Theoretical Framework

4. Integrating theory

We have seen that three “ingredients” come together in learning, testing and assessing relevant mathematical skills.

I propose the following conceptual framework:

ICT tool use

Algebraic skills

Assessment

ICT tools that facilitate algebraic skills

ICT tools for assessment

Assessing algebraic skills

Learning

Learning, testing and assessing relevant mathematical skills with ICT can be scrutinized in three different ways:

• 
  As tool use . Through instrumental genesis learning is facilitated.
  
• 
  As ‘learning of algebra’. We look at symbol sense for ‘real learning’, and basic skills for algorithms.
  
• 
  As assessment. Whereby pre-emptive formative assessment not only shows the learning result but also aids learning. Feedback is an important factor in this type of assessment.
  

We aim to integrate the assessment of algebraic skills into one prototypical design, thus giving us insight into these three aspects and their dependencies. This ‘triangulation’ helps in creating a ‘body of knowledge’ on acquiring algebraic skills with the help of computer tools. Hopefully this will lead to common ground within the research fields of algebra, assessment and tool use.

We will now sum up the implications of this conceptual framework, and underlying theoretical frameworks for the three topics, for our prototype.

5. Choosing content that makes symbol sense

In this chapter we will discuss how we choose questions for acquiring, testing and assessing relevant algebraic skills. Together these questions will make up the content for our first prototype. First we describe what we mean by relevant algebraic skills, then we use several sources to choose relevant questions for our prototype and describe why they are relevant for our research.

To choose our content we first have to determine what our algebraic focus will be. As we are interested in ‘symbol sense’:

• 
  What skill level does a question have?
  
• 
  What type of ‘symbol sense’ does the question address?
  

Classification of skill level

In the NKBW project (2007) a classification for skill level of a question was used, involving the letters A,B and C. This classification was used earlier in the Webspijkeren project (Kaper, Heck, & Tempelaar, 2005), and drew on work by Pointon and Sangwin (2003).

Building on the general schema of Bloom's Taxonomy, Smith (Smith, Wood, M., & Stephenson, 1996) came to eight categories.

Group A	Group B	Group C
1. Recall factual knowledge	4. Information transfer	6. Justifying and interpreting
2. Comprehension	5. Application in new situations	7. Implications, conjectures and comparisons
3. Routine use of procedures		8. Evaluation

 Using this classification scheme several exams and questions were classified . Some of them were already analyzed in the NKBW project. From all these questions a selection was made, based on the skill level of the question. Level “C” is almost impossible to cater for with ICT tools. We decided to choose level B questions or level A questions with an adapted level B approach.

It is interesting to point out the similarities of this approach to, for example, the assessment pyramid by de Lange (1999) used for PISA, but also the TIMMS framework uses equivalent skill levels:

These three levels are:

1. 
   Reproduction, definitions, computations. (lower level)
   
2. 
   Connections and integration for problem solving. (middle level)
   
3. 
   Mathematization, mathematical thinking, generalization, and insight. (higher level)
   

We then analyzed, also with the help from work from NKBW project, documents that are closely related with the expected algebraic skills going from secondary to higher education:

• 
  Several entry exams from university and HBO;
  
• 
  An exit exam for secondary education;
  
• 
  The book ‘basiswiskunde’ by Rob Bosch and v/d Craats;
  
• 
  Articles on symbol sense
  

Keeping in mind the skill levels we listed several suitable ‘symbol sense’ type questions. It was now time to determine what to focus on.

Algebraic focus

Now the collection of questions had to be narrowed down. Final questions were selected by focusing on Arcavi’s ‘flexible manipulation skills’ and the ‘choice of symbols’

In appendix Athe chosen questions are categorized into Arcavi behaviors and a rationale is provided what every question is all about. Per question we answer:

1. 
   Why is this an interesting question?
   
2. 
   What skill or behavior is assessed here?
   
3. 
   What answers do we expect?
   
4. 
   What could be obstacles in answering this question?
   
5. 
   What feedback could be given for this question?
   
6. 
   What tools could be used to model this question?
   

As we use Arcavi’s categorization it is also a good idea to acknowledge his instructional implications

• 
  symbolic manipulations should be taught in rich contexts which provide opportunities to learn when and how to use these manipulations.
  
• 
  give a complex function (and graphing tool) and ask the function.
  
• 
  informal sense-making makes sense.
  
• 
  use algebraic symbolism early to empower symbols.
  
• 
  make use of post-mortem analysis of problem solutions
  
• 
  classroom dialogues and what if questions.
  

These implications will be used when we finalize the prototypical design and formulate a didactical scenario