Deakin University, Geelong-Victoria
Assessing students learning of problem solving in the mathematics classroom is definitely challenging for teachers, as they are required to give more attention to assess not only mathematical content, but also the ways that used at various contexts and applications in solving mathematical problems. Therefore, for teachers, assessment should be considered as an integral part of teaching and learning. Assessment of problem solving, in addition, should encourage both learning and reflection on that learning.
According to Groundwater-smith et al, ‘assessing students learning involves teachers, learners in the task of investigating, describing and judging learning outcomes’ (2003:268). In problem solving tasks, teachers assess students’ competency and cognitive skills in analysing the problem, students’ performance in the classroom either as an individual, or as a member of group during co-cooperative work, and the procedures they use to solve the problem (Mousley 2007).
In order to achieve these goals, performance based-assessment is presumed as an alternative assessment of problem solving, that involving evaluation of the actual process of learning mathematics and the actual application of knowledge to solve the problem (Slater n.d). The purpose of this essay is to discuss the use of portfolios as the authentic performance-based assessment to assess student learning of problem solving, and to give detailed information about the benefits of portfolio assessment as an educative assessment tool, including my relevant experiences of using portfolio assessment in my context as a secondary mathematics teacher in Aceh Province, Indonesia.
OVERVIEW OF EDUCATIONAL ASSESSMENT
Assessment is employed for many purposes, but perhaps the most important is how it assists the educational process. The word assessment is derived for the Latin assidere, meaning “to sit beside or with” (Weggins 1993 cited in Lorna 2003). A. English and H. B. English (1958) define the assessment as ‘a method of evaluating personality in which an individual, living in a group under partly controlled physical and social conditions, meets and solves a variety of lifelike problems, including stress problems, and is observed and rate’ (cited in Payne 2003:5-6).
Payne defines educational assessment as ‘the interpretive integration of application tasks (procedures) to collect objectives-relevant information for educational decision making and communication about the impact of the teacher-learning process’ (2003:9). It means assessment should be designed to improve students’ learning and performance, as well as to provide useful feedback about learning to students, teachers, and students’ parents. Therefore, the term ‘educative assessment’ is embodying for the ‘improvement’ and ‘feedback’ (Groundwater-smith et al 2003). The former, assessment is seen as the way for teachers to improve students’ learning and to encourage students to take responsibility for their improvement. The latter, assessment is the media to acquire and measure the quality of teachers in teaching, as well as to provide information to students’ parents and school administration.
The Purpose and Principles of Educational Assessment
According to Wiggins, the main aim of assessment is to ‘educate and improve student’s performance, not merely to audit it’ (1998:7). In the context of students learning, the purpose of assessment is to provide feedback to improve students’ learning, while in the context of certification and quality, assessment is seen as the tool to grade students and provide feedback to the educational system and curriculum.
There are several principles suggested by researchers in designing the characteristics of good practice for assessing student learning. Perhaps the most important is that assessment should be integrated in the teaching and learning process, focused on how students learn and their motivation, promoted commitment to achieve learning goals, encouraged students become self-evaluated, self-reflective, and self managing. In addition, such assessment should be recognised as essential professional skills for teachers in designing appropriate assessment; observing, analysing, interpreting evidence of learning, and giving feedback to the students (Qualification and Curriculum Authority n.d).
Typically, for problem solving tasks, assessment should be seen as a tool to diagnose students’ strengths, weaknesses, and misconceptions. Therefore, teachers should be able to select assessment tasks which are appropriate for those goals. There are some characteristics of assessment for problem solving. First, enhance the ability of students to use a particular mathematical concept. Second allow students to start, but has increasingly difficult extension questions. Lastly, allow the teacher to see the limitations of each student. (Mousley 2007)
ASSESSING PROBLEM SOLVING
Learning Theories Perspectives on Assessment of Problem solving
How students learn mathematics is referring what the cognitive and constructivist theories have said which stress active learning with growing complexities of mental structures. Piaget’ theory of cognitive development applies to the general learning of mathematics. This theory suggested thinking processed through qualitative changes (in stages) due to increasing biological maturity of mental structure with age and environmental interactions. The contribution of the cognitive development theory is well known in education to improve classroom practice and aid student progression (Hill 2001).
In the context of assessment of learning, the Piagetian approach has provided teachers with realistic views of what can be expected from students at the appropriate developmental times (Edwards & Morrow 2004).
In addition, students’ performance on any assessments can be understood in term of the language and culture of that student, particularly in learning mathematics problem solving where students are often encouraged to do tasks in collaboration with their peers. Therefore, social-constructivist theory provides an alternative assessment approach that compatible with a ‘constructivist’ view of students, with its emphasis more on learning that instruction from teachers (Estrin 1993).
