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Problem solving strategies of eight pre-service secondary school mathematics teachers (PSSMTs) were examined in this study. A case study research design was employed and clinical interview technique was used to collect the data. Materials collected for analysis consisted of audiotapes and videotapes of clinical interviews, subjects' notes and drawings, and researchers' notes during the interview. Strategies used by PSSMTs to solve a problem called "the fencing problem" were identified. Findings of the study suggest that the subjects used combinations of different strategies to solve problems. The implications of the finding were also discussed.

Mathematics word-problems continue to be an insurmountable challenge for many middle school students. Educators have used pictorial and schematic illustrations within the classroom to help students visualize these problems. However, the data shows that pictorial representations can be more harmful than helpful in that they only display objects or persons while neglecting the spatial relationships between those components (Hegarty & Kozhevnikov, p. 686). Research supports schematic diagrams that highlight spatial relationships and focus on related information within the problem (Hegarty & Kozhevnikov, 1999; van Garderen & Montague, 2003). In addition to these strategies, educators have used heuristics (systematic scripted procedures to solve word problems) as a way to provide structure and routine to a variety of word problems. This review of the literature found varying results for the use of some heuristics being implemented in present-day classrooms. On the other hand, both visual-spatial diagrams and heuristics have demonstrated value in middle school special education classrooms. These findings support the further use and research of visual-spatial representations and problem-solving heuristics in order to solve mathematics word problems.

It is important for teacher educators to understand new alternative certification middle and high school teachers' mathematical problem solving abilities and perceptions. Teachers in an alternative certification program in New York were enrolled in a proof-based algebra course. At the beginning and end of a semester participants were given a problem solving examination and were required to reflect upon their students' and their own problem solving abilities. There was a significant improvement in problem solving abilities for the teachers over the course of the semester, and there was a direct correlation between standardized content knowledge test scores and problem solving examination scores. Furthermore, teachers perceived that although their students had improved somewhat in their problem solving over the course of the semester, student problem solving abilities were generally weak due to not understanding how to start a problem, lack of persistence, and poor literacy skills. Teachers perceived that they shared similar problems as their students, but the algebra course and their teaching of problem solving helped them with their own problem solving abilities.

The second in a series of four papers on computer use in mathematics education, this paper reviews some of the pedagogical rationales and research evidence related to the use of computers as a problem-solving tool in mathematics classes. A discussion on philosophies and objectives, effects of computer programming on mathematics achievement, motivation and attitudes, and influence on problem-solving ability are included. For other papers in this series, see SE 016 289 through SE 016 292. (Author/DT)

Technology can play a pivotal role in mathematics education. Qualitative methods were used to report on how a Year 1-2 teacher incorporated "Conveyance technology" (iPads), to support the teaching and learning of mathematics through student reasoning and problem-solving. The study was conducted in Australia. A single case was chosen, reporting a lesson where students engaged with a 12-cube task. The findings contribute to the broader discourse on the role of technology in early mathematics education, suggesting that conveyance technology supported by effective teaching practices and open-ended problems, fosters students' reasoning and problem-solving skills.

Technology can play a pivotal role in mathematics education. Qualitative methods were used to report on how a Year 1-2 teacher incorporated "Conveyance technology" (iPads), to support the teaching and learning of mathematics through student reasoning and problem-solving. The study was conducted in Australia. A single case was chosen, reporting a lesson where students engaged with a 12-cube task. The findings contribute to the broader discourse on the role of technology in early mathematics education, suggesting that conveyance technology supported by effective teaching practices and open-ended problems, fosters students' reasoning and problem-solving skills.

Technology can play a pivotal role in mathematics education. Qualitative methods were used to report on how a Year 1-2 teacher incorporated "Conveyance technology" (iPads), to support the teaching and learning of mathematics through student reasoning and problem-solving. The study was conducted in Australia. A single case was chosen, reporting a lesson where students engaged with a 12-cube task. The findings contribute to the broader discourse on the role of technology in early mathematics education, suggesting that conveyance technology supported by effective teaching practices and open-ended problems, fosters students' reasoning and problem-solving skills.

Technology can play a pivotal role in mathematics education. Qualitative methods were used to report on how a Year 1-2 teacher incorporated "Conveyance technology" (iPads), to support the teaching and learning of mathematics through student reasoning and problem-solving. The study was conducted in Australia. A single case was chosen, reporting a lesson where students engaged with a 12-cube task. The findings contribute to the broader discourse on the role of technology in early mathematics education, suggesting that conveyance technology supported by effective teaching practices and open-ended problems, fosters students' reasoning and problem-solving skills.

