*Students studying mathematics — ETH main building*

The issue that some students fail to “see” and to “learn” abstract ideas is a major challenge for mathematics education, and all the more pronounced in university mathematics. This project brings together expertise in the learning sciences (Kapur, D-GESS) and mathematics (Hungerbühler, D-MATH; Keller, D-MATH) to design preparatory and reflective activities that create opportunities for students to “live” abstract mathematical ideas. These activities are centered around moments of failure in problem solving, because research indicates that failure can act as a catalyst that primes students for learning.

In *Linear Algebra: Problem Solving as Preparation for Learning*(Hungerbühler & Kapur), we focus on pre-instructional interventions, where students work on challenging worksheets as preparation for instruction. In *Principles of Productive Failure at the Assessment Level*(Keller & Kapur), we focus on post-instructional interventions, where students reflect on and revise incorrect answers after instruction.

By modifying existing classroom structures (worksheets, assessments), we are creating an adaptable template for mathematics university education based on the principles of Productive Failure . Potentially, this research will benefit students across a range of (mathematics) courses.

**Linear Algebra: Problem Solving as Preparation for Learning**

Linear algebra is one of the more abstract, and therefore challenging, foundations of university mathematics. Because modern STEM disciplines make extensive use of linear algebra, advancing instruction in this area is not only of academic interest to education researchers, but also of practical interest to ETH as an institution of learning. As part of the Future Learning Initiative (an ETH+ initiative), this project brings together expertise in the learning sciences (Kapur, D-GESS) and mathematics (Hungerbühler, D-MATH) to design preparatory activities for the teaching of linear algebra. The research goals of the project are to: a) design preparatory, problem-solving exercises based on learning sciences principles, and b) implement and evaluate the pedagogical effectiveness of these exercises in the teaching of linear algebra to first-year students.

The project builds on Prof. Kapur’s research on productive failure, which has been successfully implemented in schools in Singapore, Canada, Germany, and elsewhere. The research contribution of the project is the implementation and evaluation of productive failure at the level of university mathematics. On a practical side, the aim of the project is to further improve mathematics education at ETH. Overall, this project aims at simultaneously advance both the science of learning and ETH as an institution of learning.

In practical terms, we have introduced the following changes to the instruction of linear algebra: After having identified areas of conceptual difficulty in first year linear algebra, we have developed worksheets targeting these concepts. To follow the spirit of productive failure, students complete these worksheets prior to the formal instruction given in the lecture. At the end of each semester, students have the opportunity to take a single-choice-test, which assesses their learning-outcomes. Thereby, they obtain valuable individual feedback on their performance. Participation in the preparatory activities, which are given as homework, is voluntary.

The first implementation of the project has taken place during the fall semester 2018 and the spring semester 2019. Three additional iterations are planned for the academic years 2019/2020 until 2021/2022.

An example of linear algebra exercises can be found here.

## Meike Akveld

## Vera Baumgartner

## Alexander Caspar

## Simona Daguati

**Principles of Productive Failure at the assessment level**

This second part of the project “Mathematics Education”, itself a part of the bigger ETH+ project “Future Learning Initiative”, takes up the same fundamental principles of Productive Failure as the above described first part, but in contrast to the first part focused on the post-instruction phase.

Over the last years, Multiple-Choice-Question became more and more an important part of almost all assessment levels ranging from weekly exercises to exams.

As a particular feature of such Multiple-Choice-Questions, Dr. Laura Keller implemented feedback-rich mechanisms in her first years lecture in mathematics for student in chemistry and medicine.

But so far, no rigorous evaluation of such feedback-rich Multiple-Choice-Questions has been done, and there is only little knowledge of how students deal with wrong answers and of how effective the given feedbacks are.

In the present project we will on one hand redesign these feedback-rich Multiple-Choice-Questions according to the principles of Productive Failure in such a way that the students not only receive feedback on the items they solved incorrectly, but also that they are given the opportunity to revise their answers. More precisely, correct answers given after revision will still give partial credit. These combined mechanisms are intended to deepen the understanding of complex and abstract concepts, to connect insight and to foster the ability of the students to adapt to new situations and to learn from errors — a critical 21st century competency required in all our graduates.

On the practical sine, the implementation of these feedback-rich Multiple-Choice-Questions with partial credit feature will be done primarily on Moodle.

On the other hand, we will evaluate these feedback-rich Multiple-Choice-Questions in assessments (possibly also at the exam level) in order to gain insight into how students learn and how optimally designed Multiple-Choice-Questions can enhance their learning experience and learning efficiency.

Findings from this project will thus potentially feed directly into the teaching of mathematics – and other disciplines where abstract concepts are crucial – throughout ETH. The evaluation will potentially be based on various evaluation tests (classical test theory, item response theory,…).

The current members of this research project are Prof. Dr. Manu Kapur, Dr. Laura Keller, Dr. Dragan Trninic and Vera Baumgartner.

An example of such a feedback-rich Multiple-Choice-Question can be found below.

First wrong answer

*Right answer after revision and partial credit*