Overview of Problem Solving in Teaching Mathematics
According to the literature of mathematics, a problem is a task for which the person confronting it either wants or needs solution; has no readily available procedure for finding a solution; and must make an attempt to find a solution.’ (Charles & Lester 1982 cited in Clements & Ellerton 1991:7). The National Council of Teachers of Mathematics (NCTM) 1989 defined ‘problem solving as the process of applying previously acquire knowledge to new and unfamiliar situation’. (cited in Clements & Ellerton 1991). The example of problem solving portfolios is shown at the appendix A.
Problem solving skills should be the main concern of mathematics classrooms. In Indonesia, the mathematics curriculum is associated with a syllabus or GBPP (Garis-garis Besar Program pengajaran or the Guidelines of Instructional Program) which suggests the strategies for teaching mathematics, including ‘active learning, mentally, physically, and socially’. To achieve the goals, teachers are required to encourage student-centered learning, as well as problem solving skills by giving divergent problem with different possible solutions. (Depdikbud 1994 cited in Hadi 2002).
For mathematics teachers to be able to integrate problem solving, they should recognise good mathematical problems and the appropriate solutions. However, it is essential to note that problem solving is more than just obtaining the answer. It entails more than thinking skill and strategies to solve the problem. Particularly for students, the objective of problem solving is to acquire an appreciation for understanding the nature of problem solving, likewise, developing specific abilities.
Assessment of Problem Solving
Assessing problem solving is not simply to grade or to score how students’ work on the problems, but to see and observe how students approach the solution, and to the strategy and method they use. In addition, assessment is employed by teachers to encourage, recognize, and support’ students’ mathematical thinking (Suurtamm et al 2010). In other words, the types of assessment that allow students to express and record their progression to obtain the solutions are presumed to be the best assessment mode for problem solving. Such an assessment, indeed, is recognised as an authentic assessment where the teacher can directly examine student performance on worthy intellectual tasks (Wiggins 1990).
Authentic assessments provide students with real assessment tasks because they are relevant to their life experience in a direct way (Wiggins 1990). As a part of alternative assessment, authentic assessment has the common notion of a meaningful performance or product (Estrin 1993). According to Hamayan (1995), ‘alternative assessment refers to procedures and techniques that can be used within the context of instruction and can be easily incorporated into the daily activities of the school or classroom.’ (Cited in Custer n.d).
There are some characteristics of authentic assessments: Firstly, they require students to be active in acquiring knowledge. Secondly, they present the students with a different range of tasks, such as conducting research, writing, revising, discussing, and presenting paper either by individual or collaborating as a group. Thirdly, they provide adequate opportunities for students to plan the responses on typical tests. Lastly, they achieve validity and reliability by emphasizing and standardizing the appropriate criteria for scoring such (varied) product. The validity in addition, should depend in part upon whether the test simulates real world ‘tests’ of ability (Wiggins 1990:2).
Perhaps, the most important aspect in authentic assessment is it provides broad opportunities for students to demonstrate their knowledge, skills and abilities in the context of daily life work and routine by solving problems, doing mathematical computations, writing mathematical journal, conducting research and presenting their findings, and assembling portfolios of representative work (Meisels 1997).
According to Paulson and Meyer, ‘a portfolio is a purposeful collection of student work that exhibits the student’s effort, progress, and achievements in one or more areas. The collection must include student participation in selecting contents, criteria for selection, the criteria for judging merit, and evidence of student self-reflection’ (1991:60).
Portfolios have long been recognised as an authentic assessment which are used in many mathematical classes to show students progress. According to Groundwater-Smith et al, ‘portfolio assessment is founded upon teachers and students making judgments about learning based upon a rich array and variety of evidence’ (2003:280). Furthermore, they stated that in learning, portfolios may be a collection of evidence of students’ learning progress, in a specific or more general curriculum area. On the other hand, portfolios may record the achievement that celebrates students learning. Initially, portfolios are best characterised as working portfolios, formative, and diagnostic in nature which may contain reflective statements made by students regarding to their approaches on specific key learning area and goal. Lastly, portfolios are documents that record the students work and finished products (Jasmin 1995 cited in groundwater-smith et al 2003).
Originally, portfolios are defined as a historical record of students work. This type of assessment, however, is presumed as an authentic evidence of students’ work and therefore, they must be systematically written and well-organised rather than just a collection of papers in a folder.
Krulik and Rudnick (1998) contend that portfolios are best used for diagnostic and self-assessment purposes and therefore should not be graded. Nagle (1992), on the other hand, points out that such modes of assessment have been adopted by several states in the USA for state-wide assessment (cited in Mousley 2007:5). In the USA, students’ portfolios have become a major mode of assessment in the mid-1990s, since the writings and acceptance of cognitive psychologist (Marsh 1996).
Characteristics of portfolios
Portfolios play an important role as an intersection of learning and assessment therefore teachers should find the way so that the two processes can work together. Some characteristics of good portfolios are suggested by Paulson and Meyer. What follows are the summary of the effective portfolios for learning and assessment.
1. The portfolio offers the student an opportunity to learn about learning. Thus, the end product must include information showing that a student has engaged in self-reflection.
2. The Portfolio is something that is done by the students, not to the students. Such assessment, in addition, offers a concrete way for them to learn to appreciate their own work and value themselves as learners.
