Teaching Mathematics Practices according to NCTM Standards and Positions: “Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.” There is a demand for students to clarify the idea behind each method, justify the adequateness and examine the limitation of each method. “To facilitate such discussion, Japanese teachers use blackboard as a visual aid for students to participate in discussion throughout all the grades while considering student level of understanding of mathematics as well as their communication skills” (Takahashi, 2006). Hence, one of the major components of lesson planning is developing a plan for using the blackboard.
D. Routine and Non-routine Problem SolvingProblem solving is defined as the analysis and solution of tasks or situations that are complex or ambiguous and that pose difficulties or obstacles of some kind (Mayer & Wittrock, 2006, as mentioned by Seifert, K; Sutton, R. 2009). A person engages in problem solving when he tries to solve a problem whose solution is not directly available, “a question he cannot answer or a situation he is unable to resolve using the knowledge immediately available to him (Kantowski, 1977)”.
Problem solving has two aspects; process and product (Kantowski, 1977). Process refers to a set of behavior or activities that direct the search for the solution while product refers to the actual solution. In order to improve students’ problem solving ability, students must be exposed to a variety of word problems (Garelick, 2013 “Problem Solving, Moving from Routine to Non-routine and Beyond”) i. Routine ProblemsThe problem or situation is a standard task (routine task) if a student can immediately recognize the necessary procedures, given to the students to provide practice on a particular mathematical technique which they have just learned in class (Pehkonen, Navari, and Laine 2013; Schoenfeld, 1992 as mentioned by Mwei, 2017). Generally, routine problems are described as being well-structured; the tasks are clearly formulated with the goal of determining students’ ability to use standard algorithm. Algorithms can be seen as: rules for calculating; computational procedures for deriving a solution to a given problem; and logical step-by-step procedures for solving a mathematical problem (Problems and Problem Solving, Ministry of Education 2011). However, knowing the procedures doesn’t necessarily mean being able to successfully solve the problem since solution requires at least one of the four arithmetic operations.
The crucial part is identifying what arithmetic operation (or operations) is (are) needed to solve the problem. ii. Non-routine ProblemsTIMSS defined non-routine problems as problems which are very likely to be unfamiliar to students. Non-routine problems can be real-life problems or purely mathematical. “They make cognitive demands over and above those needed for solution of routine problems, even when the knowledge and skills required for their solution have been learned”(Mullis, et. al., 2011).
According to Schoenfeld, there are tasks which can be challenging tasks to some but may just be routine exercises for the others (Schoenfeld, 1983). Solving non-routine problems will require them “alternative strategies, thinking processes and creative thinking” (Mahlios, 1988 as mentioned by Ozcan, Z.C., et. al.
2017). In non-routine problems, the mathematical model (or solution path) is not readily available to the students. According to Garelick, “Problems that are extensions of routine problems require students to synthesize ideas and procedure from two or more areas of learned procedures are frequently multistep.” The first task is to look for strategies to discover a solution. Though the students may arrive at the same answer, their solutions may vary.
Because there is no clear method or solution, it allows the students to develop critical thinking (as students are required to think of their own rational approach to arrive at solutions), mathematical reasoning, and if done in groups, encourages exchanging of ideas and collaboration among students. Non-routine problems maybe in the form of word problems (which are mostly based on real-life situations) or open problems.Example of non-routine problem: Census Taker ProblemA census taker approaches a house and asks the woman who answers the door “How many children do you have, and what are their ages?” The woman replies “I have three children, the product of their ages are 36, the sum of their ages are equal to the address of the house next door.” The census taker walks next door, comes back and says “I need more information.” The woman replies “I have to go, my oldest child is sleeping upstairs.
” Census taker then says “Thank you, I now have everything I need.” What are the ages of each of the three children?(Problem and Problem Solving, Ministry of Education 2011 Kingston Jamaica) Open problems are “incomplete” problems (Becker, et.al, 1990) which usually produce different answers because of the ambiguity or the ill structure of the problem.
Because the students are forced to make their own assumptions, their own interpretation of the problem may be different from the others. It aims to develop students’ understanding of mathematical inquiry as they work individually or in small groups. “Open-ended mathematics problems provide students tasks which are enjoyable and challenging which in turn gives them desire to solve the problems because they have an opportunity to find their own way of thinking (Ninomiya, Pusri, 2015)”. Example of closed problem:For A triangle has an area of 24 square inches. Its base is 12 inches. What is its altitude? Example of open problem:Choose three numbers from the list below to complete the following sentence:A triangle has an area of _____ square inches.
Its base is ______ inches and its altitude is ______ inches.8 32 12 4 3(Problem and Problem Solving, Ministry of Education 2011 Kingston Jamaica) Non-routine problems can either be “Process-Constrained” or “Process-Open” (Cai, 2002). A process-constrained task can be solved using “standard algorithm” however students need to figure out the appropriate algorithm. Algorithms include rules for calculating, computational procedures for deriving a solution to a given problem; and logical step-by-step procedures for solving a mathematical problem.