This chapter provides a summary of some popular model-based deep learning methods and their extensions. Section 8.1 briefly describes classical model-based methods and their benefit as well as limitations. Section 8.2 describes how deep learning can help in overcoming some limitations of classical model-based methods. Section 8.3 discusses how to incorporate a pre-trained deep network as a regularizer using the plug-and-play approach. Section 8.4 describes end-to-end training using a model-based deep learning framework. This section also discusses some benefits and limitations of end-to-end training. Section 8.5 and 8.6 describe unsupervised model-based deep learning approaches when a clean training dataset is not available. Section 8.6 considers model mismatch issues as well as the joint design of acquisition and reconstruction frameworks.

Introduction

Education: that which reveals to the wise, and conceals from the stupid, the vast limits of their knowledge. (Mark Twain)

Problem-based learning (PBL) was introduced as a teaching method in undergraduate medical education at McMaster University Medical School, Ontario, in 1969. Since then it has been integrated into medical curricula worldwide and has been endorsed by bodies such as the World Health Organization (Spencer & Jordan, 1999); it was first adopted in Europe by the University of Maastricht Medical School in 1974. In response to the recommendations made by the General Medical Council (1993) in Tomorrow's Doctors, in 1996 the University of Glasgow was the first UK medical school to adopt PBL as the core educational method in the medical undergraduate curriculum. Around 30% of medical schools in the UK have since moved to a PBL curriculum.

Considerable research has examined the effectiveness and outcomes of PBL curricula. Although it seems early to suggest that the approach encourages recruitment into psychiatry, there is evidence that PBL has specific advantages in the acquisition of skills relevant to the specialty and that it may improve performance in psychiatric examinations.

What is problem-based learning?

Strategies for PBL were developed as part of an exploration of educational theories of adult learning. There is an emphasis on active learning, facilitated by a constructive, self-directed, collaborative and contextual learning process (Dolmans et al, 2002). Key concepts within PBL are that its use with small groups should enable students to explore their existing knowledge, identify gaps, carry out independent learning and return to the group to discuss and share this new knowledge. This allows students to establish links more readily between different learning experiences and the process equips the student with skills that can be applied to self-directed, lifelong learning.

Despite the strong theoretical basis behind PBL, its use in medical schools has been shown to be heterogeneous, in terms of both the format used and the amount of PBL within various curricula (Maudsley, 1999; Kelson & Distlehorst, 2000). Regardless of these differences, three common features of PBL occur across medical undergraduate curricula:

Problem-based learning (PBL) is a student-centred pedagogic approach in which learners work collaboratively to solve problems and reflect on their experiences. It was pioneered and used extensively at McMaster University in Canada. It has also been widely applied in the UK within medical education.

PBL draws on social-cultural and constructivist theories of learning. Instead of presenting learners with information and asking them to understand and memorize it, PBL means the learners find out for themselves. Learners often find this approach more interesting and interactive and there is evidence that it leads to deeper learning, allowing learners to construct their own understanding.

Key characteristics of PBL include:

Introduction

In the practice of engineering education, there is a wide variety of implementations of problem-based or project-based learning (PBL). In this chapter we aim to explain the relationships between different types of problem-based and project-based learning to help teachers and educational managers make innovative choices and provide benchmarks for educational researchers. We present a combined understanding of problem- and project-based learning, the theoretical and historical background, and the different models of PBL that can capture the existing practices, ranging from small- to large-scale practice, from classroom teaching to institutional models, and from single-subject to interdisciplinary and complex knowledge construction.

It is well known that one-way dissemination of knowledge by means of lectures is not very effective in achieving learning (van der Vleuten, 1997). In higher education concepts such as “self-directed-learning,” “case-based learning,” “inquiry based learning,” “experiential learning,” “service learning,” “project-based service learning,” “active learning,” CDIO (Conceive, Design, Implement, and Operate), “project-based learning,” and “problem-based learning” were introduced in the decades after the Second World War. All these new learning concepts come under the umbrella of learner-centered or student-centered learning models. Problem-based and project-based learning, both known as PBL, originate from the reform universities, and the new educational models, established between 1965 and 1975. In problem-based learning, problems form the starting point for students’ learning emphasizing a self-directed learning process in teams. The educational model problem-based learning was introduced at curriculum scale at the medical faculty of McMaster University, Canada, followed by Maastricht University in the Netherlands and many others. Project-based learning shares the aspect of students working on problems in teams, but with the added component that they have to submit a project report completed collaboratively by the project team. The problem- and project-based/project organized model adopted at Aalborg University and Roskilde University, Denmark, was inspired by the critical pedagogy in Europe after the student revolts of the 1960s. At Aalborg University both models of PBL were eventually combined in problem-based project organized learning, which was practiced at all faculties – the Faculty of Engineering and Science being the largest. This combined approach is the central point of reference for this chapter, as the pedagogical development in engineering education indicates that both educational practices are successful in their own way and the abbreviation PBL is here defined as including both practices.

Abstract

This literature review examines the synergy between problem-based learning and problem solving in mathematics education in 26 selected research reports. Focusing on what mathematics teachers can do to promote students’ problem-solving competencies, the study sought to develop an account of the critical role of problem-based learning in learning academic content necessary to apply in problem solving. The review is framed in the context of progressivism, reconstructionism and real-world problems. The reading and analysis of the studies revealed five themes: knowledge for real-world problem solving; hard scaffolds as an instructional strategy; soft scaffolds as an instructional strategy; interdependency for improved performance; and reflection for enhanced actions. The review's findings suggest that problem-based learning can be used as an approach for learning academic content to apply in problem solving. Hence the authors suggest fostering problem-based learning instructional strategies in pre-service mathematics teachers’ preparation to enhance movement towards teaching problem solving in school mathematics in South Africa.

Framing the Problem

Life is about solving problems, and people solve problems each day of their lives. Therefore, ‘real-world problem solving extends from habit to the intellectual to the social and cultural sphere. Hence today's society requires people to be able to solve complex problems’ (Van Merriënboer 2013:153).

However, ‘the challenge is that problem solving is an extremely complex cognitive process about which little is known’ (Van Merriënboer 2013:153).

The South African Department of Basic Education [DBE]'s (2011) Curriculum and Assessment Policy Statement (CAPS) considers ‘problem solving as an essential aspect of mathematical teaching and learning’ (p. 4).