Integrated Assessment on Mathematics 12-16

INTEGRATED ASSESSMENT ON MATHEMATICS 12-16

ABSTRACT. In this paper we give an account of an integrated package of assessment, describing its conceptual posing as well as its formats , functions, use and effects. The package introduces a diversity of materials searching for connecting and integrating different contributions in a multidimensional scheme.

INTRODUCTION

One of the purposes in actual curricular reforms, is to include a design of assessment which contributes to regulate the docent practice and should enhance mathematics learning. In this context, we have tried to implement the experience on assessment carried out during the past decade with the new trends on the subject (Niss 1993 and NCTM 1993), and with the contrasting experience carried out already on the basis on a recent development of a complete curricular project in the field of Mathematics for the age group 12-16 designed for the Department of Education of the Catalan Government by Alsina, Fortuny, Giménez (1992). The BDM 12-16 (translated as Good Morning Maths 12-16 ) Project is developed on the basis of activities designed for a progressive and interactive discovery of mathematical contents. These contents can be assimilated by means of complementary activities which offer the students a chance to practice and diversify the situations previously introduced.

The main aim of our research is to design and validate an "integrated package of assessment", exemplifying the different necessary components for a reflection in a regular teaching practice. The validation of such design uses a systematical observation of teacher / student interrelations (Ball 1991). Our proposal try to give to the teacher basic indications for building an accurate student' profile and help the diversity treatement, by reflecting different tasks' results .

INTEGRATED PACKAGE

The integrated package of mathematical assessment , named IPMA, has been developed throughout the whole cycle 12-16 into distinct formats. The scheme (see figure 1) shows how the integrated package of assessment was designed with four key elements (PJ) Work Projects, (AS) Progress activities, (AU) Self-regulators elements of attitudes and values, and (OB) Observation. It also shows which were the criteria for making the student's work (with balance of adjustement) together with the corresponding justifications and decisions.

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With such elements, teacher and students control the general competences of mathematical ability (Bell 1993), the specific range of abilities is revisted (Resnick 1987, De Lange in press) in different type of tasks, and interaction and attitudes is also analyzed (Leder 1993). Diagnosis activities are not presented here as separated, because of extension (see Fortuny,Giménez and Alsina 1994). The control of diagnostic aspect is included as a starting point from the result of progress activities.

PROGRESS ACTIVITIES AND RANGE OF ABILITIES

By progress activities we mean our approach to the process of controling over the students' learning process . We relate a 5x3 matrix of different type of activities (concepts, algorithms, problem solving, visual-language and checking reasoning) with their range of elaboration (low, intermediate, high). We suggest to the teachers the use of different strategies of presentation of items in a test-form or more open activity (NCTM 1992). Also different languages could be used ; verbal, pictures, combination of them, etc. Progress activities give the oportunity to present different type and range as we show in the following example (see AS matrix in figure 1 ).

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Range indicates the variability of complexity of mental operations implied in a mathematical activity, and describe the degree of inter-connections among concepts and structures. A possible representation of range relates directed and radial conceptual systems and procedural networks (Resnick 1987, Kaput 1987, Schoenfeld 1992, De Lange in press). The range should be considered as an expression of the amount and progressive quality of such networks. Range of abilities requires different type of abilities as: mathematising, interpreting, processing, algorithmisation, recursivity...

The existance of variability in the range does not mean taking into account different levels of difficulty, but considering the distinct complexity of the mental processes involved activity. According to this principle, we used such ranges in the implementation process in the classroom and in service teaching courses. The activities of lower range presupose, among others the application of technical routines, standard algorithms, solving problems of a given type; whereas the activities of an intermediate range are connected to problem solving and to working with processes such as relating or integrating. Finally, the activities of a high range requires the analysis of complex situations under different prospectives and moreover, they imply decision tanking, structuring, creativity and critical analysis (Alsina,Fortuny, Giménez 1992, De Lange in press).

OBSERVATION AND SELF - REGULATION

It seems important for one of the components of the general assessment process to be determined by the multiple and systematic observations the teacher keeps under each of the students (motivation, interaction, ... ). The teacher should follow different steps when observing the students: (a) analyze everyday work with open activities, problem solving, self observation, classroom journal, or other activities (b) find characteristics of students according to appropriate categories, (c) write corresponding grids accurately prepared.

PROJECT WORK

Project activities is an open activity which provides the opportunity to present general framework of student' ability to design, uses of mathematization, etc. and contribute to show planification ability and work organisation. A project is a work to be done on a specific topic at non-school hours. This project must follow certain steps proposed by the teacher, though its structure must not be totally constrained; students should be allowed to make some kind of choice. Some advice at the starting point, formal constraints and the composition of the final work are some of the difficulties the teacher will encounter and which will try to overcome. He/she will either give relevant information or provide hints to facilitate the students' autonomy. Moreover, the teacher will either have to supervise, and probably revise, the preliminary plan or to tolerate an absolute freedom of action. The proposal of work projects facilitates the control over the procedural elements, contributes to the acquisition of those procedures and makes the work with transverse elements of the curriculum (consumption, environment, leisure, sport, etc.) easier.

RESEARCH QUESTIONS

The main questions investigate in this study are the folowing:

1. Could we find a set of simple instruments (package of assessment) of assessment which integrate enough elements to enhance mathematics learning as an integral part of teaching -learning process, to engage the students, and to ptovide opportunities to reflect and improve their work ?
2. Do the teachers use such set of instruments to extent their knowledge and integrate in their regular classroom ?

METHOD

In view of the questions, the research was carried out in three phases: Design of the package, Control of classroom processes ( by different elements of orientation and analysis of teacher's observations ), and recognition of results.

