Enhancing Learning through Assessment

Singapore secondary students are familiar with two part linear inequalities such as ax + b > cx + d,  three part inequalities ax + b ?cx + d <ex + f and quadratic inequalities ax2 + bx +c > 0.  However compound linear inequalities with connective words such as “and” and “or” do not receive much attention.  Such inequalities include ax + b > c and dx + e ?f  or  ax + b > c or dx + e ?f . Depending on the conditions, some compound linear inequalities with the connective word “and” may not have any solutions and those with the connective word “or” may have solutions which are union of the two sets or the entire number line.
The objectives of this study are twofold. The first is to identify whether students were aware of these differences and if they were not, whether they can be helped to construct new understanding of compound inequalities.  The study comprising two phases was conducted to identify the difficulties students had with compound linear inequalities. In Phase One, 57 Secondary Three Express provided written responses to a 14-item Written Task.  In Phase Two, 20 of the 57 students were interviewed. The objective of Phase Two was to determine whether these students were able to construct a better knowledge of compound linear inequalities.
The study found that although students were able to solve compound linear inequalities involving connective word “and”, there was floor effect with compound linear inequalities involving the connective word “or”. However through appropriate prompts students constructed a better understanding of compound inequalities involving the connective words “and” and “or”. The study showed that the affordance of number line aided students in their construction of knowledge.  The study argued that teachers and textbook writers should address students’ difficulties with compound linear inequalities.

Developed by Albert Bandura (1977; 1986), self-efficacy refers to students’ beliefs about their ability to accomplish certain tasks and it is one of the most prominent theory about human learning (Ormrod, 2008). Students’ self-confidence increases when they taste success in a given task. Researchers have indicated that higher self-efficacy is predictive of higher performance (Bong & Skaalvik, 2003). Equally important are lesson planning, selection of resources, lesson delivery and feedback. Providing timely feedback to students so that they become aware of what they know and the actions they need to take to move on to the next level makes learning visible to them (Hattie & Timperley, 2007).
This paper focuses on the impact on students’ self efficacy and achievement in Mathematics when regular feedback is provided. Bite sized instruction and assessment with immediate feedback were used to increase students’ confidence and self-efficacy in Mathematics. During lessons, learning outcomes were clearly stated and articulated in a form that can be easily understood by the students. They became aware of what they will be learning during the lesson and were also referred to these learning outcomes during the lessons by their teachers so that they were able to keep track of their learning. Bite-sized quizzes were used to check for students’ understanding at the end of the lesson which were marked and returned on a regular basis. Level of difficulty of questions given in the bite-sized quizzes was pitched at a level that can be reached by the students and the difficulty level was increased/decreased appropriately so that students find it meaningful to attempt the questions.
Data analysis shows that the refined assessment and feedback modes have improved students’ academic achievement in Mathematics. Students found the new process of testing and receiving immediate feedback useful and their self-efficacy towards the subject increased. The implementation of the refined assessment and feedback modes had a transformative effect on the students’ learning attitude especially those who were not doing well in Mathematics. Feeling the joy of learning has indeed motivated the students.

This paper provides a three-stage hierarchical teaching framework, Provision-Demand-Stretch (PDS), that can be used to deliver a developmental lesson or a summative lesson, at the end of a unit, or the construction of mathematical tasks.  The PDS framework is based on Klein’s (1996) practical work, which is underpinned by Feuerstein’s theoretical framework of Theory of Mediated Learning Experience.
At the Provision stage of a developmental lesson, teachers provide learners with specific examples to illustrate the concepts that lead learners to acquire the basic concepts.  For example, consider a lesson of addition and pattern recognition. Find the sum of three numbers: (a) 5, 12, 13, (b) 9, 10 and 11, (c) 8, 10 and 12. All three examples have a sum of 30.  The latter two examples are ‘special’ in that the three numbers are consecutive and the sum is three times the middle number.
At the Demand stage, teachers assess the learning of the learners.  Demand can be a two-way process. The teachers ‘demand’ learners to provide examples that help demonstrate how well they the learners have internalised the concept presented at the Provision stage. Initially learners are likely to provide examples similar to those presented in the Provision stage. However, learners, who have yet to grasp the concept, could also demand the teachers to provide further examples to help them learn the content.
The intent of the Stretch stage is to assess whether learners could transcend the examples from the Provision Stage and is characterised by requiring learners to provide examples that address specific conditions.  There are two levels of Stretch, Horizontal and Vertical.  Tasks at Horizontal stretch are at the same level of complexity as those presented at the Provision stage.  Vertical stretch tasks challenge learners to provide examples that fulfil conditions that are beyond those presented at the Provision stage.
In this paper, a range of examples, primary and secondary, are used to illustrate the Provision-Demand-Stretch framework across the curriculum.