There are many factors that affect student engagement in science, technology, engineering and mathematics (STEM). Underlying this are the views of the broader community--and parents in particular--about the relevance of STEM, and the approach to the teaching and learning of STEM from the early years and continuing throughout schooling. Connected to this is the way industry articulates the importance of STEM related-skills that extend beyond traditional STEM occupations. University admissions policies also have a strong influence on student choices in the senior secondary years. The purpose of the strategy is to build on a range of reforms and activities already underway. It aims to better coordinate and target this effort and sharpen the focus on the key areas where collaborative action will deliver improvements to STEM education. The national strategy is focused on action that lifts foundational skills in STEM learning areas, develops mathematical, scientific and technological literacy, and promotes the development of the 21st century skills of problem solving, critical analysis and creative thinking. It recognises the importance of a focus on STEM in the early years and maintaining this focus throughout schooling. This document outlines two goals (ensure all students finish school with strong foundational knowledge in STEM and related skills; and ensure that students are inspired to take on more challenging STEM subjects), five areas for national action, and guiding principles for schools to support STEM education. [Cover title varies: "National STEM School Education Strategy, 2016-2026."]

One of the five goals proposed in the National Council of Teachers of Mathematics' (NCTM) "Curriculum and Evaluation Standards for School Mathematics" is for students to become "mathematical problem solvers." To achieve this goal, the NCTM proposes fundamental changes in both the content of mathematics curricula and the pedagogy of the mathematics classroom that outline a broad role for problem solving. Textbooks and other published curricula are believed to play a significant role in the classroom instruction of mathematical problem solving. Described and compared are the conceptions of problem solving found in one commonly used elementary mathematics textbook ("Addison-Wesley Mathematics") and three distinctive elementary mathematics curricula ("Real Math", "Comprehensive School Mathematics Program", and "Math in Stride"). Although in each, problem solving was professed to be of central importance to the development of mathematical understanding and competence, how problem solving is defined and taught varied amongst the four published curricula. The different ways that problem solving is defined, formulated, and taught are explored, and the pedagogical and epistemological assumptions that underlie these differences are also discussed. The four curricula are compared on various perspectives of problem solving, the levels at which it was incorporated into the curriculum and how the authors of each curriculum viewed "mathematics as problem solving." (14 references) (MDH)

One of the five goals proposed in the National Council of Teachers of Mathematics' (NCTM) "Curriculum and Evaluation Standards for School Mathematics" is for students to become "mathematical problem solvers." To achieve this goal, the NCTM proposes fundamental changes in both the content of mathematics curricula and the pedagogy of the mathematics classroom that outline a broad role for problem solving. Textbooks and other published curricula are believed to play a significant role in the classroom instruction of mathematical problem solving. Described and compared are the conceptions of problem solving found in one commonly used elementary mathematics textbook ("Addison-Wesley Mathematics") and three distinctive elementary mathematics curricula ("Real Math", "Comprehensive School Mathematics Program", and "Math in Stride"). Although in each, problem solving was professed to be of central importance to the development of mathematical understanding and competence, how problem solving is defined and taught varied amongst the four published curricula. The different ways that problem solving is defined, formulated, and taught are explored, and the pedagogical and epistemological assumptions that underlie these differences are also discussed. The four curricula are compared on various perspectives of problem solving, the levels at which it was incorporated into the curriculum and how the authors of each curriculum viewed "mathematics as problem solving." (14 references) (MDH)

Problem solving continues to be a major focus of K-12 mathematics education reform. This case study shows how a teacher's middle school mathematics instruction evolved from teaching problem solving as computation exercises to teaching how to select and implement combinations of problem solving strategies. The teacher's thinking about problem solving instruction also evolved to encompass different understandings about what a problem is, what problem solving means, and what teachers' and students' roles are in teaching and learning problem solving. The result of this evolution was that the teacher's instruction became more closely aligned with reform goals for problem solving. This finding suggests that reform efforts should incorporate strategies and support for teachers to revise and refine their own instruction through reflection and dialogue with colleagues as they try out approaches and activities in their classrooms for teaching problem solving. (Contains 1 table.)

This paper reports a study conducted in a laboratory classroom with the capability to record classroom social interactions in great detail using advanced video technology. The social interactions of student groups during collaborative problem solving were analysed based on transcript data. This analysis suggests that meaning negotiation in mathematics classrooms can be usefully distinguished as social, sociomathematical, or mathematical. We suggest that all three modes coexist in an entangled form in the negotiative interactions documented in the mathematics classroom and we envisage all three as constitutive of learning.

