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An example formative assment to identify specific starting points for teaching algebra
Brigitte GrugeonAllys
IUFM of Amiens
DIDIREM, University Paris 7
France
HYPERLINK "mailto:grugeon@clubinternet.fr" grugeon@clubinternet.fr
In France as in other countries, one of the most important current problems and challenges pertaining to the teaching and learning of mathematics at the lower secondary is to solve the enormous range in student mathematics achievement. So, many teachers test difficulties of managing the heterogeneity of knowledge of the students and so, difficulties of differentiating teaching in a given field, in particular in the field of the elementary algebra. What tools can help teachers to diagnose students competence and knowledge? How achieve a better regulation of learning by organising tailored activities in classrooms? This paper presents some of didactic research findings [1, 4 and 5] and uses them to raise some questions for discussion.
Algebra is a crucial field as regards the relationships students develop with mathematics. For a lot of these, and for most adults in society, algebra is the domain where, abruptly, mathematics became a non understandable world. Faced with such evident teaching and learning difficulties, didactic research has been very active during the last twenty years. It firstly tried to better understand learning processes in algebra and explain the breach mentioned above. These attempts were successful in identifying some decisive factors, such as those linked to the discontinuities existing between arithmetic and algebraic thinking modes and the specificity of algebraic semiotic practices. Grugeon [4] established a multidimensional model of students expected algebraic competence in secondary schools (students aged 13 to 15 years). The diagnosis of a students competence intends to situate the student along four dimensions: (1) relationship between arithmetic and algebra, approached through the meaning of letters (unknown, variable, generalized number, abbreviation or label) and the status of equality sign, (2) algebraic calculus, (3) translation between various representations (graphical, geometrical, algebraic, natural language) and (4) type of justifications (proof by example, proof by algebra, proof by explanation, proof by incorrect rule).
From this model, an adaptative diagnosis tool to help teachers to diagnose students competence and knowledge in algebra was built. The aim of this diagnosis is to personalize the learning exercices offered to students. A first software Ppite was experimented in real settings (see HYPERLINK "http://pepite.univlemans.fr" http://pepite.univlemans.fr) [2]. The key point of our assessment approach is that students answers to mathematical problems are not simply interpreted as errors or as lack of skills but as indicators of incomplete, naive and often inaccurate conceptions that the students themselves have built. Because, they are strongly rooted in personal experience and could not be tested using available tools, faulty conceptions endured. A fine analysis of the students work is required to understand the coherence of the personal conceptions, to achieve a better regulation of learning by organising tailored activities in classrooms in order to develop or to strengthen right conceptions, and to destabilise wrong or unsuitable ones that interfere with and sometimes prevent learning.
The diagnosis tool offers patterns of exercices according to three types described starting from the Mathematics TimSS USA 2007. The tool and object dimensions are evaluated: While functional relationships and their uses for modeling and problem solving are of prime interest, it is also important to assess how well the supporting knowledge and skills have been learned:
1. Modelling situations, generalizing pattern relationships, solving problems
 To model situations using algebraic expressions,
 To generalize pattern relationships in a sequence using algebraic expressions and prove them,
 To solve problems using equations/formulas and functions.
2. Comparing algebraic expressions, solving equations
 To identify sums, products, and powers of expressions containing variables.
 To evaluate expressions for given numeric values of the variable(s).
 To simplify or to compare algebraic expressions to determine their equivalence.
 To solve simple linear equations and inequalities, and simultaneous (two variables) equations.
3. Recognizing and generating representations
 To recognize and to write linear equations, inequalities, simultaneous equations, or functions that model given situations.
 To recognize and to generate equivalent representations of functions as ordered pairs, tables, graphs, or words.
The software Ppite automatically builds a rich student cognitive profile from data collected after the student solved a set of tasks especially designed for that purpose. These exercices involved preformatted answers and openended answers. The diagnosis is a three stage process.
First, each students answer is coded according to a set of criteria on six assessment dimensions: (i) Validity (correct, partially correct or non optimal, incorrect, not attempted, not coded), (ii) Meaning of Letters (unknown, variable, generalized number, abbreviation or label), (iii) Algebraic Writing (e.g. correct usage of parenthesis, incorrect usage of parenthesis, incorrect identification of + or x), (iv) Translation (ability to switch between various representations: graphical, geometrical, algebraic, natural language), (v) Type of Justifications (proof by example, proof by algebra, proof by explanation, proof by incorrect rule), (vi) Numerical Writing. This local diagnosis, provides, for each students answer, a set of codes referring to the different criteria involved in the question.
