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Contents
Part I Rationale
Part II Conceptual Framework
1. Algebraic skills
2. ICT tool use
Technology and Mathematical activity
Technology and practice
Technology and curriculum
Instrumental approach
Anthropological approach
3. Assessment
4. Integrating theory
5. Choosing content that makes symbol sense
6. ICT Tools for assessment
7. Designing the prototype and instruction
Part III Methodology
Appendix A
Appendix B
Appendix C
DITwis
Algebra Tutor
Math Xpert:
Aplusix
L’Algebrista
webMathematica
AiM: Assessment in Mathematics
CABLE
Hot potatoes
Question Mark Perception
Wintoets
Moodle quiz module with extensions
Wallis
WebWork
Appendix D
Maple TA
TI interactive
Digital Mathematical Environment
Activemath
STACK
Wims
Appendix E
References
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 partIIis 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)
- Meaning from the problem context.
- 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 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.
2. ICT tool use
In this chapter I focus on the use of ICT[1]tools in (mathematics) education.
A. Problem statement
It is important to study the way in which tools can be used to facilitate learning. How are tools used and what characteristics do they have to have.
B. Theoretical overview
First I will sketch a general overview, ending in a description of the instrumental approach of tool use. Here I will use the construct of figure 1, looking at how tools affect aspects of the teaching and learning of mathematics.
/ From the Second Handbook of Research on Mathematics Teaching and Learning(Lester jr., 2007)
Not only mathematical activity, students, teachers and curriculum are affected, also the relationship between these aspects.
Technology and Mathematical activity
Many people use tools all the time. The Vygotskian notions on tool use(Vygotsky, 1978)sees a tool as a mediator, a " new intermediary element between the object and the psychic operation, directed at it" . In mathematics, tools have to have certain characteristics to be beneficial. The Handbook on Research mentions three important issues:
Externalization of representations
Heid (1997)also mentions this. The important question remains: how is mathematical activity influenced or changed by tool use? Feedback is mentioned. Otherwise time-consuming "production work" as well. Unlike the physical tool a cognitive tool provides a "constraint-support system" (Kaput, 1992)for mathematical activity.
Mathematical fidelity
A representation must be faithful to the underlying mathematical properties; this is mathematical fidelity (Dick, 2007). In essence this means that a tool can represent maths incorrectly. This also has to do, in my opinion, with the difference between "use to learn" and "learn to use"(cTwo, 2007), as the latter means one has to know the shortcomings of a tool, but this knowledge could also lead to a better understanding of a concept, thus a tool is used to learn.
Another aspect is the underlying machine code for a certain tool. It often is the case that certain extreme values yield strange results. So an important question is: "is the mathematical fidelity of a math system good enough to support maths at secondary school?". It will almost surely be a trade-off between this and the amount of time needed to improve the system.
Cognitive fidelity
This is the "degree to which the computer's method of solution resembles a person's method of solution.". To make sure that transfer of knowledge or skill takes place
Technology and students
An important distinction in type of activity is between exploratory and expressive tools and activities (Bliss & J., 1989). They reside on a continuum. So when a procedure is described it´s exploratory but choosing one’s own procedure is expressive (albeit somewhat limited). Initial play with a technological tool is often beneficial: it stimulates expression but also builds a purposeful relationship with the tool , and thus instrumental genesis (Guin & Trouche, 1999)can take place. However, structured guidance is often necessary, as to avoid the "play paradox"(Hoyles & Noss, 1992). This means that " playing" with a tool sometimes enables students to accomplish an activity without learning the intended concepts. To solve the paradox "reflection" on the task at hand is advised.
When studying student use of a tool the construct of a 'work method' could work: Guin and Trouche(1999)see five work methods: random work method, mechanical work method, a resourceful work method, a rational work method and a theoretical work method.
The combination of the type of activity (exploratory, expressive) and work method should enable us to deduct what students are thinking.
Technology and practice
In it important that there is pedagogical fidelity in tool use. This means that students actually learn what the teacher has intended. So here we consider the match between technology and practice..
An interesting choice is whether sometimes "privileging" is appropriate: using tools when basics are known and rules are "internalized" (in a sense trivialized). Before that, tool use is prohibited. The concept of privileging can also apply to certain mental activities, like proofs etc. This coincides with the white box versus black box discussion (Buchberger, 1989), which states that “privileging” with tools –meaning that tools may only be used when a concept is understood- is necessary. This prevents students from just “executing an algorithm" without knowing what they are doing.
Using technology in practice also means that the teacher role could change. This is also an aspect that can be studied throught the construct of teacher role, e.g. Counselor and Technical Assistant (Zbiek & Hollebrands, 2007).
The use of technology changes this role. But: this could very well clash with teacher's expectations. Beaudin and Bowes (1997), and later Zbiek and Hollebrands introduced the PURIA model for CAS implementation:
personal Play,
personal Use,
Recommendation,
Implementation,
and Assessment.
This could also be applied to the implementation of DME[2] use in the Netherlands.
Technology and curriculum
There are several reasons why technology is adapted:
Representational fluency:technology makes it possible to move easily between several representations. This also belongs to good design principles for technological environments(Underwood et al., 2005). For example, the applet Algebra Arrows has a multirepresentational aspect. One could think of the distinction: context-table-graph-formula.