Theoretical Framework

Rationale

Research question(s)

Algebraic skills

(chapter 1)

Tool use

(chapter 2)

Assessment

(chapter 3)

Content choice

(chapter 5)

Tool choice

(chapter 6)

Prototypical design

(chapter 7)

Methodology

Part I

Part II

Conceptual framework

(chapter 4)

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:

1. 
   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).
   
2. 
   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
  

G
Algebraic Skills
rouping 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.

Basic skills:

algebraic calculation, procedural routine

Symbol sense: algebraic reasoning, strategic skills (Arcavi, 1994)

One can not do without the other. Both should be trained, making use of several influential models on learning mathematics.

Or as Zorn (2002) puts it: "By symbol sense I mean a very general ability to extract mathematical meaning and structure from symbols, to encode meaning efficiently in symbols, and to manipulate symbols effectively to discover new mathematical meaning and structure."

Symbol Sense
The notion of “actual understanding” of mathematical concepts has been given different names. Hoch called this "structure sense"  at the beginning of 2003. Arcavi (1994) used the term "symbol sense" , analogue to the term "number sense". It is an intuitive feel for when to call on symbols in the process of solving a problem, and conversely, when to abandon a symbolic treatment for better tools.

Drijvers (2006) sees an important role for both basic skills and symbol sense. The declining algebraic skills of students is concerning. As Tall and Thomas (Tall & Thomas, 1991) put it: "There is a stage in the curriculum when the introduction of algebra may make things hard, but not teaching algebra will soon render it impossible to make hard things simple."

Several problems with symbol sense are:

• 
  process-object duality: a student thinks in terms of activity rather than objects.
  
• 
  visual properties of expressions
  
• 
  lack of flexibility
  
• 
  lack of meaning of algebraic expressions
  
• 
  lack of exercises
  

Building on this last observation Kop and Drijvers (Kop & Drijvers, in press) have suggested a categorization of “symbol sense” type questions. This source will –together with other sources- provide a starting point for designing a prototype.

Impact of technology
Technology has an impact on mathematics education. Research with calculators (Ellington, 2003) has shown that the pedagogical role of tool use should not be underestimated. The use of tools seems to strengthen a positive attitude towards education, showing that there is more to learning than just practicing and testing. van Streun (2000), Lagrange, Artigue, Laborde and Trouche (2001) all determined enriched solution repertoires and a better understanding of functions, especially through the use of multiple representations. However, use should not be haphazard, but for prolonged use.

The next step in using tools for algebra was in the use of Computer Algebra Systems (CAS). The first large-scale study on the use of CAS was by (1997)

It is also important to stress the changing roles of students and teachers. Guin and Trouche (1999) noticed that students have different "styles" of coping with problems: random, mechanical, rational, resourceful and theoretical.

The modes of graphing calculator used by Doerr and Zangor (2000) could also be applied to the use of applets: computational, transformational, visualizing, verification and data collection and analysis tool.

Finally, the advent of computing technology has also strengthened believe that multiple representations of mathematical objects could be fully integrated in mathematics curriculum. This could provide a valuable source of implicit feedback, making sure that the added value  of (formative) assessment could be greatly enhanced.

According to Lester (2007) three factors are important in technology-related studies concerning algebra: time, the nature of the task and the role played by the teacher in orchestrating the development of algebraic thought by means of appropriate classroom discussion. One extra factor has to do with the instrumental genesis of the tool used. Transfer of what has been learned has to take place. Therefore the relation between tool use and pen-and-paper has to be taken into account. More on this in the second chapter.

C. Algebraic skills in this research

We want to study whether algebraic skills, and in particular symbol sense, can be improved by using an ICT tool.