing. Making sense of the content later is also difficult because many students do not include the lecturer’s oral explanations in their notes. One approach to addressing these problems can be the use of guided notes: a modified version of the instructor’s notes with certain blanks the students have to fill in during the lecture. We investigated to what extent guided notes can support students in their note-taking in mathematics lectures in a study using a mixed-method design. This study provides on the one hand quantitative data suggesting that guided notes are perceived as beneficial by many students for several aspects of their note-taking. On the other hand, it offers qualitative data illustrating how the use of guided notes can influence these aspects. The results indicate in particular that the use of guided notes can address some of the problems concerning students’ note-taking in traditional mathematics lectures, while it can also lead to new problems that one needs to be aware of.Humboldt-Universität zu Berlin (1034)Peer Reviewe

C' (x) = C(x+1)-C(x)? - Students' connections between the derivative and its economic interpretation in the context of marginal cost

International audienceThe derivative concept plays a major role in economics. Therefore, students of economics should have a proper understanding of the concept and its application in economics. One important competence for these students is to interpret the derivative in economic contexts. In books of economics the derivative is commonly interpreted as amount of change while increasing the production by one unit. However, from a mathematical point of view, this interpretation does not directly correspond to the derivative. In the study presented here, it was investigated to what extent students can make an adequate connection between the derivative as a mathematical concept and its economic interpretation mentioned above

Theoretische und empirische Analysen zum benötigten, gelehrten und bei Studierenden der Wirtschaftswissenschaften erreichten Verständnis des Ableitungsbegriffs

Das Ableitungskonzept spielt in der Ökonomie eine große Rolle. Daher sollten Studierende der Wirtschaftswissenschaften ein mathematisch adäquates Verständnis der Ableitung erwerben, das aber auch die Praxis des Umgangs mit ihr in der Ökonomie mitberücksichtigt, damit die Studierenden die Ableitung dort verständig verwenden können. In dieser Dissertation wird untersucht, inwiefern dies unter gegenwärtigen Bedingungen gelingt. Dazu wird zunächst auf Basis von didaktischer Literatur zur Beschreibung von „Verständnis der Ableitung“ und Analysen ökonomischer Lehrbücher zum Umgang mit der Ableitung in ökonomischen Kontexten ein Rahmenmodell zur Beschreibung eines möglichen adäquaten Verständnisses der Ableitung in der Mathematik für Wirtschaftswissenschaftler entwickelt. Dann erfolgt im Vergleich dazu die Untersuchung des praktisch gelehrten und des bei Studierenden erreichten Verständnisses der Ableitung in vier empirischen Studien (Lehrbuchanalysen, Vortest, Analyse von Klausuraufgaben und eine Interviewstudie). Die Studien zeigen, dass Studierende der Ökonomie ein umfangreiches Verständnis der Ableitung benötigen, und dass ein solches in der Mathematik für Wirtschaftswissenschaftler auch vermittelt wird. Die untersuchten Studierenden (Uni Paderborn) verfügten zum Studienbeginn noch nicht über ein solches Verständnis. Sie hatten aber auch größtenteils nach Besuch ihrer Lehrveranstaltung „Mathematik für Wirtschaftswissenschaftler“ kein adäquates Verständnis der Ableitung, obwohl ein solches dort vermittelt wurde. Auf Basis der ermittelten Schwierigkeiten beim Verständnis der Ableitung werden schließlich zahlreiche Konsequenzen für die Behandlung der Ableitung in der Mathematik für Wirtschaftswissenschaftler vorgeschlagen.The concept of derivative plays a major role in economics. Hence, students of economics should gain an understanding of the derivative, which is adequate from a mathematical point of view on the one hand, but also includes the way the derivative is used in economic contexts, so the students will later be able to use it in economics in a reflective manner. In this PhD-Thesis it is investigated to what extent this is achieved in mathematics for economics students under current (teaching) conditions. First, a theoretical model to describe a possible adequate understanding of the derivative that also includes the way the derivative is used in economic contexts is developed. This model is based on didactic literature about students' understanding of the derivative as a mathematical concept and on an analysis of economic textbooks focusing on the way the derivative is used in economic contexts. Afterwards, the state of the art concerning the understanding of the derivative taught to and acquired by economics students before and after their mathematics course is investigated in several empirical studies (textbook analyses, administration of a pretest, analysis of exam tasks and an interview study). The studies show that economics students need a broad understanding of the derivative, and that a broad understanding of the derivative is also taught in mathematics courses for economics students. The economics students investigated (at the University of Paderborn) did not have such an understanding when entering university. However, many did also not have an adequate understanding of the derivative after their mathematics course, although such was taught there. Finally, on the basis of the students' difficulties concerning the understanding of the derivative found out in the studies, several possible consequences for the teaching of the derivative concept in mathematics courses for economics students are suggestedvon Frank Feudel ; Betreuer: Prof. Dr. Rolf Biehler, Prof. Dr. Hans M. DietzTag der Verteidigung: 19.12.2018Universität Paderborn, Dissertation, 201