The design of the package is presented in the introduction and started in 1991-92. Four secondary school teachers in two different schools in Barcelona area participated in the development and experimentation for two years (1992-1994).

INSTRUMENTS

We consider as elements of orientation, all the instruments that help the initial control about planification (phase 1) , implementation (phase 2) and regulation (phase 3). With the planification elements, one must constantly take control over one's aims by means of explicitation of the pedagogical intentions (see figure 2 for "week planification gride" as an example) .

For the second and third phases of implementation and regulation, different kinds of instruments will be used such as: observational grids, detailed analysis of the research tasks, valuation of class journals, cooperative assessment, valuation of projects , etc.

These are tools which help the teacher to draw what it is known as "the students' profile" and to observe the development of the students' learning process. We also register the students' attitude, their intentions and their mathematical contents. We do not evaluate only a final project but we assess the students' progress by setting them tasks of different grades of difficulty. The results will be also registered in several grids form and discussed with students.Let's look an example. In progress activities above mentioned, double reading of control (of types of skills and of their range ) allows a double collective valuation (figure 3) which can be illustrated in two tables giving to the teacher more information than usual.

As teacher observations, we collect the observations that a teacher did about the abilities in specific problem situations. We are using as a method for observation the reflective thinking of different parts of the package from accurate analysis and we will explain some case studies.

RESULTS AND DISCUSSION

Teachers revealed three levels of observation in the continuous assessment : (a) local aims as specific attainment of the level of communication and interpretation , (b) general aims as description of the answers with regards to the consecution of specific - general purposes, (c) intentional, attitudinal and confidence as improvement of classroom environment, reinforcing the need of research and individual participation.

Observing open and progress activities

Open activities gaves us the possibility to know by the observation the specific connotative and intentional purposes of the students. In different situations the teacher could observe that communication is not totally appropriate for a properly description of some shapes the students had drawn. Thus, we do not only show, in the corresponding grid, the students' skills, but also how the power of metacognitive reasoning fits in the framework. This way, teacher write down in the final observational grid sentences like "Peter has gone through the process of interiorization and adaptation in a satisfactory way".

The teachers involved in the experiences (discussing the assessment process) established that the aims of the activity we will observe were going to be the following: Interpretation of data (indirect measurements in a scheme) and discussion (meaning of coefficient and assignment of volume units), elaboration process (working with a scale and planning with conditions) , communication processes (incorporation of criticisms), consolidation and recursive understanding (writing a report stating the remaining ideas).

In the next sample activity (figure 4) the students should, first of all, be able to recognize a real situation (the value of the "coefficient of building land ") and showing a mathematical ability (geometrical design and construction).

Figure 2. Translation of a first part of an activity sheet in the project BDM 12-16.

Let's look at one teacher's observational exemple to illustrate one of this levels of observation from the above presented activity . Eric's distribution (figure 5) was generally rejected because of "non-rectangular buildings" , but Jordi admited that he had tried to do something different. The student remarked: "If I had tried to use an oblique rectangle I would also have made a mistake !" But other mistakes were generally observed by the students in the class-discussion: "All the designs except Melisa's did not fit into a realistic situation " (a lot of students drew their buildings too close to the border).

Generally, the students had all tried to be original but they decided Melisa's project was the best one because she had distributed the buildings as in a real situation". A lot of people forgot the scale indication. With this "way of doing " the Students revealed their consecution of local mathematical objectives, and the teacher can analyze it and situate in the student profile. Progress activities could be also considered as a good instrument. In the experimental situations, teachers confirmed that students with same global scores were separated by two different categories: level and type of activities. The experiences with 13 to 15 years old students gave the oportunity to show that the range discriminate the students by levels.

In the example Florencia and Boyana (13 years old students in a private school in Barcelona) have small differences on scores. Both can solve either a closed problem or direct inference but cannot solve an open situation. Florencia using an algorithmical way of doing in all the situations (see figure 6), tried to solve by equations the open situation, she made a lot of mistakes in language & visualization situations (represent the building figure as 2 dimensional) and cannot explain completely the 2nd range problem about similar triangles. Nevertheless, Boyana is more "regular" , and made mistakes in all third level questions solving the others. The teacher observed immediately how the differences of quality means that Boyana is situated regularly in a 2nd level , but Florencia still remain in 1st level in some activities. Florencia has more difficulties to establish theoretical frameworks and her language of explanations is less fluent than other students. The teacher involved in such experiece said: " I never thought that such differences could appeared".

About General Tasks

Use of classroom journals and note-books was considered very important observational tools to recognize what happens in classroom activities. Nevertheless, they require a disciplinate effort and dedication on the teacher part since each student must be controlled, at least, twice a year. The following categories should be included in the collection of information from the group-class: reflecting the whole work, formulation and expression, integration, invention, group work, attitudes of progress, norms, self-organization, reflection of action, use of technological resources, including accurate comments.

Following with the example of coefficient of builidng land, the teacher A wrote down the following comments:

About Melisa - "Adjusted interpretation of data in drawing activities".

About Eric - "Extremely Interested . Original purposes. No incorporation of data assignment ".

About Jordi - "Bad interpretation of scale drawings".

After analyzing such particular observational process , the teachers wrote down in their journals some general comments , which we have translated as follows :

A - The general planning was quite successful. Nobody gave the sheet without an answer. (Acceptation as an intentional issue) . Everybody suggested that the suitable volume was the "area of the pentagon multiplied by 0,4) and nobody made a mistake on the "scale reproduction" needed in requirement 2.

B - Some students recognize immediately the missing data, but 60% couldn't solve that part of the problem because they didn't use a scheme to establish relations on measurements.