pedagogical remedial scenariossubversion.math-bridge.org/math-bridge/public/wp01... · pedagogical...

Pedagogical Remedial Scenarios

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ECP-2008-EDU-428046

Math-Bridge


Deliverable number D-1.3

Dissemination level Public

Delivery date April 30th 2010

Status Final

Author(s)

Prof. Dr. Rolf Biehler (University of Paderborn) Prof. Dr. Reinhard Hochmuth (University of Kassel) Pascal Rolf Fischer (University of Kassel) Thomas Wassong (University of Paderborn)

eContentplus

This project is funded under the eContentplus programme1, a multiannual Community programme to make digital content in Europe more accessible, usable and exploitable.

1 OJ L 79, 24.3.2005, p. 1.


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Project ref. no. ECP-2008-EDU-428046

Project title MATH-BRIDGE European Remedial Content for Mathematics

Deliverable dissemination level Public

Contractual date of delivery April 30th, 2010

Actual date of delivery April 30th, 2010

Deliverable number D 1.3

Deliverable title Pedagogical Remedial Scenarios

Type Deliverable

Status & version Final

Number of pages 39

WP contributing to the deliverable

WP 1

WP / Task responsible Prof. Dr. Rolf Biehler/ Michael Dietrich

Author(s) Prof. Dr. Rolf Biehler (University of Paderborn) Prof. Dr. Reinhard Hochmuth (University of Kassel) Pascal Rolf Fischer (University of Kassel) Thomas Wassong (University of Paderborn)

EC Project Officer Marcel Watelet

Keywords Remedial Scenarios


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Table of Content

1 INTRODUCTION ....................................................................................................................................... 4

2 GOALS AND SCENARIOS FOR USING THE MATH-BRIDGE MATERIAL ................................. 4

3 THE POTENTIAL OF THE ACTIVEMATH SYSTEM ........................................................................ 5

4 THE NEED FOR COMPLEX LEARNING OBJECTS IN ADDITION TO ATOMIC LEARNING OBJECTS .............................................................................................................................................................. 6

4.1 PEDAGOGICAL STRUCTURE OF THE VEMA-MATERIAL ............................................................................. 6 4.2 TYPIFICATION OF CLOS ............................................................................................................................ 8

5 TECHNICAL FEATURES OF MATH-BRIDGE WITH REGARD TO IMPLEMENTATION IN VARIOUS REMEDIAL COURSE SCENARIOS ........................................................................................... 10

5.1 MANUAL BOOK GENERATION ................................................................................................................. 10 5.2 MODIFIED FORMALIZED PEDAGOGICAL SCENARIOS FROM LEACTIVEMATH AND AUTOMATIC BOOK

GENERATION ..................................................................................................................................................... 11 5.2.1 LearnNew (as in LeActiveMath) ................................................................................................... 12 5.2.2 Rehearse ........................................................................................................................................ 14 5.2.3 Workbook ...................................................................................................................................... 16 5.2.4 TrainCompetency .......................................................................................................................... 18 5.2.5 ExamSimulation ............................................................................................................................ 20

5.3 FORMALIZED PEDAGOGICAL SCENARIOS ON THE BASIS OF CLOS ........................................................... 22 5.3.1 Order of the CLOs ......................................................................................................................... 22 5.3.2 Optional scenarios for the selection of the CLOs ......................................................................... 24

5.4 LEARNING ADVICE COMPONENT ............................................................................................................. 26 5.4.1 General Learning Advice Component ........................................................................................... 26 5.4.2 Formalized Learning Advice Component (Self-Assessment Component) ..................................... 27

5.5 EMBEDDING MATH-BRIDGE IN LEARNING MANAGEMENT SYSTEMS ....................................................... 30

6 SUMMARY AND FUTURE WORK ....................................................................................................... 30

7 REFERENCES .......................................................................................................................................... 32

8 ANNEX ....................................................................................................................................................... 33

8.1 REMEDIAL SCENARIOS OF THE MATH-BRIDGE PARTNERS ....................................................................... 33 8.1.1 Remedial Scenarios at KS and UP ................................................................................................ 33 8.1.2 Remedial Scenario at UC3M ........................................................................................................ 35 8.1.3 Remedial Scenarios at UV ............................................................................................................ 35 8.1.4 Remedial Scenarios at TUT .......................................................................................................... 36 8.1.5 Remedial Scenario at UM2 ........................................................................................................... 37 8.1.6 Remedial scenarios of OUNL ........................................................................................................ 37


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1 Introduction

This deliverable describes the pedagogical remedial scenarios that will be used for Math-Bridge. Since these scenarios will be relevant for all partners in order to use Math-Bridge for their individual remedial courses, we first will have a look at the goals and scenarios for using the Math-Bridge material.

On the other hand, Math-Bridge will be based on the intelligent tutor system “ActiveMath” for which some pedagogical scenarios were already developed and implemented in context of the project LeActiveMath. Therefore it is necessary to briefly describe the potential of the ActiveMath system in the second chapter.

Analysing the interactive content that will be provided from the content partners, we found out that it is necessary to introduce a new structure element, which is described in chapter 4: The Need for Complex Learning Objects in Addition to Atomic Learning Objects (CLO in the sequel).

Based on these preliminaries we will describe in the next chapter all technical features of Math-Bridge with regard to implementation in various remedial course scenarios. This chapter – which is central for the deliverable – describes all formalized pedagogical scenarios that will be adapted from the project LeActiveMath on one hand, on the other new scenarios on the basis of CLOs will be defined. Moreover we describe the need of a learning advice component as well as some necessary feature for the embedding of Math-Bridge in learning management systems.

The final chapter contains a summary and future work with regard to the findings of this document.

2 Goals and Scenarios for Using the Math-Bridge Material

Goals of Math-Bridge

Math-Bridge aims at designing, using and testing learning material for the remedial repetition of school mathematics in preparation of mathematics at university level. This learning material can be used in two general formats of learning: In terms of self-directed learning of individuals or in the context of the remedial scenario of a “bridging course” that can be found at most European universities. These courses are often offered in the form of different kinds of blended learning scenarios. Math-Bridge will support both formats of learning.

Math-Bridge Partners’ Remedial Course Scenarios

Having a look at the current use of bridging courses in the project partner’s universities1, we identified blended learning scenarios with a varying extent of eLearning. All types of courses of the partners use learning material. Usually this is non-dynamic in their way of presenting and structuring the content. Partners use books or multimedia-enriched learning material. The chapters of the material are ordered in a predefined and constant matter designed by pedagogical expert knowledge. But the course scenarios differ in the learners’ handling of the material.

In the blended learning scenarios with an extensive eLearning or distance-learning part the students decide about the intensity, the sequencing and the time allocation independent of the 1 The descriptions of the partners’ remedial course scenarios can be found in the annex.


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teacher. Here, the tutors just support the learner in structuring his learning and provides feed-back. Only the selection on the chapters to be learned is done by the tutor by just providing the necessary learning material.

In the scenarios with a reduced eLearning part but an extensive or exclusive attendance part the tutors decide which content will be learned at which moment and in which intensity. They also control the order of the material and the time spent on different chapters. The students have almost no possibility to decide on these aspects of learning themselves.

Beside the structure and handling of the learning material we identified another typical element of the current course scenarios: the use of a Learning Management System (LMS). The LMS is used on one hand for communication, for providing and presenting the learning material and on the other hand as a common learning place.

As a consequence the Math-Bridge material must be usable for all kinds of course scenarios as described above and either has to provide necessary features of LMS or must be integrable in LMS.

