Theoretical Framework

Contents

Part I Rationale 2
Part II Conceptual Framework 3
1. Algebraic skills 5
2. ICT tool use 8
Technology and Mathematical activity 8
Technology and practice 9
Technology and curriculum 10
Instrumental approach 11
Anthropological approach 11
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 28
Part III Methodology 29
Appendix A 31
Appendix B 36
Appendix C 38
DITwis 38
Algebra Tutor 39
Math Xpert: 40
Aplusix 41
L'Algebrista 42
webMathematica 43
AiM: Assessment in Mathematics 44
CABLE 45
Hot potatoes 46
Question Mark Perception 47
Wintoets 48
Moodle quiz module with extensions 49
Wallis 50
WebWork 51
Appendix D 52
Maple TA 52
TI interactive 53
Digital Mathematical Environment 54
Activemath 55
STACK 56
Wims 57
Appendix E 58
References 59


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





1. Algebraic skills

In this chapter we focus on algebraic skills and symbol sense. For this it
is important to sketch a general outline of the subject at hand. In recent
years

A. Problem statement
Algebraic skills of students are decreasing. We want to make sure that
students really understand algebraic concepts, so just testing basic skills
is insufficient. What defines real algebraic understanding?

B. Theoretical overview
In a historical context al-Khwarizmi, Vieta and Euler considered algebra to
be a "tool for manipulating symbols and for solving problems." In the 80s
Fey and Good (1985) argued that the "function concept is at the heart of
the curriculum". More recently Laughbaum (2007) sees ground for this
statement in neuroscience.

To get a clear picture of algebraic skills and the purpose of algebra we
have to look into the theoretical foundations.
Meaning of algebra
Radford (2004) sees several sources of meaning in algebra:

1. Meaning from within mathematics, which can be divided into:
a) Meaning from the algebraic structure itself, involving the letter-
symbol form.
This is also referred to as "structure of expressions" or "structure
sense" (Hoch & Dreyfus, 2005). I would like to use the term " symbol
sense" here, in line with Arcavi (1994) and Drijvers (2003).
b) Meaning from other mathematical representations, including multiple
representations. This corresponds with the "multirepresentational"
views of Janvier (1987), Kaput (1989) and van Streun (2000)
2. Meaning from the problem context.
3. Meaning derived from that which is exterior to the mathematics/problem
context (gestures, bodily movements, words, metaphors, artifacts use)
Ideally, all these sources would be addressed in an instructional sequence.

To focus more on the actual concepts that are learned Kieran's (1996) GTG
model combines several theories into one framework. In this model three
activities are distinguished: Generational, Transformational and
Global/Meta-level activities.
In upper secondary and college level these activities apply:
- Generational activity with a Primary focus on the letter-symbolic
form: form and structure (Hoch & Dreyfus, 2005) and parameters.
- Generational activity with multiple representations: functions and
their meaning, symbolic and graphical representations hand in hand.
- Transformational activity related to notions of equivalence.
- Transformational activity related to equations and inequalities.
- Transformational activity related to factoring expressions.
- Transformational activity involving the integration of graphical and
symbolic work.
Global/Meta-level activity involving problem solving
- Global/Meta-level activity involving modelling.
Algebraic activities in school
It is essential to have a clear view on what activities in secondary
education have to with algebra. A non-limitative list of activities
include:
- implicit or explicit generalization
- investigation of patterns and numerical relations
- problem solving though applying general or specific rules
- reasoning with unknown or undetermined quantities
- arithmetic operations with literal variables
- symbolizing numerical operations and relations
- tables and graphs represent formulas and are used to investigate them
- formulas and expressions are compared and transformed
- formulas and expressions are used to describe situations in which
measures and quantities play a role
- solution processes contain steps based on rules, but without meaning
in the context
Grouping these activities one can distinguish two dimensions of algebraic
skills: basic skills, including algebraic calculations (procedural) and
symbol sense (conceptual). The latter is "actual understanding" of
algebraic concepts.





One can not do without the other. Both should be trained, making use of
several inf