Theoretical Framework

- -

Contents

Part I Rationale 2

Part II Conceptual Framework 3

1. Algebraic skills 5

2. ICT tool use 8

3. Assessment 14

4. Integrating theory 22

5. Choosing content that makes symbol sense 23

6. ICT Tools for assessment 25

7. Designing the prototype and instruction 30

Appendix A 33

Appendix B 38

Appendix C 40

DITwis 40

Algebra Tutor 42

Calmaeth 43

Math Xpert: 44

Aplusix 45

L’Algebrista 46

webMathematica 47

Wiris 48

AiM: Assessment in Mathematics 49

CABLE 50

Hot potatoes 51

Question Mark Perception 52

Wintoets 53

Moodle quiz module with extensions 54

Wallis 55

WebWork 56

TI interactive 57

Cognitive Tutor 58

Algebra Buster 59

Appendix D 60

Maple TA 60

Digital Mathematical Environment 61

Activemath 62

STACK 63

Wims 64

Appendix E 65

Appendix F 68

71

References 72

Part I Rationale

For several years now the skill level of students leaving secondary education in the Netherlands has been questioned. Lecturers in higher education often complain of an apparent lack of algebraic skills, for example. I was personally confronted with this challenge when I redesigned the entry exam of the Free University from 2001-2004. This problem has only grown larger in the last few years. In 2006 a national project was started to address and scrutinize this gap in mathematical skills, called NKBW. In the same period use of ICT in mathematics education has also increased. It is our conviction that ICT can be used to aid bridging this gap.

Therefore this research will focus on two relevant issues in mathematics education in secondary schools in the Netherlands: on the one hand signals from higher education that freshmen students have a lack of algebraic skills, on the other hand the use of ICT in mathematics education.

Relation to current curricular developments in math education

These developments have to be seen in a larger context. In 2007 the cTwo commission (commission on the future of mathematics education) published a vision document (2007) which has all the ingredients for this research.

First of all the importance of numbers, formulas, functions, change, space and chance are stressed (viewpoint 4). On an algebraic level this corresponds with the sources of meaning (Radford, 2004) for algebra. Activities are: modeling, manipulating formulas.

Also, the role of ICT in this process is described (viewpoint 10). ICT should be "use to learn" and not "learn to use". This strict dichotomy will be difficult to accomplish, as they go hand in hand. This will be elaborated on in the chapter on tool use.

In viewpoint 14 a specific case is made for the transition of students from secondary education towards higher education. Again, it is stressed that this transition needs more attention.

Viewpoint 15 stresses the importance of assessment of algebraic skills.

Finally, viewpoint 16 mentions the pen-and-paper aspect of mathematics.

Why with a computer tool?

But why should we use a computer in learning algebra? We contend that computers can aid understanding of algebra by providing a learning environment that enables you to practice algebra anytime, anyplace, anywhere, because:

• 
  Randomization of exercises means there are many more questions.
  
• 
  It is possible to use several representations
  
• 
  The applets can be used anyplace, anytime, anywhere.
  
• 
  Automated feedback can help in this process
  
• 
  Students tend to be more motivated
  

We will elaborate on this in our conceptual framework.

Part II Conceptual Framework

This research focuses on the question:

In what way can the use of ICT in secondary education support learning, testing and assessing relevant mathematical skills?

First it is useful to analyze our question word-for-word.
In what way. To us it is not a question whether ICT can be used to support learning, testing and assessing mathematical skills, but how this should take place.
Secondary education. In this research we focus on upper secondary education and in particular students preparing to go on to higher education.
Learning, testing and assessing. Not only grades and scores are important, but also the way in which mathematical concepts are learned and tested diagnostically. We specifically aim to find out more about all three aspects.
Relevant mathematical skills. When students leave secondary education they are expected to have learned certain skills. Here we focus on algebraic skills, with particular attention given to “real understanding of concepts”, symbol sense.

So following a pragmatic approach three key issues are part of this research question: skills, assessment and ICT tool use.

The structure of part II is as follows:
First we discuss the three key concepts algebraic skills, assessment and tool use in chapters 1, 2 and 3. Every section starts with a problem statement, then gives an overview of relevant literature and ends with some words on the implications for my research.
In chapter 4 we integrate these concepts into one framework for my research.
Based on this conceptual framework two major decisions have to be made:

• 
  Which ICT tool to use for assessment. For this we will formulate criteria based on the conceptual framework and give an overview of available ICT tools for assessment.
  
• 
  What content to use for learning, testing and assessing algebraic skills. Per question we will motivate why the question is relevant for this research.
  

In chapters 5 and 6 these two decisions are explicated.
Together they will make up the design principles for our first prototype, which will be summarized in chapter 7. In part III we then discuss the methodology we use