Course Scenario of the Mathematics Learning Centre (MLC) at Memorial University (Newfoundland, Canada)

Although Memorial University is not a project partner, part of the Math-Bridge team met with Dr. Sherry Mantyka and Theresa Ricketts and discussed remedial scenarios on the background of the courses at the MLC. The result of the discussion can be found in Mantyka & Ricketts 2010. The scenario presupposes other kinds of learning objects than are available from the content partners of the MathBridge project. In particular, the extended training of technical competencies where MLC is focussing on needs a huge number of adequate LOs. We consider the inclusion of MLC scenarios for some topics as part of future work, when all LOs of all partners are technically available in the ActiveMath system.

3 The Potential of the ActiveMath System

For Math-Bridge the project partners have chosen ActiveMath as the central eLearning system. ActiveMath provides some particular features that have to be mentioned here.

In the ActiveMath system, the content is formally represented in (atomic) Learning Objects (LO). These objects are enriched with metadata regarding for instance related topics, the prerequisites and the type of the LO. All LOs regarding one domain are structured in collections, internally they are stored in a database. These LOs can be organised and structured in books in two different ways:

The author and the tutor (and the learner) can build their own books by ordering the LOs in a particular way and group them into pages, sections and chapters. Here fore pedagogical expert knowledge is necessary.

ActiveMath also provides the automatical generation of books according to certain principles, called “formal learning scenarios”. The system uses the metadata for the selection of the applicable LOs. Therefore learning scenarios have to be identified; they define the formal structures of the automatically generated books and should be based on pedagogical insights. The existing learning scenarios for ActiveMath are discussed in Reiss et al. (2005).

The learner gets access to the learning material through predefined books and through automatically generated books according to the learning goals he has specified. Besides he can use the extensive semantic search to get access to specific LOs.


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Using the scores of exercises a learner has solved, the system collects information about the learning progress of the learner. The ActiveMath learner model consists of a complex structure of domains and competencies. The evaluation of the exercise results depends on the Item-Response-Theory and the Transferable-Belief-Model (cf. Faulhaber & Melis 2008). The learner model is used in ActiveMath for selecting adequate examples and exercises regarding the learning progress of the individual learner.

4 The Need for Complex Learning Objects in Addition to Atomic Learning Objects

Math-Bridge will use modified meta-data and slightly formal learning scenarios and can build on the ActiveMath system, which will be described later in detail. In order to use their content for Math-Bridge, all content partners need to implement it into the ActiveMath system. Therefore the content is actually being sliced into atomic learning objects such as examples, exercises, definitions, interactivities etc. and will afterwards be recombined by the system into books as briefly described above.

In this section we will first describe the structure of the VEMA Material as provided by the partners from Kassel and Paderborn (cf. Fischer & Biehler 2006; http://www.mathematik.uni-kassel.de/vorkurs [27.04.2010, 12:00]). With this structure we can prove the necessity of the introduction of Complex Learning Objects.

4.1 Pedagogical Structure of the VEMA-material

The VEMA-Material contains learning material for nearly the whole school mathematics from grades 6 to 13, except for geometry, stereometry, and stochastic. The material is structured in 5 chapters from basic arithmetic over functions up to calculus and vector analysis. Every chapter contains several subchapters and each subchapter contains modules. Modules are self-contained learning units about a certain mathematical domain, for instance binomial formulae or the trigonometric functions. Every module is structured in exactly the same manner (cf. Fischer & Biehler 2006). This structure is illustrated by the following diagram:

Structure of a module within VEMA


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Every module starts with an introductory overview page, where the major topics are listed. These major topics are very abstract and are just used to get an overview of the topic. The second page is an introduction to the domain. The introduction uses discovering, inductive and exemplary approaches to get started. Here the content is presented on a concrete level for the learner. The introduction starts from knowledge that is prerequisite and develops from this basis the new domain. In some cases the content on this page is enriched with some exercises. These two pages form the overview and introduction into a module.

The main part of the module starts with the Info-page. Here the definitions, theorems and algorithms of the module are listed on one page. These are the central concepts of the module so that this page can be used as reference book. The Info-page presents the content on an abstract mathematical level so that the pure definitions, theorems and algorithms are presented without examples or exercises.

The following page is Motivation / Interpretation / Explanation. In this page the central definitions, theorems and algorithms are networked among each other and illustrated with examples. In case of theorems we can find here plausible arguments or proofs for their correctness. This page combines the definitions with concrete examples and explanations. Here you can find interactive exercises as well as flash-films and animations the user can interact with and that help him to develop a deeper understanding of the concepts. This way the user can also haptically experience the concepts and hence the interactions provide information, develop comprehension and enhance the memorisation of knowledge. On the following page some inner- and outermathematical applications are shown and discussed. These concrete examples show the connection of the actual domain to other mathematical and non-mathematical domains.

On the next page some typical mistakes are collected in terms of exercises. The learners are invited to find mistakes in a given solution or argumentation, to correct them and to explain possible didactical reasons for them. These exercises ought to train the diagnostic competencies of the learners and to depict misconceptions in order to prevent them. The learners can check their answer by reading the corrected sample solutions and compare them with their own solutions afterwards.

The last page is the exercises-page. Exercises are important for the learner to check their understanding of the learned subjects and to give possibilities of practicing the concepts. For every exercise a sample solution is available, which the learner can use to compare it with his own solution. These sample solutions are also helpful to get an initial idea to solve an exercise or can be used as support as soon as the solution process gets stuck.

Some modules have an additional part with further two pages: The first one, visualizations, lists all interactive visualizations that are used in the module. Learners, who needs more input concerning the specific domain, can optionally go to the second page, the supplements. Here, content is available in form of examples, definitions, interactivities or exercises that is not obligatory, but might be interesting for a deeper insight of the domain.

Analysing the learning material from the project VEMA however shows the need for the introduction of a new element for structuring the content. Within the learning material of the VEMA-material the sequence of atomic learning objects are not (always) freely exchangeable. Some sequences of atomic learning objects belong strictly together due to findings of didactical subject matter analysis. For instance, an “introduction” or an “explanation” forms a holistic unit of learning. They contain several atomic learning objects and intermediate text. The individual learning objects are reusable in other contexts, but it would be helpful to keep the whole learning unit as well, which could be reused as well in other contexts. We call these units “complex learning objects”.


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This means that within these pages, there are certain atomic learning objects that build a coherent learning complex whose structure is derived from didactical needs. Hence the structure of these complexes should not be broken up. Moreover this content specific order of atomic learning objects cannot be generalized. This is important to underline, as every type of the current “formal learning scenarios” from LeActiveMath has its specific but general formal structure, as will be described later. It is not possible to use this structure universally without destroying well-evaluated bits of coherent learning units. Therefore it is on one hand indispensable to keep these complexes of learning objects as an entity; on the other it is impossible to indicate a general structure for the atomic learning objects within these entities.

Nevertheless these entities can always be related to a certain definition or theorem and can be typified with regard to the module-pages they refer to. Vice versa the module-pages of the VEMA material can be generally described by a sequence of such “complex learning objects”.

Since we are convinced that slicing and recombination of the content of other partners will need to define such complexes of learning objects too, we introduce a new structure element called “complex learning object” (CLO) that can be implemented within ActiveMath as subgroups and can be used as atomic learning objects themselves. With this new element, for instance the VEMA material can be sliced and recombined without loosing the subject matters’ ideas of certain content elements.

We now want to typify the most important CLOs with regard to the material of VEMA.