Supporting the use of study skills in large mathematics service courses to enhance students’ success – one example

International audienceMany university freshmen have difficulties in first year mathematics courses due to inappropriate study skills and learning strategies. In large mathematics service courses the effect of poor study skills is even worse because individual support is hardly possible. Past research shows that it is not sufficient to introduce new study skills because students often do not use these; they need first to be supported when working with them. In this paper we describe one possible support with a coaching program for a tool that aims to assist students in learning mathematical concepts. We describe the aims, the design, and the implementation of the coaching program supporting the students’ work with the tool provided, as well as some evaluation results concerning reported students’ benefit from the program, their attitudes towards the tool, and the frequency of use of the tool at the end of the coaching

Students’ Understanding of the Derivative Concept in the Context of Mathematics for Economics

The derivative concept plays a major role in economics. However, its use in economics is very heterogeneous, sometimes inconsistent, and contradicts students’ prior knowledge from school. This applies in particular to the common economic interpretation of the derivative as the amount of change while increasing the production by one unit. Hence, in calculus courses for economics students, learners should acquire an understanding of the derivative that is mathematically acceptable and connected to their prior knowledge, but which also takes into account its practical use in economics. In this paper we first develop a theoretical model describing such an understanding of the derivative for economics students. We then present an exploratory study investigating the extent to which economics students have such an understanding after their calculus course. The results indicate that many of them might not have acquired this kind of understanding, in particular concerning the common economic interpretation of the derivative. The study furthermore yields possible gaps in students’ understanding and possible misconceptions.Die Ableitung spielt in der Ökonomie eine wesentliche Rolle. Der dortige Umgang mit ihr ist aber sehr heterogen, teilweise inkonsistent und wi- dersprüchlich zum Vorwissen der Studierenden zur Ableitung aus der Schule. Dies trifft insbesondere auf die in der Ökonomie oft verwendete Interpretation der Ab- leitung als Zuwachs der Funktion bei Erhöhung der Produktion um eine Einheit zu. Daher sollten Studierende der Ökonomie in ihren mathematischen Lehrveranstaltun- gen ein Verständnis der Ableitung erwerben, welches mathematisch akzeptabel ist und an ihr Vorwissen anknüpft, das aber auch den praktischen Umgang mit ihr in der Ökonomie mitberücksichtigt. In diesem Beitrag entwickeln wir zunächst ein Modell, das ein solches Verständnis theoretisch beschreibt. Anschließend stellen wir eine ex- plorative Studie vor, in der wir untersucht haben, inwieweit Studierende über ein solches Verständnis der Ableitung nach ihrem Kurs der Mathematik für Wirtschafts- wissenschaftler verfügen. Deren Ergebnisse lassen vermuten, dass viele Studierende nach dem Kurs nicht über ein entsprechendes Verständnis verfügen, insbesondere in Bezug auf die oben beschriebene ökonomische Deutung der Ableitung. Sie liefert außerdem Hinweise auf mögliche Verständnislücken und Fehlvorstellungen.Peer Reviewe