The Open-Ended Approach: A New Proposal for Teaching Mathematics

On Teaching Problem Solving in School Mathematics
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The aim of open-approach teaching is to foster both the creative activities of the students and their mathematical thinking in problem solving simultaneously.

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In other words, both the activities of the students and their mathematical thinking must be carried out to the fullest extent. Then, it is necessary for each student to have the individual freedom to progress in problem solving according to his or her own abilities and interests. Finally, it allows them to cultivate mathematical intelligence. Class activities with mathematical ideas are assumed, and at the same time students with higher abilities take part in a variety of mathematical activities, and also students with lower abilities can still enjoy mathematical activities according to their own abilities.

go to link In doing so, it enables the students to perform the mathematical problem solving. It also offers them the opportunity to investigate with strategies in the manner they feel confident, and allows the possibility of greater elaboration within mathematical problem solving. As a result, it is possible to have a richer development in their mathematical thinking, and at the same time, foster the creative activities of each student.

This is the idea of the " Open-Approach " , which is defined as a method of teaching in which the activities of interaction between mathematics and students are open to varied problem solving approaches. This has been explained from three aspects:. Characterizations of the " Open-Approach " problem and method.

We should become more aware of the information processes that consist in the "Open-Approach", that are the relationships between the problem and method. Don't have an account? Your Web browser is not enabled for JavaScript. Some features of WorldCat will not be available. Create lists, bibliographies and reviews: or.

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Close-ended questions usually end conversations. Open- ended questions that are too general or unfocused may be difficult for the students to respond to and may also end the conversation. There are five advantages that can be summarized, based on what Sawada mentioned in The Open-Ended Problem Solving provides free, responsive, and supportive learning environment because there are many different correct solutions, so that each student has opportunities to get own unique answers.

Therefore, students are curious about other solutions, and they can compare and discuss about their solutions with each other. As students are very active, it brings a lot of interesting conversation to the classroom. Since there are many different solutions, students can choose their favorite ways towards the answers and create their unique solutions. Activities can be the opportunities to make comprehensive use of their mathematical knowledge and skills.

Through the comparing and discussing in the classroom, students are intrinsically motivated to give reasons of their solutions to other students. It is a great opportunity for students to develop their mathematical thinking. Research Instruments Research instruments were nine lesson plans on functions, formative test and summative test.

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There were nine lesson plans on functions integrated in constructivist approach to improve mathematical reasoning. The lesson plan scope and sequence were shown in Table 1. TABLE 1. Lesson plans on functions Lesson plans Period No. Contents 50 minutes 1 Definition of a function 1 2 The values of functions 1 3 Kinds of functions 1 4 Increasing and decreasing functions 1 Reflection 1 Formative test 6 written items 1 5 Constant functions and linear functions 1 6 Quadratic functions 2 7 Using graphing and quadratic functions to solve problems 2 Reflection 2 8 Exponential functions and step functions 1 9 Absolute value functions, polynomial functions, logarithmic functions and 1 rational functions.

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Summative test 6 written items 1 Total 13 Table 1 showed contents and time to spend in each lesson plan with five steps in teaching. The steps included: 1 orientation, 2 elicitation of the prior knowledge, 3 turning restructuring of ideas clarification and exchange of ideas, construction of new ideas and evaluation of the new ideas , 4 application of ideas, and 5 review.

See a Problem? The Open-Ended Approach: A New Proposal for Teaching Mathematics (): Jerry P. Becker, Shigeru Shimada: Books. The open-ended approach a new proposal for teaching mathematics. Material. Type. Book. Language English. Title. The open-ended approach a new proposal.

In implementing each lesson plan, the researcher emphasized reasoning through constructivist approach by open- ended questions and cooperative learning think-pair-share. Each formative test and summative test consisted of six written items.

Formative and summative tests were written tests. The total score of each item was 4. Table 2 showed scoring criterion of each test item.