4.2 Typification of CLOs

Introduction CLO

This CLO intends to introduce the learner to a certain concept. It may be designed as introduction to several definition or theorems. The introductory CLO usually is based on other conceptual prerequisites. It may consist of examples, exercises, interactivities, problems or motivations in a didactically well-chosen manner, where no formal generalization is possible (such as start always with an example, then a counter-example, then an exercise etc.)

Motivation/ Interpretation/ Explanation CLO

The Motivation/ Interpretation/ Explanation CLO is designed to develop the learner’s comprehension of a certain definition or theorem. It is usually written for just one definition or theorem and may consist of examples, explanations, counterexamples, proofs and interactivities.

examples* exercises*

motivation* problems*

interactivities*

Introduction CLO


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Application CLO

Application CLOs shall give an insight of a concept’s inner-mathematical or real world applications. It can be designed for just one definition or theorem but may also refer to a combination of several of them. The application CLO can contain examples, exercises or interactivities.

Misconception CLO

The misconception CLO consists of examples, exercises or interactivities that show typical mistakes that typically occur when using a concept.

Supplement CLO

The Supplement CLO provides additional learning material that is not obligatory for a certain topic but interesting for learners who want to deepen and expand their knowledge. The CLO may consist of definitions, explanations, examples, exercises, problems, proofs or interactivities.


interactivities* problems*

definition*

proof*

explanation* Supplement CLO


interactivities*

Misconception CLO


interactivities*

Application CLO

example*

interactivity*

counterexample*

proof*

explanation* Motivation/ Explanation/ Explanation CLO


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Practice CLO

A sequence of well chosen exercises according to pedagogical and content criteria, in order to provide opportunities to practice the knowledge and competence about a set of certain concepts. As the system allows to switch off the feedback components of exercises Practice CLOs can also be used for assessment purposes.

These types of CLO are derived from the analysis of the VEMA material. Though these types describe the most important and widely-used aspects of comprehensive learning, the project partners may need further types of CLOs, which can be easily added to the above lists of topics. This will be chosen in the next step of development.

Having introduced and typified the CLOs, it is possible to describe the formalized pedagogical scenarios for using the material of VEMA. These scenarios can be used for other CLOs as well as for atomic learning objects with the same subject matter. On the other side, scenarios which are designed for atomic learning objects can also use CLOs with the same didactical intentions.

Using both, atomic and complex learning objects we can now describe the technical features of Math-Bridge with regard to the implementation for different remedial course scenarios.

5 Technical Features of Math-Bridge with regard to Implementation in various Remedial Course Scenarios

In this chapter we will describe different technical features that are necessary for the creation of remedial course scenarios in all kinds of formats with the MathBridge material. We will start with the manual book generation as an instrument for individual book design. Then we will describe modified formalized pedagogical scenarios from LeActiveMath and automatic book generation, which are designed for atomic learning objects. Afterwards we will introduce two new formalized pedagogical scenarios on the basis of CLOs before we come to a description of the learning advice component we suggest to be developed and implemented for Math-Bridge. This component includes special technical services and features as well as a “formalized learning advice component”. Finally we describe the need for features that allow the embedding of Math-Bridge in learning management systems.

5.1 Manual Book Generation

It is already possible for teachers as well as for students to manually generate books within ActiveMath by assembling learning objects. For Math-Bridge, this interface and “assembly tool” needs to be available for atomic learning objects as well as for CLOs. Moreover we need an understandable user guide for enabling teachers and learners to use this feature, which is currently not available in ActiveMath.

This is for instance necessary for teachers in blended learning scenarios who want to structure

exercise 1* exercise 2*

exercise 3*

Practice CLO


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the content for their learners themselves. Reasons for this may be that they want to integrate Math-Bridge only at a certain stage of their courses and only concentrate on specific part of the content. E.g. they might want to let students just prepare for a specific lesson and hence only need a specific selection of an introduction without presenting the definitions which will be presented within the lessons. Another example could be that they do not want to overcharge their students with a larger extent of adaptivity that may be counterproductive in the context of a difficult content. Therefore they might want to specifically choose learning objects for their students and forbid them to get adaptively changed content.

On the other side, students may also want to design individual books without using certain formalized pedagogical scenarios, for instance if they want to collect certain learning objects for designing a sort of individual learning diary that can be used as a personal reference book or they might find out that the formalized pedagogical scenarios do not fit their needs so that they want to design individual books themselves.

Giving the opportunity of manual book generation to both, students and teachers, can inversely be prolific for the project: Analysing the structure of these individual books may reveal the need for the implementation of new formalized pedagogical scenarios.

Since not all learners or teachers have sufficient didactical and pedagogical capabilities for designing such books it is necessary to design a user interface that supports the manual design of books and that is user-friendly and that provides sufficient pedagogical advice to enable a learner to assemble elements of a workbook according to his learning goals and according to pedagogical principles.

Manual Exam Simulation

The design of assessment tests is a very important and sensitive issue. Usually an assessment test is designed with regard to a specific learning group and their individual learning history. Besides it serves as verification of the achievement of specific learning goals. Hence it is very difficult to design a general assessment test for all kinds of learners in different countries. The development of assessment tests is part of the workpackage 1 and is prepared and discussed within the Deliverable D1.2 “Content and Assessment Tools” (cf. Biehler et al. 2010).

Taking into account the individual needs of different teachers or learners for individual test design, it is necessary to enable a manual design of an exam simulation. This feature must enable the teacher/ learner to individually design a test not only with regard to the topics to be tested but also concerning test time, the difficulty, the competencies and other metadata. He may be able to choose specific items as well as types of items that are randomly picked for a test. Besides the teacher/ learner must certainly be able to switch off the feedback of the system during the test in order to simulate a real exam situation. As an example we relegate to the Artist Assessment Builder (cf. https://app.gen.umn.edu/artist/user/login.asp [27.04.2010, 12:00]).

Hence learners and especially teachers can use Math-Bridge to design tests for their individual learning scenarios on their own.

5.2 Modified Formalized Pedagogical Scenarios from LeActiveMath and Automatic Book Generation

We will now describe those formalized pedagogical scenarios which derive from the project LeActiveMath and which will be adapted for Math-Bridge. The system ActiveMath already uses different formalized learning scenarios that combine atomic learning objects into books with different learning goals. These pedagogical scenarios are described precisely in the


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publications of Reiss et al. (2005), Ulrich et al. (2006) and Ulrich (2008).

In this section we only will briefly describe the learning scenarios that we will use for Math-Bridge. The first three scenarios will exactly be used as they already exist within LeActiveMath. The last two scenarios need to be adapted for Math-Bridge and need some changes as indicated in the sections below. The following description is a descriptive summary of these scenarios and refers to Reiss et al. (2005).

5.2.1 LearnNew (as in LeActiveMath) This scenario is exactly as in LeActiveMath.

Objectives The first learning scenario was implemented for learners who want to learn an unknown mathematical concept. Therefore, all necessary learning material concerning a chosen topic is offered to the learner in order to give him potentially a full understanding of the concepts. (cf. Reiss et al. 2005, p. 19)

Description As illustrated in the figure below, LearnNew consists of different stages of learning starting with an introduction, proceeding with the development of the concept, the practise section, the connection and transfer and finally the phase of reflection. In this last stage the user may consult the Open learner model which gives feedback about his actual performance in content knowledge and competencies (cf. Reiss et al. 2005, p. 20). In the right column of the diagram you can find the sorts of learning objects, respectively the system activity that are relevant for each stage.