3. The files and document compiled in the portfolio is the kind of documents which students take on new meaning for certain purposes. For instance, a portfolio for problem solving activities should be separated from other portfolios such as a portfolio on mathematics projects or classroom exercises on certain topics in mathematics.
4. The portfolio must include student’ actual activities either explicitly or implicitly. For example, the purpose and goal of a portfolio, content, standard, and judgment. Similarly to the content of a portfolio must be relevant for the purpose.
5. The portfolio should contain information that illustrates students’ growth and improvement, likewise their weaknesses and strengths.
(Paulson and Meyer 1991:61-62)
Advantages of portfolios
There are several benefits of using a portfolio suggested by Marsh (1996) and Payne (2003) which relates to problem solving. First, it is more natural since it is collected in the classroom and shows the actual students’ work on solving problems, including the ways and strategy they use to obtain the solution, and the explanation. Moreover, it provides information on students’ growth on a certain period of time (e.g. every semester). Additionally, the portfolio also encourages student self-evaluated skills and allows realistic assessment of proficiency.
On the other hand, there are some drawbacks regarding the use of portfolios to assess student learning of problem solving. Particularly for the teacher, they are likely to encounter difficulties to attain acceptable levels of reader/scorer/assessor reliability of portfolios. Furthermore, it takes more time for teachers to assess portfolios since they have to make selective criteria of assessment using rubrics. (Payne 2003).
I personally think that such drawbacks can be minimised by the teacher, if only s/he is consistent in the use of portfolios as assessment tools and willing to spend some time to create appropriate rubric to assess student’ portfolios on problem solving.
Types of Portfolios
There are several items of portfolios suggested by Crowley (1993:545) which I think are appropriate for problem solving lessons such as samples of journal writings, mathematics problem, an elegant proof, student-formulated problems, student-made concrete representations, and group projects (cited in Bobies et al 1999).
Portfolio assessment is apparent as essential, not only to improve the complex task of student assessment, but also to contribute to a more positive attitude and enjoyment toward learning (Thomas et al 2004). From my point of view, the portfolio is the sort of assessment that best suits problem solving. It provides evidence of students’ ability to interpret problems, to analyse them in sensible ways, to select, apply and integrate mathematics knowledge and content in a variety context. In some cases, portfolios can be used to show-case student’s work (Mousley 2007). Often we see additional comments by the teacher and students about the pieces of work in the portfolios collections. For some purposes, I think that the portfolio is not necessarily graded, typically in the task of problem solving. Portfolios can be used as a diagnostic assessment to identify student misconceptions on problem solving tasks and to encourage students’ self-assessment (Refer to appendix B). Portfolios, indeed, are recognised as flexible and provide tremendous opportunities for learning mathematics problem solving.
Nevertheless, it does not preclude that portfolios can be employed by the teacher both as formative and summative assessment. With the former, portfolios provide specific feedback from students, thus enabling the teacher to recognise what they know, believe, and understand. With the latter, portfolios can be assessed at the end of semester as the major criterion to further education. However, I would like to argue that portfolios of problem solving should be seen either as a diagnostic or formative assessment, rather than summative assessment, because in the summative assessment, students unconsciously will put aside the essential value of problem solving and may tend to focus on how to create a high-quality portfolio appearance for grading purposes.
Like any other performance assessment, portfolios might be evaluated using any number of approaches such as checklists, rating scales, and rubrics. As far as I am concerned, portfolios are best evaluated using a rubric of some sorts. Provided that problem solving employs many ways in approaching the solutions, teachers are required to design an appropriate rubric to describe specific criteria that match to the problem and the process of getting the solution. The limitation of this essay is excluding a discussion on the appropriate rubric for assessing portfolios of problem solving tasks.
Assessing learning is the most important part in education for the purpose of creating an effective learning environment. The assessment system should match the curriculum and learning process. The purpose of assessment covers every aspect of educational fields: learners, teachers, parents and school administrative. Assessment of problem solving extends to various contexts such as students abilities in interpreting and analysing the problem, and strategies they use to solve the problems.
A portfolio is recognised as the powerful tool of assessment suitable for assessing problem solving. It provides the record and track of students work and progress. In learning of problem solving, teachers can diagnose students’ strengths, weaknesses, and misconceptions using portfolios in classroom activities, whereas, portfolios can be valuable for students as reflective feedback for their improvement, as well as providing a record of their growth.
In Indonesia, portfolio assessment is seen as the new model of assessing mathematics problem solving. Creating effective portfolios, in addition is likely challenging, considering their purpose, particularly for problem solving, as they must include relevant information on how students approach the solution, ways and strategy they use to obtain the solution, likewise the explanation. Therefore, portfolios must be well-written and organised.
Indeed, assessing portfolios can be an alternative to diagnostic and formative assessment, but perhaps, the most important of assessing learning is how the teacher can see assessment tools as another way to encourage students learning, rather than simply evaluating how they learn.
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