Mathematics is closely connected to economics, commerce and business modelling, as well as systems for military weapons. Due to the widespread of its use, it was noted that students in the U.S. were beginning to perform a little worse in mathematics than children from other countries worldwide. Mathematical knowledge among citizens was considered a very important factor for a country to be a leading world power. Assessment activities have been a continuing focus of academic research for more than twenty-five years. In that period, there have been new tools developed. In addition, the curriculum has shifted its focus to the results of learning. The shift of focus in the theory of learning to constructivism from behaviourism has greatly influenced the learning and teaching of mathematics. Conventional tests are only centred on the mathematical procedures and skills of students. Thus, application of authentic tools for assessment to measure the learning of students is seriously required in mathematics. This paper entails a discussion about the appropriate tools; which is submitted to assess the performance of students in mathematical posing of problems and solving them. The 3 tools chosen are: problem posing and problem solving, authentic assessment and mixed method study.

Introduction

1. Problem Posing and Problem Solving

Problem-posing and problem-solving have become significant cognitive practices in learning and teaching mathematics. According to many researchers, the traditional assessment method do really reveal students' knowledge and learning acumen. From the viewpoint of learning and teaching mathematics, problem-posing and problem-solving have become essential instructional techniques to use in class. These practices can be applied in finding out the way people learn mathematics based on the constructivist theory of learning. Nonetheless, meaningful, unique and adaptive evaluations are still not fully developed for the assessment of students' learning, especially for the posing of problems.

2. Authentic Assessment

The alternative suggestion for this purpose was authentic assessment; a rubric for performance is a suitable tool for assessing the ability of students to pose and solve mathematical problems (Rosli, Goldsby, & Capraro, 2013). The practice community needs to move forward to improve the tools for assessment. It also needs to shift from traditional assessment to authentic methods to measure the overall learning of students (Rosli, Goldsby, & Capraro, 2013).

3. Mixed Methods Study

Mixed-methods study involves elements of case study, action research and surveys in examining a child at risk. Through a diagnostic examination, in-take interview, think aloud interview in the clinic and error analysis, the study found the main presenting difficulties of the child (Mundia, 2012). Mixed method designs for research have many disadvantages and advantages;...
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It involves the strong sides of both quantitative and qualitative techniques;
ii. It gives a more complete perspective of the phenomena or problem under study; and iii. It does not restrict the information being gathered.

Findings

1. Benefits and Implications

Authentic assessment supposes that students are bound to give something that shows a wide, compartmentalized recurrence of the things offered to them, but a combined scholarship connecting their learning based in other subjects and presented in a situation that matches that where the learning may be potentially most significant (Tanner, 1997). Authentic assessment may entail a combination of authentic assessment and authentic learning experiences. First, one needs to come up with activities which need students to integrate, synthesize and apply skill and knowledge in a way that shows the actual world and goes beyond the classroom. Job study programs and apprenticeships are types of techniques that vocational teachers have applied that are found in environments of authentic learning (Mcalister).

Qualitative perspectives of a mixed design method of research can be used together with quantitative research information to create a statement of belief about the presence of a qualitative study finding (Curllette, 2006). A mixed research method incorporates both qualitative and quantitative methods of research. One advantage of applying a mixed research method is that integrating the two techniques sharpens our knowledge of the study results (Reilly, 2007). A rubric for performance is a major factor in numerous interpretations of authentic assessments. The performance rubric is used through criterion-referenced techniques. Through interpretations that are criterion-referenced, Reynolds et al. stated that assessments of students to a particular performance level within a rubric, which is determined on their knowledge and skill. Analytic, holistic and process rubrics are among the scoring types of rubrics that a practice community usually uses. It qualifies as mathematical since it has several criteria based on students' overall procedural and conceptual insight along with problem-solving procedures. Scores from each criterion are added up to get an overall point as an indicator of performance (Rosli, Goldsby, & Capraro, 2013).

2. Factors Contributing the Use of Assessment Types

The open characteristic of problem-posing and problem-solving has caused an increase in the alternative assessment demand. A rubric of performance is more inclusive for the assessment of students' mathematical processes, concepts, disposition and procedures of mathematics. It changes the application of conventional assessments which are only determined by the…