Formulating tasks to develop HOTS for first-year calculus based on Brookhart abilities

Abstract

We describe an approach to develop higher-order thinking skills (HOTS) among first-year calculus students. The ideas formulated by Brookhart to develop HOTS were used to identify from the literature three core abilities that should be targeted. Then eight expected learning outcomes for the development of HOTS were documented, in the context of the study of first-year university calculus. Those expected outcomes were used to formulate sample tasks that were designed to target the development of the eight abilities. A pilot study was done to determine whether the tasks had the high mathematical demand envisaged. It was found that about 37% of the participants did not give any response to the tasks. Further it was found that about 31% of the participants were able to critically evaluate a given possible solution to a problem and make a value judgement. It is recommended that to promote HOTS among students, the formulation of tasks should focus on developing the following abilities: interpreting a general definition or statement in the context of a given model; translating a worded or graphically represented situation to relevant mathematical formalisms; identifying possible applications of mathematics in their surroundings; identifying linkages between groups of concepts and interpreting these linkages in the context of a model; working systematically through cases in an exhaustive way; critically evaluating one’s and others’ presented solutions to a problem; interpreting and extending solutions of problems; and using with reasonable skill available tools for mathematical exploration.

Flipped Classroom Strategy Based on Critical Thinking Skills: Helping Fresh Female Students Acquiring Derivative Concept

“…As Tall (2008) described it in the second stage of his theory TWM, which he called the Proceptual-symbolic, where the tasks used to build the relationship between the different representations of concept, and to form a clear conceptualization of concepts Calculus. That result coincided with the Maharaj & Wagh's (2016) study in the application of tasks that rely on higher thinking skills that helped students to develop their understanding of calculus.…”

“…It also includes conceptual and procedural knowledge of how to deal with the situation. Maharaj and Wagh (2016) pointed out that building a conceptual structure in mathematics must be structured and logical to students, who should be taught mathematics for relational understanding of the mathematical concepts of Calculus. Accordingly, the Conference Board of the Mathematical Sciences (CBMS, 2016) recommended that the higher education institutions in general, and mathematics department faculties in particular, should integrate active learning in classrooms based on integration between science, technology, engineering, and mathematics; and need to be well informed about active learning and related topics.…”

“…The conducted researches asserted that the kinds of tasks applied in Calculus situations in class like examples, homework assignments, and all other projects must focus on conceptual understanding. As Maharaj and Wagh (2016) recommended the application of tasks based on Higher-Order Thinking Skills (HOTS) to help students during studying Calculus to focus on understanding and interpreting mathematical definitions, translating a problem verbally formulated or graphically represented to the formal mathematics. These tasks also identify the possible applications of mathematics in reality, and focus on identifying the links between sets of concepts and interpreting these links in mental schemes, working regularly from special cases to the general formula, and focusing on a critical assessment of other solutions.…”

“…These tasks apply based on (HOTS) to help students follow the assumptions in the task to reach known results from the concept as the classification between validity of the outcome or error in relation to the facts given and focus on understanding and interpreting mathematical definitions. These tasks also help students to build new knowledge implemented in the future (Maharaj & Wagh, 2016;Strayer & et al, 2016). The third phase concentrate on the relationship between assignments and tasks for the construction of meaningful concepts in a formal form based on definitions of known objects, and show the conceptual structure (Tall, 2008).…”

The Algebraic Thinking Process in Solving Hots Questions Reviewed from Student Achievement Motivation

“…Prospective mathematics teachers need to develop the extent of mathematical thinking ability in solving HOTS problems. Based on research (Maharaj & Wagh, 2016) suggest getting used to and practicing HOTS questions among students so that HOTS thinking patterns are no longer something new but must be used to be done in the learning process in the classroom. The test instrument or commonly called a question is one of the measuring instruments used to detect students' algebraic thought processes.…”