Justification This learning scenario enables the user to learn the concepts in a comprehensive way. The introduction serves as motivation to the learner as well as it supports the comprehension of the concept. Using examples and explanations for the definitions supports the user in understanding the concept. Practicing on the other hand helps do incorporate the knowledge and to train competencies needed for the concept. Showing theorems, lemmas and remarks and theorem’ proofs support the learner in networking the definitions he learned (cf. Reiss et al. 2005, p. 19). The final phase of reflection trains the learner’s ability in self-estimation and self-regulation.

Adaptivity In this scenario, adaptivity is realised for instance using the learner model to identify prerequisites to a concept that are already unknown to the user. These prerequisites will be optionally available for the learner. Another example for adaptivity in this scenario is the way of representation of the content: If a user has a higher competency level, then he will only get the definition but not any additional explanation or example (cf. Reiss et al. 2005, p. 19).


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Figure is taken from Reiss et al. 2005, p. 191.

1 In LeActiveMath the Learner Model was called OLM. Actually ActiveMath uses the Learner Model SLM (cf. Faulhaber & Melis 2008). For Math‐Bridge we will adapt the SLM in accordance with the competence model for Math‐Bridge (cf. Biehler et al. 2009).


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5.2.2 Rehearse This scenario is exactly as in LeActiveMath.

Objectives

This learning scenario was not developed for novices but for those who want to rehearse certain concepts in order to fill gaps in their knowledge. The learning scenario therefore gives a complete and detailed repetition comparable to the scenario LearnNew but with higher learner autonomy and without introductory texts. The student is enabled to adapt the content individually, e.g. he can ask for up to three further examples in order to get different perspectives to one concept (cf. Reiss et al. 2005, p. 21).

Description

Rehearse starts with the concept repetition and proceeds with an understanding phase including examples and counterexamples to the concepts. Afterwards applications of the concept are shown in form of theorems and remarks. The scenario ends with a self-test that gives feedback to the student’s knowledge and knowledge gaps and shall thereby also train his self-diagnostic abilities on meta-cognitive level (cf. Reiss et al. 2005, pp. 21).

Justification

Skipping introductory or motivational learning elements is possible since the student knows the concept in a certain degree and since he is motivated to learn it knowing about his deficits. Therefore it is sensitive to directly start with the repetition of the definition. Still the phases for developing understanding and giving insight of applications are necessary to network the concept and to develop comprehension. The final self-test gives feedback to the student and indicates, if he managed to increase his knowledge. On meta-cognitive level the learner’s ability of self-diagnose is trained (cf. Reiss et al 2005, p. 22).

Adaptivity

This scenario has a higher level of adaptivity than the scenario LearnNew. The learner can ask for up to three further examples in the phase of concept understanding. Concerning the self-test, the system not only gives adaptive feedback and updates the learner model, in addition to the default set of ten exercises, the learner can ask for another set of five exercises via the button “additional exercises”. Having finished the test, the learner may ask for further examples or exercises to the concept or may switch to another concept (cf. Reiss et al. 2005, p. 22).


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5.2.3 Workbook This scenario is exactly as in LeActiveMath.

Objectives

This scenario is mainly a collection of exercises and examples that can be freely chosen and arranged by the learner himself in order to train a certain concept (cf. Reiss et al. 2005, p. 27).

Description

First the student has to choose the fields he’s interested in and has to decide if the exercises are of higher level from the start on or not. Moreover he has to indicate if he wants to concentrate on a specific competency or wants to work on all competencies. Afterwards the system generates a course which provides at least three exercises of different competencies and formats beginning with one competency. The level should be adequate to the learner’s actual competency level. The user may work on all exercises but can switch certain exercises as well. Examples are always optionally available for the learner. Having trained all chosen competencies on a certain level, additional examples can be chosen by the learner if needed (cf. Reiss et al. 2005, p. 27).

Justification

The scenario aims at an individual training which is specified to the learner’s needs which is made possible by giving him many options for individual changes to his workbook. Moreover it can be used for classroom teaching in practice phases (cf. Reiss et al. 2005, p. 28).

Adaptivity

The student has many options within this learning scenario. He can initially choose the topics and competencies to work on and can tell the system to show additional examples during his learning if needed. The exercises’ competency levels are adaptively chosen by the learning system with respect to the learner’s competencies and his initial decisions (cf. Reiss et al. 2005, pp. 27).


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5.2.4 TrainCompetency This scenario is formally the same as in LeActiveMath, however, it has to be modified as the competencies, the achievement level and the difficulty assignments are different from LeActiveMath. The scenario in LeActiveMath gets information about the current “competency level” of the student and selects exercises of the respective “competency level”. We will use the same formal scenario but the meaning of “competency level” has to be redefined according to the new Math-Bridge competence model (cf. Biehler et al. 2009).

Objectives

In the scenario “TrainCompetency”, the learner can train and enhance a certain competency in order to get a deeper understanding of certain concepts (cf. Reiss et al. 2005, p. 25).

Description

The student first gets an overview of his actual performance from the learner model together with a suggestion which competency should be trained. After his choice of the competency to be trained, the learner (with let’s say competency level x) gets a definition and afterwards an example and an exercise on competency level x-1. After the successful performance of the exercise, the system shows an example and an exercise on level x and so on (cf. Reiss et al. 2005, p. 25). Hence the student’s competency level can be trained and enhanced with respect to his actual competency performance (cf. Reiss et al. 2005, p. 25).

Justification

This scenario enables the learner to specifically train certain competencies either concentrating on a specific topic or not. With this, the learner gets a holistic view on mathematics and can develop a deeper understanding of certain topics.

By starting with a lower competency level than his actual enables the learner to develop his competencies with respect to his individual performance (cf. Reiss et al. 2005, p. 25).

Adaptivity

The number of exercises that are presented in the practice phase depends on the learner’s success rate and his time spending on the exercises. Before proceeding with the next level, the learner can optionally choose to see an example for the next level first. The learner can always proceed with a higher level if available independent of his performance (cf. Reiss et al. 2005, pp. 25).


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5.2.5 ExamSimulation

This scenario differs from the scenario with the same name in LeActiveMath by the following features: Instead of single exercises also Practice CLOs can be chosen. Instead of taking into account the Levels I-IV of LeActiveMath the Math-Bridge system will use difficulty level and achievement level accordingly. The ExamSimulation feature has also to take into account the Math-Bridge assessment system (see Deliverable D1.2 of the Math-Bridge project (Biehler et al. 2010)).

Objectives

The fifth and last learning scenario is for those students who have already learned certain concepts and want to test themselves under exam conditions.

Description

The student can only select the concepts to be tested and a time limit for the whole test. The exercises are automatically chosen by the system not with regard to the learner’s competencies but with increasing difficulty level. During the test the learner does not get any feedback at all. Afterwards the system gives feedback on the learner’s test performance and makes revision suggestions (cf. Reiss et al. 2005, p. 29).

Justification

The possibility of external testing is a central point in all learning contexts since this provides an overview on the actual performance and therefore on the next learning steps to be done. This scenario supports the learner in the preparation for an exam but is on the other hand not only “teaching to the test” since the learner needs to think about the feedback and diagnoses they get in the feedback stage (cf. Reiss et al. 2005, p. 29).

Adaptivity

Adaptivity plays a subsidiary role in this scenario. The learner only has the possibility to choose a concept to be tested and gets automatic and individual feedback of the system at the end of the test. This feedback consists of test results and suggestions for the future work and is fitted to the individual learner (cf. Reiss et al. 2005, pp. 29).