This paper presents the thesis that pair problem-solving serves as an aid to students in developing their metacognitive skills. Metacognition has been defined as knowledge and cognition about cognitive objects and as any kind of monitoring. The ability to monitor one's thoughts and actions is greatly facilitated through the pair problem-solving method. Briefly, the method calls for one student to listen to another student solve a problem aloud. The listener may ask questions so as to understand each step and all the reasonings of the solver's solution. Modeled after the clinical interviews of Piaget, students learn self-monitoring skills while learning math concepts. Examples of verbal protocols from college freshmen illustrate the emergence of metacognition during clinical interviews and during pair and group problem-solving in class. Finally, a description of a remedial math program at a major university provides practical applications of metacognition in the classroom. (Author/PK)

This paper presents the thesis that pair problem-solving serves as an aid to students in developing their metacognitive skills. Metacognition has been defined as knowledge and cognition about cognitive objects and as any kind of monitoring. The ability to monitor one's thoughts and actions is greatly facilitated through the pair problem-solving method. Briefly, the method calls for one student to listen to another student solve a problem aloud. The listener may ask questions so as to understand each step and all the reasonings of the solver's solution. Modeled after the clinical interviews of Piaget, students learn self-monitoring skills while learning math concepts. Examples of verbal protocols from college freshmen illustrate the emergence of metacognition during clinical interviews and during pair and group problem-solving in class. Finally, a description of a remedial math program at a major university provides practical applications of metacognition in the classroom. (Author/PK)

This paper presents the thesis that pair problem-solving serves as an aid to students in developing their metacognitive skills. Metacognition has been defined as knowledge and cognition about cognitive objects and as any kind of monitoring. The ability to monitor one's thoughts and actions is greatly facilitated through the pair problem-solving method. Briefly, the method calls for one student to listen to another student solve a problem aloud. The listener may ask questions so as to understand each step and all the reasonings of the solver's solution. Modeled after the clinical interviews of Piaget, students learn self-monitoring skills while learning math concepts. Examples of verbal protocols from college freshmen illustrate the emergence of metacognition during clinical interviews and during pair and group problem-solving in class. Finally, a description of a remedial math program at a major university provides practical applications of metacognition in the classroom. (Author/PK)

This paper is based on the concept that lively and interactive math classes are possible by incorporating rich tasks to meet the needs of students operating at different levels in the classrooms. A study was carried out to find out the impact on learning and motivation of using rich tasks at secondary level in the maths class by incorporating co-operative learning. Qualitative research paradigm was opted for the study using an action research approach and the data were collected through two semi-structured interviews conducted at the onset of the research and after the intervention. Few important findings indicate that rich tasks demand different levels of challenge and extend opportunities to those students who need them.

This paper is based on the concept that lively and interactive math classes are possible by incorporating rich tasks to meet the needs of students operating at different levels in the classrooms. A study was carried out to find out the impact on learning and motivation of using rich tasks at secondary level in the maths class by incorporating co-operative learning. Qualitative research paradigm was opted for the study using an action research approach and the data were collected through two semi-structured interviews conducted at the onset of the research and after the intervention. Few important findings indicate that rich tasks demand different levels of challenge and extend opportunities to those students who need them.

Considering the requisite skills and comparing commonalities in historical development, the disciplines of physics and mathematics can be viewed as related to and supportive of each other, yet it remains a curiosity that few, if any, cross-connections exist between school physics and school mathematics in the area of research on problem solving. This study synthesizes the up-to-date information available from research reports, books, and monographs with the intent of providing a base for interdisciplinary research on the comparison of problem solving in school physics and mathematics and for the promulgation of informed pedagogical considerations concerning the teaching of problem solving within these related fields at the middle school and high school levels. The studies reported here are organized into three groups as they relate to problem solving as both a general and specific activity, to the learner and his/her association with problem solving, and to the teaching of problem solving. A list of 66 references is included. (JJK)

This paper presents another effort in determining the difficulty of engineering students in terms of solving word problems. Students were presented with word problems in algebra. Then, they were asked to solve the word problems orally; that is, before they presented their written solutions, they were required to explain how they understood the problem, and to give the processes they wanted to use in order to obtain the answer. Responses of students for each word problems would be noted. Discussions were recorded so that all responses were accounted for. Using NEA (Newman's error analysis), student's problems on reading, comprehension, transformation, and process skills can be determined by the teacher before the encoding of the solution is done. Also, the teacher directly addresses whatever misconceptions are made by the student in the process as well as of other students who are thinking the same way. More than 70% of the errors found were comprehension and transformation errors. Thus, students were given remedial classes to minimize their comprehension and transformation errors. (Contains 4 tables.)

This study investigated how to improve the teaching of problem solving in a large Melbourne secondary school. Coaching was used to support and equip five teachers, some with limited experiences in teaching problem solving, with knowledge and strategies to build up students' problem solving and reasoning skills. The results showed increased confidence by all teachers in the range and use of problem solving strategies, and for students increased use of strategies and improved reasoning skills to solve problems.

