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Critical Thinking And Problem Solving In Math

Critical Thinking And Problem Solving In Math

















































Critical Thinking

When examining the vast literature on critical thinking, various definitions of critical thinking emerge. Here are some samples:

  • «Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action» (Scriven, 1996 ).
  • «Most formal definitions characterize critical thinking as the intentional application of rational, higher order thinking skills, such as analysis, synthesis, problem recognition and problem solving, inference, and evaluation» (Angelo, 1995, p. 6 ).
  • «Critical thinking is thinking that assesses itself» ( Center for Critical Thinking, 1996b ).
  • «Critical thinking is the ability to think about one’s thinking in such a way as 1. To recognize its strengths and weaknesses and, as a result, 2. To recast the thinking in improved form» (Center for Critical Thinking, 1996c ).

Perhaps the simplest definition is offered by Beyer (1995). «Critical thinking. means making reasoned judgments» (p. 8). Basically, Beyer sees critical thinking as using criteria to judge the quality of something, from cooking to a conclusion of a research paper. In essence, critical thinking is a disciplined manner of thought that a person uses to assess the validity of something (statements, news stories, arguments, research, etc.).

Characteristics of Critical Thinking

Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking, «Ambiguity and doubt serve a critical-thinking function and are a necessary and even a productive part of the process» (p. 56).

Another characteristic of critical thinking identified by many sources is metacognition. Metacognition is thinking about one’s own thinking. More specifically, «metacognition is being aware of one’s thinking as one performs specific tasks and then using this awareness to control what one is doing» (Jones & Ratcliff, 1993, p. 10 ).

In the book, Critical Thinking, Beyer elaborately explains what he sees as essential aspects of critical thinking. These are:

  • Dispositions: Critical thinkers are skeptical, open-minded, value fair-mindedness, respect evidence and reasoning, respect clarity and precision, look at different points of view, and will change positions when reason leads them to do so.
  • Criteria: To think critically, must apply criteria. Need to have conditions that must be met for something to be judged as believable. Although the argument can be made that each subject area has different criteria, some standards apply to all subjects. «. an assertion must. be based on relevant, accurate facts; based on credible sources; precise; unbiased; free from logical fallacies; logically consistent; and strongly reasoned» (p. 12).
  • Argument: Is a statement or proposition with supporting evidence. Critical thinking involves identifying, evaluating, and constructing arguments.
  • Reasoning: The ability to infer a conclusion from one or multiple premises. To do so requires examining logical relationships among statements or data.
  • Point of View: The way one views the world, which shapes one’s construction of meaning. In a search for understanding, critical thinkers view phenomena from many different points of view.
  • Procedures for Applying Criteria: Other types of thinking use a general procedure. Critical thinking makes use of many procedures. These procedures include asking questions, making judgments, and identifying assumptions.

Oliver & Utermohlen (1995) see students as too often being passive receptors of information. Through technology, the amount of information available today is massive. This information explosion is likely to continue in the future. Students need a guide to weed through the information and not just passively accept it. Students need to «develop and effectively apply critical thinking skills to their academic studies, to the complex problems that they will face, and to the critical choices they will be forced to make as a result of the information explosion and other rapid technological changes» (Oliver & Utermohlen, p. 1 ).

As mentioned in the section, Characteristics of Critical Thinking. critical thinking involves questioning. It is important to teach students how to ask good questions, to think critically, in order to continue the advancement of the very fields we are teaching. «Every field stays alive only to the extent that fresh questions are generated and taken seriously» (Center for Critical Thinking, 1996a ).

Beyer sees the teaching of critical thinking as important to the very state of our nation. He argues that to live successfully in a democracy, people must be able to think critically in order to make sound decisions about personal and civic affairs. If students learn to think critically, then they can use good thinking as the guide by which they live their lives.

Teaching Strategies to Help Promote Critical Thinking

The 1995, Volume 22, issue 1, of the journal, Teaching of Psychology. is devoted to the teaching critical thinking. Most of the strategies included in this section come from the various articles that compose this issue.