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5.3 Formalized Pedagogical Scenarios on the Basis of CLOs

The following pedagogical scenarios need to be implemented for Math-bridge and are based on the learning scenarios from the project VEMA (cf. Fischer 2007). Hence they use complex learning objects but could also be used for atomic learning objects that have the same didactical intentions as the CLOs.

Since the technical implementation of the CLO has not been done yet, the process will show which CLOs will be adaptive and to which extend the adaptivity will be realized. Adaptivity here means in which way CLOs can be substituted as a whole or adaptively modified by exchanging specific atomic LOs that are part of the CLO. The adaptation may use information from the learner model. This presumes also a decision about the structure of metadata concerning CLOs. We assume that the CLOs’ structure of metadata will be comparable to the structure of metadata of the LOs.

Therefore the adaptivity of the CLOs and its structure of metadata have to be discussed in Task 7.1 “Implement remedial scenario and decision rules” and concretised in Deliverable 7.1 “Decisions for remedy implemented”.

5.3.1 Order of the CLOs The order of CLOs relates to the structure of the learning material from VEMA. As the diagram below shows, the complete order starts with a brief overview of the central concepts to be learned. This does not include the formal definitions or theorems, but a brief description of the main concepts. By this the learner is supported in keeping an orientation of the central aspects of the following content to be learned. The next page is the Introduction page which consists of the Introduction CLO. Afterwards, all definitions and theorems are briefly listed in a fixed order within the Info page. On the following Motivation/Interpretation/Explanation (MIE) page all definitions and theorems are explained by several MIE CLOs that may contain examples, explanations, counterexamples, proofs and interactivities.

In the next step, the learner gets an insight of Applications to the concepts by one or several Application CLOs. This CLO type refers to at least one definition or theorem, but may also refer to a combination of them. There do not exists Application CLOs for all definitions or theorems.

Afterwards Typical mistakes to the concepts are presented in form of Misconception CLOs. The Exercise page contains exercises as atomic learning objects or Practice CLOs in different formats: Exercises with an automatical scoring are possible as well as exercises with self-evaluation by the learner. In each case a sample solution will at least be presented to the learner after his solution in order to enable him to diagnose himself and to check his own solution approach in comparison.

On the final Supplements page learning material is presented that is not obligatory. This page consists of Supplement CLOs. The Supplements are designed for ambitious learners that want to learn more than just the pure requirements of their fields of study.


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Order of the CLOs

major topics

definition1/ theorem1

Overview

Introduction

Motivation/ Interpretation/ Explanation

Info


...

Introduction CLO

MIE CLO


…


Misconception CLOs

Application

Typical Mistakes

Application CLOs


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5.3.2 Optional scenarios for the selection of the CLOs Based on the order of the CLOs as presented above, we now introduce scenarios that can be individually changed by the learner. Like in most of the scenarios described above, the learner first of all has to choose a topic he wants to learn. Then the system shows him the complete order of pages that are available for the topic as shown in the diagram below.

The learner can now individually design his learning by choosing those pages that he wants to learn with or use one of the scenarios that are predefined: Select All, Select Basic (with intro), Select Basic (without intro), Select Rehearse. These prestructured scenarios activate different pages as described below and are designed in order to support the learner in structuring his learning. All of these scenarios can freely be exchanged by the learner by activating further pages or deactivating chosen ones. Hence the student is supported in his choice but he is free to change it individually before starting the book generation.

We will now describe these four preselections:

Select All

This scenario actives all pages. Hence the user gets a complete insight to the topic which can be useful for learning an unknown concept or for a deeper comprehension of the topic. Therefore the definitions and theorems can be experienced within the Introduction on a non-formal level in the first place in order to get an intuitive idea of them, which will be further formalized and mathematized later. By this the learner shall be enabled to develop an intuitive and problem-related understanding of a concept idea before getting a formal representation of it.

Supplement CLOs

Exercises

Supplements

Practice CLOs


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Select Basic (with intro)

In this scenario the learner starts with an Overview and the Introduction page, proceeds with the Info page and the Motivation/ Interpretation/ Explanation page. Finally he can practice his knowledge with the Exercises page.

Skipping the Application, Typical mistakes and Supplement page the learner only gets the basic content that is most important for the repletion of known concepts. The Introduction supports the comprehension of the topic.

Select Basic (without intro)

This scenario is exactly the same as the Select Basic (with intro) except for the Introduction, which is skipped, too. Skipping the introduction shortens the learning material. Instead the learner starts in a deductive approach with an overview of all definitions and theorems which are explained and justified afterwards.

Nevertheless the learner has full access to all central aspects of learning. Since this abbreviates the learning, this scenario is especially for those students who briefly want to repeat a certain topic, but do not need any introduction to the topic.


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Select Rehearse

This scenario is the shortest scenario, it only selects the Info page with the definitions and theorems and the Exercise page. The learner does not get any introductions, explanations, examples or proofs and therefore can briefly check the central concepts and practice them with the exercises.

5.4 Learning Advice Component

5.4.1 General Learning Advice Component First-year students are used to have lessons with a teacher who selects the content and decides on curricular aspects such as restrictions to learning time, a sensitive choice of learning tempo and the sequencing of content. Hence, for self-directed learning first year university students need abilities in self-regulation as well as in self-motivation in order to design their individual learning process (cf. Niegemann et al. 2008, pp. 65). In order to select the content to be learned students also need knowledge of the mathematical requirements of their fields of study on one hand and on the other self-diagnostic abilities in order to detect their content- and competency-specific deficits.


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Since Math-Bridge is designed for blended learning course scenarios as well as for self-directed learning scenarios it is not clear if the student’s learning is embedded in a course scenario where a coach or teacher can give support in mathematical, pedagogical or simply in technical questions or not.

Therefore we suggest implementing a learning advice component that supports the learner in different respects. This component should serve as a central starting point for all learning and aims at helping the learner in self-regulating his individual learning. Therefore we strongly suggest offering an introductory How-to in this component too. This How-to describes both, the technical usage of Math-Bridge as well as pedagogical suggestions for learning with the system. Hence it should contain recommendations for self-directed learning – such as how to schedule a learning process, how to keep motivation and attention etc. as well as concrete descriptions of the formalized pedagogical scenarios, their goals and their intended usage by the learner.

In order to give specific support for learning we suggest a formalized learning advice component, which is part of this general learning advice component and can be used for specific learning advices of certain concepts. This special component is described in the following section.

5.4.2 Formalized Learning Advice Component (Self-Assessment Component)

This component supports the learner in assessing his knowledge related to a certain topic and to select adequate options of Math-Bridge to enhance his knowledge of this topic and overcome specific deficiencies. For this purpose the system offers two tools: a tool for self-evaluation and a system-provided diagnostic test that cover all four competencies of Math-Bridge.

The implementation of this component with regard to a certain content domain requires a very detailed pedagogical and mathematical analysis of this domain. Moreover, highly elaborated testing instruments for this domain have to available.

Therefore the formalized learning advice component shall be implemented in Math-Bridge for selected content domains only, where such instruments are already available in the partners’ material.

Description

The scenario starts with the users’ choice of a certain topic. Then the system shows a brief description of the major concepts and provides all definitions and theorems that will be taught within the topic.

In the next step the learner is asked to autonomously estimate his own performance concerning the chosen topic differentiated according to subtopics and competencies by indicating a score between 0% (no knowledge) and 100% (perfect knowledge).

Afterwards he is offered a diagnostic test on the chosen topic that consists of at least one exercise for each competency. This self-test can contain automatically rated exercises as well as exercises with self-scoring or exercises that need the scoring of an external tutor (if available).