NASA CONNECT is an annual series of integrated mathematics, science, and technology instructional distance learning programs for students in grades 6-8. This program is designed for students to learn about the evolution of flight. The program has three components--television broadcast, Web activity, and lesson guide--which are designed as an integrated instructional package. Students learn how the Wright Brothers became the first human beings to successfully design, construct, and fly an airplane and which method the Wright Brothers used in designing their airplane. Students also observe NASA engineers and researchers using problem-solving skills to design wings that will change their shape during flight. By conducting hands-on and Web activities, students make connections between NASA research and the mathematics, science, and technology they learn in the classroom. The educator guide features: (1) program overview; (2) hands-on activity; (3) student worksheets; (4) teacher materials; (5) interactive kite modeler; and (6) resources. (KHR)

This unit is 1 of 12 developed for the university classroom portion of the Mathematics-Methods Program (MMP), created by the Indiana University Mathematics Education Development Center (MEDC) as an innovative program for the mathematics training of prospective elementary school teachers (PSTs). Each unit is written in an activity format that involves the PST in doing mathematics with an eye toward application of that mathematics in the elementary school. This document is one of four units that are devoted to mathematical topics for the elementary teacher. In addition to an introduction to the unit and some perspectives on problem solving, the text has six activity sections titled: In Search of a Strategy; Looking for Another Strategy; To Find a Third Strategy; Now Try Your Hand; Reflection on Your Experiences; and Helping Children Solve Problems. (MP)

The article begins with a brief overview of the situation throughout the world regarding problem solving. The activities of the ProMath group are then described, as the purpose of this international research group is to improve mathematics teaching in school. One mathematics teaching method that seems to be functioning in school is the use of open problems (i.e., problem fields). Next we discuss the objectives of the Finnish curriculum that are connected with problem solving. Some examples and research results are taken from a Finnish-Chilean research project that monitors the development of problem-solving skills in third grade pupils. Finally, some ideas on "teacher change" are put forward. It is not possible to change teachers, but only to provide hints for possible change routes: the teachers themselves should work out the ideas and their implementation.

The article begins with a brief overview of the situation throughout the world regarding problem solving. The activities of the ProMath group are then described, as the purpose of this international research group is to improve mathematics teaching in school. One mathematics teaching method that seems to be functioning in school is the use of open problems (i.e., problem fields). Next we discuss the objectives of the Finnish curriculum that are connected with problem solving. Some examples and research results are taken from a Finnish-Chilean research project that monitors the development of problem-solving skills in third grade pupils. Finally, some ideas on "teacher change" are put forward. It is not possible to change teachers, but only to provide hints for possible change routes: the teachers themselves should work out the ideas and their implementation.

Our laboratory conducted an experimental evaluation of Tutorials in Problem Solving (TiPS), a computer environment representing a schema-based approach to training arithmetic and problem solving skills in remedial adult populations. Specifically, this project was designed to accomplish several objectives: (1) document the level of instructional adaptation provided by the TiPS system; (2) provide a basic evaluation of the overall TiPS system; (3) determine the amount of instructional time involved with the use of TiPS system; and (4) determine the affective nature of the TiPS learning experience, According to our evaluation project: (1) TiPS does monitor the learners' performance and adapt instructional delivery to meet their needs; (2) the average posttest score for participants in a TiPS group was significantly higher than their peers in a untreated control group; (3) the average time for completing the Tips computer instruction was 3.84 hours (SD = 0.97) and ranged from 1.83 to 6 hours (excluding test-taking time); and (4) the learners exposed to TiPS reported feeling that the instructional material, including the examples and problems, helped them to understand how to approach and solve word problems and that, overall, the instructional environment was well designed.

This study aims to obtain a picture of the perception of mathematics teachers related to students' mathematical problem-solving abilities. This research method is qualitative research with a case study approach. The subject of this research is a mathematics teacher in Sumatra. A total of 32 mathematics teachers were involved in this study. The data collected in this study used a questionnaire. The questionnaire used was validated by one mathematics education lecturer and three mathematics teachers. Then, the results of the questionnaire data were analyzed using Miles and Huberman analysis which consisted of data reduction, data presentation, and drawing conclusions. The results of this study indicate that the teacher's perception of the ability to solve mathematical problems as measured by the Polya stages ranges from 63% to 77%. In addition, there are differences in students' mathematical problem-solving abilities during online and offline learning. The impact of this research can add to existing references and be considered to improve and evaluate math problem-solving abilities in both public and private schools in Sumatra. [For the complete proceedings, see ED631021.]