  • CATS (Classroom Assessment Techniques): Angelo stresses the use of ongoing classroom assessment as a way to monitor and facilitate students’ critical thinking. An example of a CAT is to ask students to write a «Minute Paper» responding to questions such as «What was the most important thing you learned in today’s class? What question related to this session remains uppermost in your mind?» The teacher selects some of the papers and prepares responses for the next class meeting.
  • Cooperative Learning Strategies: Cooper (1995) argues that putting students in group learning situations is the best way to foster critical thinking. «In properly structured cooperative learning environments, students perform more of the active, critical thinking with continuous support and feedback from other students and the teacher» (p. 8).
  • Case Study /Discussion Method: McDade (1995) describes this method as the teacher presenting a case (or story) to the class without a conclusion. Using prepared questions, the teacher then leads students through a discussion, allowing students to construct a conclusion for the case.
  • Using Questions: King (1995) identifies ways of using questions in the classroom:
  • Reciprocal Peer Questioning: Following lecture, the teacher displays a list of question stems (such as, «What are the strengths and weaknesses of. ). Students must write questions about the lecture material. In small groups, the students ask each other the questions. Then, the whole class discusses some of the questions from each small group.
  • Reader’s Questions: Require students to write questions on assigned reading and turn them in at the beginning of class. Select a few of the questions as the impetus for class discussion.
  • Conference Style Learning: The teacher does not «teach» the class in the sense of lecturing. The teacher is a facilitator of a conference. Students must thoroughly read all required material before class. Assigned readings should be in the zone of proximal development. That is, readings should be able to be understood by students, but also challenging. The class consists of the students asking questions of each other and discussing these questions. The teacher does not remain passive, but rather, helps «direct and mold discussions by posing strategic questions and helping students build on each others’ ideas» (Underwood & Wald, 1995, p. 18 ).
  • Use Writing Assignments: Wade sees the use of writing as fundamental to developing critical thinking skills. «With written assignments, an instructor can encourage the development of dialectic reasoning by requiring students to argue both [or more] sides of an issue» (p. 24).
  • Dialogues: Robertson andRane-Szostak (1996) identify two methods of stimulating useful discussions in the classroom:
    • Written dialogues: Give students written dialogues to analyze. In small groups, students must identify the different viewpoints of each participant in the dialogue. Must look for biases, presence or exclusion of important evidence, alternative interpretations, misstatement of facts, and errors in reasoning. Each group must decide which view is the most reasonable. After coming to a conclusion, each group acts out their dialogue and explains their analysis of it.
    • Spontaneous Group Dialogue: One group of students are assigned roles to play in a discussion (such as leader, information giver, opinion seeker, and disagreer). Four observer groups are formed with the functions of determining what roles are being played by whom, identifying biases and errors in thinking, evaluating reasoning skills, and examining ethical implications of the content.
  • Ambiguity: Strohm & Baukus advocate producing much ambiguity in the classroom. Don’t give students clear cut material. Give them conflicting information that they must think their way through.
  • Angelo, T. A. (1995). Beginning the dialogue: Thoughts on promoting critical thinking: Classroom assessment for critical thinking. Teaching of Psychology, 22(1), 6-7.
  • Beyer, B. K. (1995). Critical thinking. Bloomington, IN: Phi Delta Kappa Educational Foundation.
  • Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
  • Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
  • Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
  • Cooper, J. L. (1995). Cooperative learning and critical thinking. Teaching of Psychology, 22(1), 7-8.
  • Jones, E. A. & Ratcliff, G. (1993). Critical thinking skills for college students. National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA. (Eric Document Reproduction Services No. ED 358 772)
  • King, A. (1995). Designing the instructional process to enhance critical thinking across the curriculum: Inquiring minds really do want to know: Using questioning to teach critical thinking. Teaching of Psychology, 22 (1). 13-17.
  • McDade, S. A. (1995). Case study pedagogy to advance critical thinking. Teaching Psychology, 22(1), 9-10.
  • Oliver, H. & Utermohlen, R. (1995). An innovative teaching strategy: Using critical thinking to give students a guide to the future.(Eric Document Reproduction Services No. 389 702)
  • Robertson, J. F. & Rane-Szostak, D. (1996). Using dialogues to develop critical thinking skills: A practical approach. Journal of Adolescent & Adult Literacy, 39(7), 552-556.
  • Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
  • Strohm, S. M. & Baukus, R. A. (1995). Strategies for fostering critical thinking skills. Journalism and Mass Communication Educator, 50 (1), 55-62.
  • Underwood, M. K. & Wald, R. L. (1995). Conference-style learning: A method for fostering critical thinking with heart. Teaching Psychology, 22(1), 17-21.
  • Wade, C. (1995). Using writing to develop and assess critical thinking. Teaching of Psychology, 22(1), 24-28.
  • Bean, J. C. (1996). Engaging ideas: The professor’s guide to integrating writing, critical thinking, & active learning in the classroom. Jossey-Bass.
  • Bernstein, D. A. (1995). A negotiation model for teaching critical thinking. Teaching of Psychology, 22(1), 22-24.
  • Carlson, E. R. (1995). Evaluating the credibility of sources. A missing link in the teaching of critical thinking. Teaching of Psychology, 22(1), 39-41.
  • Facione, P. A. Sanchez, C. A. Facione, N. C. & Gainen, J. (1995). The disposition toward critical thinking. The Journal of General Education, 44(1), 1-25.
  • Halpern, D. F. & Nummedal, S. G. (1995). Closing thoughts about helping students improve how they think. Teaching of Psychology, 22(1), 82-83.
  • Isbell, D. (1995). Teaching writing and research as inseparable: A faculty-librarian teaching team. Reference Services Review, 23(4), 51-62.
  • Jones, J. M. & Safrit, R. D. (1994). Developing critical thinking skills in adult learners through innovative distance learning. Paper presented at the International Conference on the practice of adult education and social development. Jinan, China. (Eric Document Reproduction Services No. ED 373 159)
  • Sanchez, M. A. (1995). Using critical-thinking principles as a guide to college-level instruction. Teaching of Psychology, 22(1), 72-74.
  • Spicer, K. L. & Hanks, W. E. (1995). Multiple measures of critical thinking skills and predisposition in assessment of critical thinking. Paper presented at the annual meeting of the Speech Communication Association, San Antonio, TX. (Eric Document Reproduction Services No. ED 391 185)
  • Terenzini, P. T. Springer, L. Pascarella, E. T. & Nora, A. (1995). Influences affecting the development of students’ critical thinking skills. Research in Higher Education, 36(1), 23-39.
  • Carr, K. S. (1990). How can we teach critical thinking. Eric Digest. [On-line]. Available HTTP: http://ericps.ed.uiuc.edu/eece/pubs/digests/1990/carr90.html
  • The Center for Critical Thinking (1996). Home Page. Available HTTP: http://www.criticalthinking.org/University/
  • Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
  • Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
  • Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
  • Ennis, Bob (No date). Critical thinking. [On-line], April 4, 1997. Available HTTP: http://www.cof.orst.edu/cof/teach/for442/ct.htm
  • Montclair State University (1995). Curriculum resource center. Critical thinking resources: An annotated bibliography. [On-line]. Available HTTP: http://www.montclair.edu/Pages/CRC/Bibliographies/CriticalThinking.html
  • No author, No date. Critical Thinking is. [On-line], April 4, 1997. Available HTTP: http://library.usask.ca/ustudy/critical/
  • Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
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Hawaii students see small gains on annual standardized test — Hawaii News Now — KGMB and KHNL