In the next step, the system gives feedback in four stages:

1. The results of the test with regard to content knowledge,

2. the results of the test with regard to the four competencies,


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3. a sample solution for each exercise to enable the user to compare his own solution with and

4. the results of his self-estimation in retrospection.

Hence the user does not only have an external assessment on his performance but can also autonomously compare this result to his own assessment.

Based upon the test results the system finally provides recommendations on learning with focus on content-specific and competency-specific suggestions.

In particular, the advice contains recommendations with regard to the pedagogical scenarios the user should choose for his further learning in the system.

From here, the learner can individually decide for the next learning scenario to be chosen. The system will not automatically adapt the content with regard to the results of this scenario.

Justification

This learning scenario can be used by the learner as preliminary stage to all pedagogical learning scenarios. It supports the learner in content selection and in structuring his learning process. Besides it trains his ability of self-assessment and self-regulation. The scenario can be used for self-directed learning as well as for blended learning.

Adaptivity

The system only gives feedback on the learner’s individual performance during the test and his self-estimation. The test results have no effect on the presentation of the next learning material, the user himself decides autonomously on the next steps to be done. Moreover the exercises that are chosen for the diagnostic test are for every learner the same and do not change with respect to the learner’s actual learner model. Only the test results will update the scores within the learner model.

The following diagram shows the structure of this component.


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Self-diagnostic Orientation:

major topics

definitions & theorems

Get Overview

Self-evaluation Task: Estimate your own performance to these topics with a score between 0% (no knowledge) and 100%

(perfect knowledge)!

Diagnostic Test Diagnostic Self-Test

Feedback Feedback on content

Feedback on C1, C2, C3, C4

Comparison with sample solution

Comparison results with self-evaluation

user selects adequate content and learning scenarios


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5.5 Embedding Math-Bridge in Learning Management Systems

As described in chapter one the partners already use online learning platforms to manage their courses efficiently. Therefore we require the possibility to integrate ActiveMath into existing Learning Management Systems (LMS) or to expand ActiveMath with some features which we identified as the following:

Communications tools (synchronous and asynchronous; unidirectional and bidirectional) are necessary to inform the learners about important news besides the presence parts of the course. They can also be used for communication between several learners and between tutors and learners to support the direct communication in eLearning or distance-learning scenarios.

In addition to communication tools we have to consider new elements of the web 2.0, such as social networking. For this we can rely on the experiences of the Finnish partners, as described in Silius et al. 2010.

The possibility to provide additional (learning) material exceeding the ActiveMath content is also necessary. Documents for homework should be supported as well as additional content or other information, which isn’t available in ActiveMath.

The Online platforms need an extensive user administration and right management to define several roles for instance for lecturers, tutors, and learners. They have different responsibilities and needs different rights regarding these responsibilities. Besides a course management is necessary to provide different books for different groups.

Regarding the privacy of the learners in combination with the communication and copyright problems the course in the LMS has to be secure with a course login or a course password.

As mentioned above it is not necessary that ActiveMath has to provide all these features. Another solution could be the integration of ActiveMath into existing LMS like moodle or Blackboard. Nevertheless some features have to be supported for this, for instance the user administration and the right and the course management. Also the learning advice component has to be concerned in the case of an integration of ActiveMath into an LMS. For instance the gradebook of moodle should be supported.

6 Summary and Future Work

Math-Bridge will be able to directly use some of the formalized pedagogical scenarios that were already implemented for LeActiveMath. As described above, some of these scenarios need to be changed for the purposes of our project first while other can be used immediately.

We introduced the Complex Learning Objects (CLO) as a new structure element and developed new formalized pedagogical scenarios for the CLOs in order to enable the project partners’ content to be implemented within the learning system reused for the individual remedial course scenarios afterwards. The development of a general learning advice component as well as a formalized learning advice component introduces a tutorial component to the system to support the learner in structuring his individual learning. Hence this component will support self-directed learning with the system but without a teacher. Embedding Math-Bridge into other learning management systems will make Math-Bridge on the other hand usable for all kinds of course scenarios with teachers.

Our deliverable concentrates on technical aspects related to the current state of the Active Math system, and is implicitly based on principles of instructional design. In future work we


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plan to make explicit the background of the remedial scenarios of MathBridge in the international literature on instructional design, for instance the well-known 4C/ID method (cf. Van Merriënboer & Kirshner 2007).

The next step in the Math-Bridge project is the implementation of the scenarios which will be done in task 7.1 “Implement remedial scenario and decision rules” and which will result in the deliverable 7.1 “Decisions for remedy implemented”.

The suggestions for implementing the learning advice components as well as the suggestions for embedding Math-Bridge into other learning platforms will be relevant for the deliverable 7.2 “Customized and personalized access”.


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7 References

Biehler, R. et al. (2009) Math-Bridge: Deliverable 1.1 - Target Competencies. Biehler, R. et al. (2010). Math-Bridge: Deliverable 1.2 – Content and Assessment Tools. Biehler, R. & Fischer, P. R. (2006). VEMA - Virtuelles Eingangstutorium Mathematik. In: Beiträge zum Mathematikunterricht 2006. Hildesheim und Berlin. Faulhaber, A. & Melis, E. (2008). An Efficient Student Model Based on Student Performance and Metadata. European Conference on Artificial Intelligence. Fischer, P. R. (2007). E-Learning als effizienteres Mittel für den Brückenschlag zwischen Schule und Universität? In: Beiträge zum Mathematikunterricht. Hildesheim und Berlin. Fischer, P. R. (2009). E-Learning zwischen Schule und Universität? Ergebnisse einer empirischen Studie zum Einsatz einer E-Variante mathematischer Brückenkurse: In: Beiträge zum Mathematikunterricht 2009. Oldenburg. Fischer, P. R. & Biehler, R. (2010). Ein individualisierter eVorkurs für 400 Studierende und mehr. Ein Lösungsansatz für mathematische Brückenkurse mit hohen Teilnehmerzahlen. In: Beiträge zum Mathematikunterricht 2010. München. [in press] Mantyka, S. & Ricketts, T. (2010). Diagnostic Rules for the Remedial Learner in Math-Bridge. URL: http://www.mun.ca/mlc/research/DFKI_Diagnostic_Rules_Math-Bridge.pdf Niegemann, H. M. et al. (2008). Kompendium multimediales Lernen. Heidelberg. Reiss, K. et al. (2005). D20 for LeActiveMath project: Formalized Pedagogical Strategies. Ulrich, C. (2008). Pedagogically Founded Courseware Generation for Web-Based Learning. An HTN-Planning-Based Approach Implemented in PAIGOS. Heidelberg. Ulrich, C. et al. (2006). D24 for LeActiveMath project: Tutorial Component. Silius, K. et al. (2010). Students' Motivations for Social Media Enhanced Studying and Learning. In: Knowledge Management & E-Learning: An International Journal (KM&EL), Vol.2, No.1. URL: http://www.kmel-journal.org/ojs/index.php/online-publication/article/view/55/39. Van Merriënboer, J.J.G. & Kirshner, P. (2007) Ten Steps to Complex Learning. Erlbaum.


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8 Annex

8.1 Remedial Scenarios of the Math-Bridge Partners

8.1.1 Remedial Scenarios at KS and UP In the following we will describe the course scenario of the bridging courses in Kassel and Paderborn. There are two different Blended-Learning course scenarios: one with an extensive presence part and one with an extensive eLearning part. The bridging course in Kassel and Paderborn is a proposal for beginning students to refresh their mathematical competencies learned in school. One goal of the course is also to enhance the competencies to a level “between” school and university. This is done with representation of the content that is more formal than in school.