Second, Ppite builds a detailed report of the students answers by collecting the same criteria across the different exercises to have a higherlevel view on the students activity. At this stage, the diagnosis is expressed by success rates on three components of the algebraic competence (usage of algebra, translation from one representation to another, algebraic calculus) and by the students strong points and weak points on these three dimensions. This level is called personal features of the students cognitive profile.
Third, Ppite evaluates a level of competence in each dimension with the objective to situate the student in a group of students with equivalent cognitive profiles that will benefit from the same learning activities. This level is called the stereotype part of students profiles [3]. Stereotypes were introduced to support the personalization in the context of whole class management, to facilitate the creation of working groups and to organise tailored activities adapted to the working groups. A summary version of different levels of competence in each dimension is included in Appendix 1.
This differentiation strategy was used with interest in training by teachers. But, teachers indicate that the excessive length of the test can be an obstacle with a broader use in classrooms for differentiating teaching. To optimize the efficiency of diagnosis, it is necessary to define a criterion to reduce the number of diagnosis exercices.
What diagnosis exercices have to be privileged in order to anticipate students competence in scholar algebra?
According to the stereotype, how achieve a better regulation of learning by organising tailored activities in classrooms in order to destabilise wrong or unsuitable conceptions that interfere with and sometimes prevent learning, and to develop or to strengthen right ones?
Trying to predict the evolution of students knowledge is a central problem of educational policy. Indeed, the specific activities of diagnosis and remediation must be early to prevent that the difficulties of learning mathematic concepts do not contribute to place the student in situation of failure.
References
Artigue M., Assude T., Grugeon B., Lenfant, A. Teaching and Learning Algebra: approaching complexity through complementary perspectives, ICMI Study Conference, Melbourne, 2132(2001)
Delozanne ., Prvit D., Grugeon B., Jacoboni P., Supporting teachers when diagnosing their students in algebra, AIED supplementary proceedings, 461470 (2003).
Delozanne ., Vincent C., Grugeon B., Glis J.M., Rogalski J., Coulange L., From errors to stereotypes: Different levels of cognitive models in school algebra, Elearn, 262269(2005).
Grugeon B., Design and development of a multidimensional grid of analysis in algebra, RDM, 17(2)167210(1997) (in french)
Kiearn C., The learing ans teaching of school algebra, Handbook of research on Mathematics Teaching an Learning, Douglas Grouws E. Macmillan pubishing company, 1992
Sfard A, Linchevski L(1994) : The gains and the pitfalls of reification  The case of algebra, Educational Studies in Mathematics, Vol. 26, pp. 191228
Summary of the Learning Assessment Framework for Algebra Thinking
ComponentCodeAimLevel of competenceUsing algebra as toolUAStudying the capacity to use algebraic symbols to represent mathematical situations (production of formulas or setting in equation) and to prove.Level 1: Availability of the algebraic tool and adapted mobilization.Level 2: Mobilization of the algebraic tool and not adapted algebraic translationLevel 3: Mobilization of the algebraic tool without coherence between the model and the situationLevel 4: Not availability of the algebraic tool to generalize, prove or model and persistent arithmetic steps.Translation of a representation to anotherTStudying the capacity to translate an expression of a register with another and flexibility to interpret a representation of a register to another.Level 1: Adapted translation.Level 2: Always not adapted translation.Level 3: At least a translation without coherence enters the model and the situation.Producing equivalent expressions and solving equationsCADeveloping fluency in producing equivalent expressions and solving linear equationsLevel 1: Fascinating algebraic treatment of account aspects syntactic and semantic of the expressions being based on adaptability in interpretation of the expressions according to uses' concerned (structural design).Level 2: Primarily syntactic treatment with recurring errors of transformation privileging a procedural design of the expressions.Level 3: Treatment being based on a pseudostructural design, bringing into play rules of incorrect formation and transformation of the concatenation type.Producing equivalent numerical expressions CNDeveloping fluency in producing equivalent numerical expressionsLevel 1: Fascinating digital processing of account aspects syntactic and semantic of the expressions being based on adaptability in interpretation of the expressions according to uses' concerned (structural design).Level 2: Primarily syntactic treatment with recurring errors of transformation related to a procedural design of the expressions.Level 3: Treatment being based on a pseudostructural design, bringing into play rules of incorrect formation and transformation of the concatenation type.
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