This study aims to obtain a picture of the perception of mathematics teachers related to students' mathematical problem-solving abilities. This research method is qualitative research with a case study approach. The subject of this research is a mathematics teacher in Sumatra. A total of 32 mathematics teachers were involved in this study. The data collected in this study used a questionnaire. The questionnaire used was validated by one mathematics education lecturer and three mathematics teachers. Then, the results of the questionnaire data were analyzed using Miles and Huberman analysis which consisted of data reduction, data presentation, and drawing conclusions. The results of this study indicate that the teacher's perception of the ability to solve mathematical problems as measured by the Polya stages ranges from 63% to 77%. In addition, there are differences in students' mathematical problem-solving abilities during online and offline learning. The impact of this research can add to existing references and be considered to improve and evaluate math problem-solving abilities in both public and private schools in Sumatra. [For the complete proceedings, see ED631021.]

This report investigated to what extent a curriculum designed to actively teach critical thinking skills resulted in students utilizing higher-order thinking skills (e.g., analysis, synthesis and evaluation). An intervention strategy was designed for a sixth grade class located in a diverse suburban community in northern Illinois. The intervention targeted the mathematics curriculum and was designed to incorporate the teaching of critical thinking skills. Students were administered a pretest to determine baseline data. The intervention included the daily use of a variety of thinking skill enhancers and guided assessment of problem-solving strategies for a twenty-week period. Results indicated success on many levels. Students displayed increased self-esteem and confidence in their abilities to problem-solve. Self-reflection of their work was also observed. Students were able to verbalize their thought processes in analyzing problems. As an indicator of success, however, it was hoped that there would have been a higher number of students displaying an increase in posttest results. Nevertheless, with over fifty percent of the students involved in the intervention showing increased scores, the time and energy expended in this project were affirmed. (Contains 13 references.) (Author/YDS)

"Blackline masters, with teacher's commentary and answers"--Cover

"Blackline masters, with teacher's commentary and answers"--Cover

The roles of cognition and metacognition were examined in the mathematical problem-solving behaviors of students as they worked in small groups. As an outcome, a framework that links the literature of cognitive science and mathematical problem solving was developed for protocol analysis of mathematical problem solving. Within this framework, each behavior is categorized by heuristic episode and cognitive level. Data were obtained from videotapes of six small groups of seventh-grade students (N=27), who attended a middle school in Queens (New York City), as they worked together to solve a mathematical problem. Three coders viewed each tape in 1-minute intervals. Watching one or two students in the group, the coders assigned one of the following episodic categories and cognitive levels: (1) read (cognitive); (2) understand (metacognitive); (3) analyze (metacognitive); (4) plan (metacognitive); (5) explore (cognitive or metacognitive); (6) implement (cognitive or metacognitive); (7) verify (cognitive or metacognitive); (8) watch (undetermined cognitive level); and (9) listen (undetermined cognitive level). Explicitly delineating the role of metacognition and cognition within the heuristics of problem solving gives researchers and mathematics teachers a tool to evaluate problem solving. Implications for cooperative group processes and classroom assessment are discussed. Three figures and one table contain the study data. The framework that was developed is outlined. (Author/SLD)

Learning loss due to online learning has caused students to experience a decrease in their ability to solve mathematical problems. One of the declines in students' mathematical problem-solving skills occurred in the statistics course. Statistics as a branch of mathematics has an important role in training students' skills in processing and analyzing data. The low ability of students to solve mathematical problems became a serious problem. So we need an appropriate learning model to overcome these problems. One of the learning models that can be used is hybrid learning. The subject of this study was the primary student study program from the Faculty of Education at UPH in Indonesia, with 110 samples divided into 54 control groups and 56 experimental groups. The control class was treated with full online learning, while the experimental class used hybrid learning. This research aimed to compare which type of learning is more effective, full-online or hybrid learning. The research method used in this study is quasi-experimental. Data were processed using SPSS, called the normality test with Shapiro-Wilk, and statistical tests using Mann-Whitney. The results showed that hybrid learning was more effective than full online learning. [For the full proceedings, see ED654100.]

The purpose of this study is to determine how Finnish pupils' problem-solving skills develop from the 3rd to 5th grade. As research data, we use one non-standard problem from pre- and post-test material from a three-year follow-up study, in the area of Helsinki, Finland. The problems in both tests consisted of four questions related to each other. The purpose of the formulation of the problem was to help the pupils to find how many solutions for a certain answer exist. The participants in the study were 348 third-graders and 356 fifth-graders. Pupils' fluency, i.e. ability to develop different solutions, was found to correlate with their ability to solve the problem. However, the proportions of the pupils (17% of the 3rd graders and 21% of the 5th graders) who answered that there were an infinite number of solutions are of the same magnitude. Thus, the pupils' ability to solve this kind of problem does not seem to have developed from the 3rd to the 5th grade. The lack and insufficiency of pupils' justifications reveal the importance of the teacher carefully listening to the pupils' ideas in order to be able to promote pupils' understanding of the concept of infinity, as well as the basic calculations.