Data Collection The data collection of this study was performed during 13 periods of instruction, as described in the previous section. During the study period, the participants received learning instruction based on a constructivist approach. Then, all participants were asked to take the formative test after finishing the fourth period. After the formative test, the teacher continued the instruction as specified in the lesson plans for resolving the problems found in the fourth period. The participants took summative test at the final period.

Then the researcher analyzed and interpreted the data. TABLE 2. Description of scoring mathematics reasoning ability Scores Levels Reasoning ability 4 Superior Deductive arguments are used to justify decisions and may result in formal proofs. Evidence is used to justify and support decisions made and conclusions reached. Some correct reasoning or justification for reasoning is presented.

No correct reasoning nor justification for reasoning is presented.

The Open-Ended Approach Framework

IOC was calculated by the formula Boonchom, For analyzing of research objective, the researcher used scoring mathematics reasoning ability from that of Exemplars TABLE 3. Results of formative test Overall score 24 points Number of students passing Item n x S. TABLE 4. Results of summative test Overall score 24 points Number of students passing above Item n x S. TABLE 5. Summary of both tests Overall score 24 points Number of students passing Item x S.

For homework and worksheets, the results showed that during the first few period of class, students adopted a definition used as a reason to answer only which could not lead to any other mathematical knowledge or their own reasoning to support answers.

After that, students developed more clear reasoning. They can summarize their own answer using definition and can adopt other mathematical knowledge or their own reasoning to support answers. In cooperative learning think-pair-share , it was found that in the first two periods, students summarized their own answers and did not discuss with their friends.

  1. Simulation and Modeling Methodologies, Technologies and Applications: International Conference, SIMULTECH 2012 Rome, Italy, July 28-31, 2012 Revised Selected Papers.
  2. Rich Task Construction: Making "Good" Problems Better | Waters | Ohio Journal of School Mathematics.
  3. Proof Theory: History and Philosophical Significance.

So, there was no final answer for class presentation. They did not review previous knowledge relevant to the presentation. After that, students started presenting and discussing with their friends before settling on a final answer. They made the rest of students understand more by reviewing previous knowledge and using mathematical knowledge to solve problems and used it to explain the answers very well. The topic used in this study was functions in Grade The researcher developed the research instruments composed of: 1 nine lesson plans that integrated five steps in teaching which are orientation, elicitation, restructuring of ideas clarification and exchange of ideas, construction of new ideas, and evaluation of the new ideas , application of ideas, and review, and 2 formative test and summative test.

The lesson plans were applied to 35 Grade 10 students with two cycles of action research; each cycle was composed of plan, do, check, and reflect.

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The formative test and summative test were of determined content validity by using index of Item Objective Congruence IOC judged by three experts. In addition, the researcher considered students' homework, worksheets and cooperative learning think-pair-share. Students showed gradual development of their reasoning ability. They could summarize their own answers using definitions and other mathematical knowledge to support answers. Students settled on final answers after considering and discussing with their friends.

They reviewed previous knowledge to explain the reasons before presenting to their friends and students settled on a final answer after discussing with their partners. Moreover, the cooperative learning think-pair-share increased their discussion, thinking, and reasoning ability. Furthermore, implementation of the constructivist approach should be intensively studied about knowledge and skills of students.

9780873534307 - The Open-ended Approach: a New Proposal for Teaching Mathematics, First Edition

In this case, training and continuous development is needed for students to develop mathematics reasoning ability. Students can learn from each other and can construct knowledge and reasoning ability. Their experiences can be shared with each other to understand and increase reasoning skill. Students were more engaged with constructivist approaches. Piaget also addressed that when students are engaged, they could easily be working with hands-on materials, while portraying independent thinking in collaboration with others on investigative, cumulative mathematics problems and they could actively be constructing their own meaning.

I would like to express my appreciation to students who participated in the research practicum school, Satriwitthaya School, Bangkok, Thailand.