In the following we describe both course scenarios. For each scenario we will discuss the structure of the course (the distribution of presence and eLearning parts, the communication possibilities, the different roles of presence and eLearning and the support of learning in a Blended-Learning-Course) (cf. Fischer 2009). The second part of each course scenario describes the roles of different actors in the learning process: the learner, the tutor and the teacher. We also discuss the role of the Learning Management System (cf. Fischer & Biehler 2010).

8.1.1.1 The Course Scenario with an Extensive eLearning Part

8.1.1.1.1 The Structure of the Course The course includes 4 weeks in September with overall 6 days; the learners are present at the university. The rest of the 4 weeks is time for learning online. The first weeks start with one or two orientation days, where the learners are confronted with the learning system the first time, get some tips of how to learn with the material and the first modules are discussed. These first two days introduce the learners to the course and its learning opportunities.

In the following at the end of every week a presence day is provided. Here the learners have the possibility to ask questions about the learned content in the first part of the morning and to select the content which is discussed in the second part of the morning. In the afternoon there is a tutorial with exercise for the content discussed in the morning. The tutorial is guided by a tutor, who supports the learners to solve the exercises and to repeat the needed content if necessary.

The rest of the course-stretch is free for learning online. Here the learners can individually learn using the approaches mentioned above, concentrating on the content they need. Open questions that appear in this process can be asked and discussed in the next presence day. Besides the orientation days the learning system supports the learner in their learning. There are the diagnostic tests with the individual feedback for the further learning. There is a list of recommended modules for every study programme, so the learners know which module they should select. There is also a text that explains the use of the material, the diagnostic tests and the role of the presence days. At least the learners can use the communication possibilities of the learning system. There are synchronous (chat) and asynchronous possibilities (forum and mail). The students can either communicate among each other or ask question to an online tutor that is present in the online learning system. He helps the learners with questions to the content, with some technical problems and corrects exercises in the open-format. He is contact person in all questions, supports and motivates the students in their individual learning.


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8.1.1.1.2 The Roles of the Actors in the Learning Process The Role of the Learner

In this scenario the learner has an active role. He plans and proves his learning self-determinate and independent and is responsible for his learn-success. He is responsible for searching for help in the case of problems. He is the main designer of his individual learning process.

The Role of the Teacher and Tutor

The teacher and the tutor have a passive role. They react in the case of asking for help, advice and attend the learners in their learning process. They support the learning process of the learners.

The Role of the Learning System

The online learning system is the synchronous and asynchronous communication-platform in this course during the eLearning-phase. Also the individual feedbacks in the diagnostic tests are given by the learning system. Also the feedbacks of the open-format-questions are given automatically after corrected by the online-tutor. The third aspect of the role of the learning system is the presentation of the learning material. The learning system is the central place of getting the learning material and using it.

8.1.1.2 The Course Scenario with an Extensive Presence Part

8.1.1.2.1 The Structure of the Course In contrast to the course scenario with an extensive eLearning part this course scenario is mainly structured and led by the teacher. Each week there are three days with compulsory attendance which contains three hours of lectures and two hours of tutorials.

The remaining two days are free for individual learning with homeworks. These homeworks consist of two parts: one exercise part with exercises to the topics that were taught in the lectures and another part with specific tasks to work individually with the modules in order to revise or to prepare scenarios for the next presence day. Therefore this course variant is extensively managed by the teacher, while the learner has a lower opportunity of individual learning. The diagnostic tests can generally be used for the course scenarios, but which specific tests are available is chosen by the teacher.

8.1.1.2.2 The Roles of the Actors in the Learning Process The Role of the Learner

Here, the learner is more a recipient of a course that is mainly structured by the teacher. Individual learning is only possible if the homework gives tolerance to the learner at this point.

The Role of the Teacher

The teacher is the manager of the whole learning process. Hence, he is the main designer of the learning process for all of his students. He chooses the content to be learned, decides on the topics to be taught in the lectures and structures the phases of self-learning using the homeworks. He is the expert of learning.


The learning system provides the material which is chosen by the teacher and is to be learnt by the student. It is possible to use the synchronous and asynchronous communication of the learning platform, but the main part of communication is held within the lectures and


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tutorials. There is no online-tutor that is usually available within the platform, the teacher or the tutors can check for questions in the platform from time to time. The platform plays a more subsidiary role in this scenario in comparison to the course variant with an extensive eLearning part.

8.1.2 Remedial Scenario at UC3M UC3M proposes an optional bridging course for first year students based mostly on a traditional presence scenario.

8.1.2.1 The Structure of the Course The course is given at the beginning of the semester and is composed of a total of 13 hours of lectures, with compulsory attendance, in 6 blocks of 2 hours a day and a 1 hour theoretic-practical examination on last day, distributed across the two first weeks of September. It has to be noticed that the second week of bridging courses coincides with the first week of regular classes. The course reviews mathematical concepts yet studied at the high school and introduces some that will be further developed during the first year in the related mathematical subjects. The course includes also 27 hours of homework in which the student has to face three kinds of problems: detailed solved problems showing the solving process, problems to be done by the student whose final solution, but not the solving steps, is given, and problems with no clue given, which have to be sent to the professor.

8.1.2.2 The Roles of the Actors in the Learning Process The Role of the Learner

Here, the learner is more a recipient of a course that is mainly structured by the teacher. Nevertheless individual learning is possible and encouraged with the homework, by means of the two types of unresolved problems.

The Role of the Teacher

The teacher is the manager of the whole learning process. Hence, he is the main designer of the learning process for all of his students. He chooses the content to be learned, decides on the topics to be taught in the lectures and structures the phases of self-learning using the homeworks. He is the expert of learning.


The learning system is just a repository in which lectures’ material and homeworks are stored. The system is also able to store students solved homeworks and provides asynchronous communication capabilities in the form of e-mail messages and forums, but no online support is given.

8.1.3 Remedial Scenarios at UV Two scenarios of applying the ActiveMath system will be designed at the University of Vienna. The intent of both scenarios is to test the system and its content. For this purpose, one or two ActiveMath courses will be generated. They will mainly be based on the material of mathe online/maths online but will be open to incorporate content provided by other partners too.

8.1.3.1 The Scenarios 1.) Complementary use of ActiveMath courses in the framework of lectures and exercises for physics and mathematics students in their first semester:


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The students are demanded to use this material whenever they feel to need some help on elementary mathematical issues that are not sufficiently covered in the lectures. By the end of the semester, the students and are demanded to give feedback regarding their experiences with the system and the content.

2.) Organized testing:

Some "testing hours" will be organized for physics and mathematics students in which they shall explore the material with respect to topics accounted by the lecturers, and immediately afterwards are demanded to give feedback. Depending on when a necessary amount of content will be available within the system, these activities can start in October 2010 or in October 2011.

8.1.4 Remedial Scenarios at TUT At Tampere University of Technology (TUT) there are three scenarios for the Math-Bridge course scenario. The first one is a combination of pretest and remedial course. It is designed for students starting their studies at the university with the purpose of measuring and improving their skills in school mathematics. The students failing in the test, will be asked to improve school mathematics by taking a MathBridge remedial course designed for this purpose.

The second scenario one emphasises individual studying and learning using web-based MathBridge learning environment to be used while studying Engineering Mathematics.

Third, a Math-Bridge course is designed for high school students for remedial high school mathematics and orientation for pre-studying of university mathematics.