This study sought to determine the effect of the problem-solving approach on academic achievement of students in mathematics at the secondary level. The secondary school students studying mathematics constituted the population of this study. The students of 10th class of Government Pakistan Girls High School Rawalpindi were selected as a sample for the study. Sample size consisted of 48 students who were equally divided into an experimental group and a control group on the basis of pre-test. Treatment of the planned problem-solving approach is the guideline of Sherreen (2006) and Polya's (1945) heuristic steps of the problem-solving approach. After the treatment, post-test was used to see the effects of the treatment. A two-tailed t-test was used to analyze the data, which revealed that both the experimental and control groups were almost equal in mathematics base at the beginning of the experiment. The experimental group outscored the control group significantly on the post-test.

The New Standards Project is a partnership of 17 states and 6 major school districts that have joined to develop alternative approaches to setting educational standards and assessing student achievement. In the New Standards Project plan, formal standard setting occurs through a process of public examination of student work products on pilot assessment tasks. This progress report describes the work on task development during the first project year, and outlines plans for pilot studies and grading exercises that will occur in the spring and summer of 1992. At that time, New Standards Project will conduct a pilot of fourth-grade performance assessments in mathematics and English/language arts. A total of 460 teachers in 23 states and over 10,000 students are expected to participate. The design of the pilot is described, and research questions are discussed. Task production for this pilot began at a workshop in 1991 where approximately 450 participants focused on developing authentic tasks. Editing and revising has occurred, with several levels of review. The development of scoring rubrics is the next step, scheduled for April and May 1992. Appendix 1 contains the 11-item teacher questionnaire for the pre-pilot assessment. Appendix 2 lists the steps in task revision as an iterative process. (SLD)

The New Standards Project is a partnership of 17 states and 6 major school districts that have joined to develop alternative approaches to setting educational standards and assessing student achievement. In the New Standards Project plan, formal standard setting occurs through a process of public examination of student work products on pilot assessment tasks. This progress report describes the work on task development during the first project year, and outlines plans for pilot studies and grading exercises that will occur in the spring and summer of 1992. At that time, New Standards Project will conduct a pilot of fourth-grade performance assessments in mathematics and English/language arts. A total of 460 teachers in 23 states and over 10,000 students are expected to participate. The design of the pilot is described, and research questions are discussed. Task production for this pilot began at a workshop in 1991 where approximately 450 participants focused on developing authentic tasks. Editing and revising has occurred, with several levels of review. The development of scoring rubrics is the next step, scheduled for April and May 1992. Appendix 1 contains the 11-item teacher questionnaire for the pre-pilot assessment. Appendix 2 lists the steps in task revision as an iterative process. (SLD)

This research aims to determine the leveling of critical thinking abilities of students of mathematics education in mathematical problem solving. It includes qualitative-explorative study that was conducted at University of PGRI Semarang. The generated data in the form of information obtained problem solving question and interview guides. The results are: (1) students who are not critical (LCTA-0) are only able to identify the given facts clearly, (2) students are less critical (LCTA-1), students can identify the facts in problems, (3) critical student (LCTA-2), students can identify the fact in problems, revealing the prerequisite knowledge appropriately, solve the problem but still less accurate in every stage, and (4) students are very critical (LCTA-3), students can identify the facts given clearly, mention the concept/theorem/prerequisite material. Based on these results, the leveling of students critical thinking abilities can be applied in a lecture at mathematics education.

The majority of research on mathematics practice in everyday situations within cultures has investigated the use of arithmetic and geometry concepts and processes. To extend this research to a situation using measurement ideas, this paper investigates the mathematics practice of a group of carpet layers in an effort to detail how ordinary people "actively give meaning to, and fashion, processes of problem solving in the midst of ongoing activities in relevant Settings" (Lave, 1988). Data were collected by observing and informally questioning the employees of a carpet laying business. Four areas of mathematics concepts used by the estimators and/or installers were observed: measurement, computational algorithms, geometry, and ratio and proportion. Two general observations were made: (1) the estimators and installers were not concerned with square footage of a room but with the square feet of carpet needed in the room, and (2) all problems in carpet laying are optimization problems. (MKR)

This document reports on the initial phase of a project investigating how to relate formal mathematical representational and problem solving skills to informal strategies that children naturally invent to solve simple addition and subtraction problems. A program was developed that allows pupils to solve word problems on a microcomputer. A pilot study was carried out with four first-grade children. The subjects were individually instructed for a series of nine 20-minute lessons. The results of the study indicated that the program is effective in teaching representational and problem-solving skills. Before instruction, the subjects consistently wrote incorrect sentences for incorrect problems and generally did not use their number sentences for their solutions. Following instruction, three of the four children consistently used number sentences to solve a wide variety of addition and subtraction problems. It is concluded that further investigation seems warranted, and that this pilot investigation suggests that microcomputers can have important roles in instruction. (MP)

This document reports on the initial phase of a project investigating how to relate formal mathematical representational and problem solving skills to informal strategies that children naturally invent to solve simple addition and subtraction problems. A program was developed that allows pupils to solve word problems on a microcomputer. A pilot study was carried out with four first-grade children. The subjects were individually instructed for a series of nine 20-minute lessons. The results of the study indicated that the program is effective in teaching representational and problem-solving skills. Before instruction, the subjects consistently wrote incorrect sentences for incorrect problems and generally did not use their number sentences for their solutions. Following instruction, three of the four children consistently used number sentences to solve a wide variety of addition and subtraction problems. It is concluded that further investigation seems warranted, and that this pilot investigation suggests that microcomputers can have important roles in instruction. (MP)

Fifty-five college students participated in a study investigating the effect of field dependence and time limitations on the solution of mathematics word problems and visual interpretation problems. Field-independent (n=25) and field-dependent (n=30) subjects represented the upper and lower thirds on the Hidden Figures Test. Field-independent subjects scored higher than did field-dependent subjects on both base and complex word problems, but there was no difference between the groups on visual problems. Although effects for time were found, they appeared to be artifactual. Results are discussed in terms of the relationship between the demands of the verbal and visual problems, and differences in restructuring abilities of field-independent and field-dependent learners. Two tables and three graphs contain study data. (Author/SLD)

In Principles and Standards for School Mathematics (2000), the (U.S.) National Council of Teachers of Mathematics recommended that students communicate their mathematical thinking in a logical manner, and use the language of mathematics to express their thinking accurately and logically. Students should not only learn mathematics content, but should learn how to generate ideas, express them in multiple ways, and justify their thinking (Carpenter, Franke, & Levi, 2003). Journals are an effective way for students to communicate their understanding of mathematics content while using the process standards of problem solving, reasoning and proof, connections, and representation. This article describes the use of mathematics journals by undergraduate Early Childhood Education majors in a mathematics methods course. In this course, pre-service teachers are asked to use journals to examine their own problem solving skills, through written and oral reflection. Samples of student reflections of their problem solving processes are included.

The present study examined the potential relationship between 1st and 5th year secondary school pre-service mathematics teachers' skills in understanding, method, modelling, verification, and extension dimensions of problem solving and their learning style characteristics. The data consisted of the skills pre-service teachers demonstrated in the solution process of open-ended problems. For this purpose, a graded scoring rubric was developed specific to each problem. Regarding the relationships between problem solving dimensions and the characteristics of McCarthy's learning styles, it was assumed that type 1 learners' skills were more dominant in the understanding dimension, type 2 learners' skills in the method and modelling dimensions, type 3 learners' skills in the verification dimension, and type 4 learners' skills in the extension dimension. On the basis of this assumption, problem-solving skills and learning style characteristics were associated and interpreted. The results obtained suggested that 5th year pre-service teachers were better in representing the skills pertaining to type 1 and type 2 learning styles, while 1st year pre-service teachers were better in representing the skills pertaining to type 1 learning style only. On the other hand, it was observed that a great majority of the pre-service teachers had a low level of the skills pertaining to type 3 and type 4 learning styles.

This review of research generates principles for the design of instructional programs that foster critical-thinking dispositions. The dispositional aspect of critical thinking may be considered part of attitudinal memory, readily activated if sufficiently strong. We describe evidence suggesting that ill-structured problem-solving can provide middle schoolers with motivating activities that strengthen critical-thinking dispositions, thus fostering sensitivity to occasions for thinking critically and the inclination for engaging in such practices. The Jasper Series and Decision Making are reviewed as cases of programs for middle level mathematics learning that afford opportunities for ill-structured problem-solving activities that incorporate five important attitude-strengthening elements. Fazio (1995) identified these elements as direct experience, sensory experience, emotional reaction, freely chosen behavior, and attitude rehearsal. We describe how the design of the two mathematics programs incorporates attitude-strengthening elements that can potentially foster critical-thinking dispositions. Based on this review, we present a set of design principles to promote those dispositions. (Contains 2 figures.)