8.1.4.1 Pretest and Remedial Course Scenario The target student group consists of the freshmen starting their studies at TUT. The goal is to support the very early steps of university level mathematics studies and to narrow the gap between school and university level mathematics.

Every new student takes the Mathematics Basic Skills’ Test in the beginning of his/her studies at TUT. The Basic Skills’ Test is a computer-assisted test and it consists of 16 high school level mathematics problems. The problems are based on teachers’ experiences and typical problems students are having in mathematics studies at TUT. The test will be carried out in MathBridge system, which is linked to Stack-environment, which contains a computer algebra system to generate the test questions and to check students' answers. The results saved in a database. The individual test results will be given to the students so that they will get feedback on their performance. The results will also be forwarded to teachers and to the responsible persons of the study programs.

Students failing the test are directed to take specially designed MathBridge-bridging course which contains lecture notes with multimedia enriched content and examples and exercises on the 16 mathematical topics. To pass the remedial course the student has to solve correctly about 80 computer generated and checked exercises from school mathematics. The student may study the course on his/her own, but the course can also be taken in a computer class where a tutor is available. The tutor can also be contacted with email or by other means of social media. Student's collaboration is supported with social media as well.

8.1.4.2 ELearning enhanced Engineering Mathematics Course Scenario Engineering Mathematics courses are mandatory for all TUT students. These courses are


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designed to be accomplished during the first year. Traditionally the courses contain lectures and weekly exercises. A student needs to work out a certain number of weekly exercises as well as an examination in the end of the course to pass the course. Rather complete command of school mathematics is needed to pass the course. This scenario applies ideas on blended learning. The students will participate lectures and exercise classes of Engineering Mathematics courses, and they can support their studies with a MathBridge course bridging engineering and school mathematics. This material can contain remedial exercises as well as some advanced extra material. The student will receive feedback of his/her learning results via e.g. automatic exercise assessment. It is also possible to study, practice and learn over the Web without guidance of a teacher in a class room. This material can contain remedial exercises as well as some advanced extra material. The student will receive feedback of his/her learning results via e.g. automatic exercise assessment. Student-student and student-teacher collaboration will be supported by social media.

8.1.4.3 High School Scenario The goal is to provide an access for Finnish high school students to Math-Bridge system so that students planning of applying to engineering studies will have an environment that provides sufficient learning materials for independent study of school mathematics and, furthermore, challenges a student with problems that are studied, e.g., at TUT. The purpose is to encourage students to study mathematics and to provide theory, examples, exercises and applications which are of interest in the engineering sciences. The courseware may be used for training mathematics for an admission test, or for a high school teacher as a possibility to deepen the high school course material and contents.

8.1.5 Remedial Scenario at UM2 The courses in UM2 will supplement normal courses, with one introductory bridging-course seen as an initialization of the student model as well as training for the online tool, both for the students and for our fellow teachers (some are quite reluctant to use it). Then, the use of the tool will be mainly self directed learning by students, with only technical help by appointed fellow 3rd or 4th year students acting as tutors at proposed specific hours. Computers will be available at all time.

The interaction will be with teachers for the introductory course, then with tutors at optional specific slots. The teachers will be invited to assign some online tasks during face-to-face learning and to make reference to it during usual teaching, like providing a correction for the online exercise and motivating the use of the online tool outside the classroom. The online survey tool will be used in order to track individually the work of each student. The teachers will be invited to explicitly make reference to it during face-to-face learning and point out the possible lack of work or on the contrary praise the good students. The compulsory use of the online tool during self directed learning phases will be checked and marked.

8.1.6 Remedial scenarios of OUNL

8.1.6.1 Learning Scenarios at the OUNL Material The education scenario of the Open University is pure individual Self-Directed-Learning. In regular university courses, they try to avoid any limitations in speed, study moments or study places, so students can study anywhere, anytime, anyplace. Most important part of most courses is the self study books/courses. These courses consist of 'study-units' and all study units have the same structure:

- introduction (presenting a question, case, etc. to wake the interest of the student, and


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offering an 'environment' for the theory)

- description of learning goals (the things students must know, or able to solve or produce, after passing the study-unit)

- study kernel (explanation, illustrations, examples, definitions, theorems, proofs, exercises, practicals, etc., also including parts of available textbooks or literature, video, interactive software, etc.)

- summary

- self test.

This structure, presented very short here, is already in use for 25 years, and very effective, theoretically well founded and well accepted by the students. Only if really needed, there are obligatory meetings (in the evening, or the weekend), for example for life discussions, presentations, etc. Often there are for each course a few voluntary meetings, where students meet teachers to get support. Till 10 years ago nearly all study materials were on paper and sent by post to students. Today more and more is electronically available, and also the communication with students is through the internet (ELO=Blackboard), not only for sending assignments or documents, but also for online meetings, using webconferencing systems (Elluminate). Everything has the characteristics of Distance education, though 'distance' has disappeared due to the internet.

8.1.6.2 The Course Scenarios at OUNL In OUNL there are two steps of bridging courses: First they do have the 'Basiscursus wiskunde' (Basic Math Course). The contents of this course cover more or less the first three years of secondary school. Then they have two parallel courses: 'Voorbereidingscursus wiskunde A' and 'Voorbereidingscursus wiskunde B' (Preparation course Math A, Preparation course Math B). These two courses cover the last 3 years of secondary school teaching. The A-course is for students wishing to continue at the University in Economics, Medicine, etc. The B-course for students is continuing in engineering, exact and life sciences. All three courses take more or less 200 study hours each.

In between (after the Basic course and before Preparation courses) they offer an online test. This is a very important and useful test, because students can discover with this test if they are well prepared to start with a preparation course or that they have to start with the Basic course, or that they should do some parts of the Basic course.

The Basic Math Course and the Preparation course Math A are split in two parts: depending on their pre-knowledge, they can choose the parts they want to follow. OUNL offers diagnostic tests, so the students can determine the need to do so.

Basic Math Course:

At this moment students can choose to do the course completely in individual Self Directed Learning, or follow a 'class' course. It's the aim to develop within the MathBridge project a third and fourth version of this course: a real interactive adaptive 'Online Basic Math Course', with or without support. So in future students will have 4 options to do the Basic Math Course: online alone, online supported, with a book, or in a class.

Today, in case of self study, students just use a printed book, well written for self study (good explanation, lots of exercises with worked out answers ...). There is no support at all from a teacher (maybe from a friend, or a neighbour, but nothing organised). There are online exercises available (soon), for training, tests ...

In the class course, the students use the same book, but they get explanations and support by the teacher, etc. Groups are small (15-20 students), so there is a lot of personal feedback.


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Students have to pay for this class course.

In future OUNL will run the Online Basic Math Course in ActiveMath. Essentially they think in two modes: completely self directed, without any interaction or support by a human being, or with some online weekly support (by webconferencing) in a fixed planning.

In case of self study with the book only, there is a lot of adaptivity: students can select the chapters they want/have to study. Exercises are available to support decisionmaking whether to study some part or not.

In case of the class course, adaptivity is limited: students can choose to follow the whole course, or just one of the two parts.

In case of the online alone version, there is again a lot of adaptivity directed by ActiveMath.

In the online supported version, there has to be a schedule, to be followed by the students, to make the webconference-meetings fruitful. So in this case adaptivity is limited.

Preparation course Math A, Preparation course Math B:

These courses are today online offered in traditional 'class versions'. About 35% of the study time, students work in a group with the teacher, using traditional secondary school books. 65% of time is use for self study, making exercises, etc. Depending on the success of the online versions of the Basic Math Course OUNL will decide to offer also online versions for these courses.