Collaborative problem solving (CPS) is an important 21st-century skill for academic and career success, and as a result, there is increased interest among businesses and educational institutions in the assessment and development of CPS skills. CPS skills are difficult to measure using traditional forms of assessment, and that difficulty has led to the use of computer environments that allow individuals to interact in complex situations and capture all actions throughout the process. In the current paper, we describe the design of a collaborative online mathematics task and explore student perceptions of the task and the extent to which the task elicits CPS skills. Results revealed areas of difficulty for students and showed that the task mostly elicited skills associated with the social dimension of CPS. We describe modifications to the task to provide better opportunities for students to display a broader range of CPS skills. An upcoming larger scale pilot study will explore whether the redesigned task can elicit the less observed CPS skills.

This paper describes a program for increasing problem solving and critical thinking skills in mathematics. The targeted population consisted of one kindergarten, one first- and one second-grade classroom in an elementary school located in an older established community 12 miles west of Chicago. The problem was documented through student performance on textbook generated tests, teacher-made tests, California Achievement Tests, teacher observations, and anecdotal comments of other teachers and parents. Analysis of probable cause data revealed that there was little emphasis on problem solving and critical thinking and too much emphasis on rote learning and drill. In addition, teachers felt that they relied too heavily on their own explanations and did not allow their students to actively generate independent solutions. Finally, it was seen that there was a lack of challenging subject matter utilizing higher order thinking skills. A review of solution strategies suggested by authorities in the field and analysis of the problem setting resulted in the selection of mathematics and children's literature with related activities as the intervention to motivate higher order thinking skills. (Contains 26 references and 4 figures of data; appendixes contain pretests, survey instruments, a 16-item annotated bibliography, ranking and graphing sheets, performance assessment forms, and activity sheets.) (Author/RS)

In the contemporary education 4.0 landscape, teachers are urged to prioritize the communication, collaboration, critical thinking, and creativity (4C) skills during teaching and learning, recognized as crucial skills for the 4.0 industrial revolution (I.R 4.0). This research aimed to develop and validate an instrument assessing mathematics teachers’ perceptions of the 4C skills through problem-solving teaching method. Employing a quantitative research design, the study utilized a questionnaire for data collection, involving four experts and 120 participants. Descriptive analysis using Cronbach's alpha and exploratory factor analysis (EFA) revealed a high reliability (Cronbach’s alpha=0.934) and factor eigenvalue exceeding 1. The KMO values (Kaiser–Meyer–Olkin) for each construct were 0.50, and Bartlett’s Test was significant (<0.5). Additionally, each item demonstrated a factor loading value above 0.50 and a variance percentage of ≥60%. The instrument comprised four sub-constructs and 16 fitting items. In summary, the study affirms the utility of this instrument in investigating mathematics teachers’ perceptions of 4C skills through problem-solving teaching methods.

In the contemporary education 4.0 landscape, teachers are urged to prioritize the communication, collaboration, critical thinking, and creativity (4C) skills during teaching and learning, recognized as crucial skills for the 4.0 industrial revolution (I.R 4.0). This research aimed to develop and validate an instrument assessing mathematics teachers’ perceptions of the 4C skills through problem-solving teaching method. Employing a quantitative research design, the study utilized a questionnaire for data collection, involving four experts and 120 participants. Descriptive analysis using Cronbach's alpha and exploratory factor analysis (EFA) revealed a high reliability (Cronbach’s alpha=0.934) and factor eigenvalue exceeding 1. The KMO values (Kaiser–Meyer–Olkin) for each construct were 0.50, and Bartlett’s Test was significant (<0.5). Additionally, each item demonstrated a factor loading value above 0.50 and a variance percentage of ≥60%. The instrument comprised four sub-constructs and 16 fitting items. In summary, the study affirms the utility of this instrument in investigating mathematics teachers’ perceptions of 4C skills through problem-solving teaching methods.

In the contemporary education 4.0 landscape, teachers are urged to prioritize the communication, collaboration, critical thinking, and creativity (4C) skills during teaching and learning, recognized as crucial skills for the 4.0 industrial revolution (I.R 4.0). This research aimed to develop and validate an instrument assessing mathematics teachers’ perceptions of the 4C skills through problem-solving teaching method. Employing a quantitative research design, the study utilized a questionnaire for data collection, involving four experts and 120 participants. Descriptive analysis using Cronbach's alpha and exploratory factor analysis (EFA) revealed a high reliability (Cronbach’s alpha=0.934) and factor eigenvalue exceeding 1. The KMO values (Kaiser–Meyer–Olkin) for each construct were 0.50, and Bartlett’s Test was significant (<0.5). Additionally, each item demonstrated a factor loading value above 0.50 and a variance percentage of ≥60%. The instrument comprised four sub-constructs and 16 fitting items. In summary, the study affirms the utility of this instrument in investigating mathematics teachers’ perceptions of 4C skills through problem-solving teaching methods.