In the second year of a tougher statewide assessment, Hawaii public school students saw their math and English scores improve.

But the majority of Hawaii students tested in the 2015-16 school year still failed to meet achievement standards in math and nearly half fell short in English, results released Tuesday show.

The Smarter Balanced Assessments were first administered in the islands in spring 2015, and are aligned to the Hawaii Common Core standards.

Officials have stressed that the new standards are tougher, and stress problem solving and critical thinking over rote memorization.

In the school year that ended in May, 51 percent of Hawaii public school students met or exceeded the achievement standards in English Language Arts. That’s up from 48 percent in the 2014-15 school year.

Meanwhile, 42 percent met the standards for math, up 1 percentage point from the previous year.

Smarter Balanced Assessments are administered to students in grades 3 through 8, and 11th grade. Altogether, more than 90,000 Hawaii public school students sat for the test in the 2015-16 school year.

Including Hawaii, some 17 states use the Smarter Balanced State Assessments, and 11 states have reported their results for the 2015-16 school year.

West Virginia performed the worst among the group, with 30 percent of students meeting achievement standards in math and 47 percent meeting standards in English language arts.

In math and English, Hawaii performed better than three of the 11 states reporting scores.

“The second year of results show incremental improvements and our second year of data provides a solid foundation for comparisons moving forward,” said schools Superintendent Kathryn Matayoshi, in a news release. “Our schools are invested in the higher standards of this test and we hope to build on our momentum each year to ensure that our students are prepared for college, careers and community life after high school.”

Among all grades, 11th graders performed the worst in math, with just 30 percent meeting achievement standards. That percentage was unchanged from the 2014-15 school year.

Third graders, meanwhile, performed best in the subject. Some 54 percent met standards, up from 50 percent the year before.

In English, a whopping 56 percent of fifth graders and 11th graders met standards. Seventh graders performed the worst, with 47 percent meeting or exceeding standards.

The state Education Department notes that 11th graders who meet the standards for math and English can be exempted from placement tests at hundreds of colleges nationally.

Based on their scores, nearly 5,600 Hawaii 11th graders already qualify for college-level courses, the DOE said.

"The University of Hawaii system is pleased that SBA scores are trending upward as it reflects the increased preparedness of our high school graduates for college,” said University of Hawaii President David Lassner, in a news release. “UH’s continued acceptance of Smarter Balanced scores for placement means that more high school graduates will have a quicker entry into college level courses, which we know improves the pace and number of college graduates."

Copyright 2016 Hawaii News Now. All rights reserved.

The children in Ms. Hunter’s Head Start classroom are fascinated whenever Ms. Winter, the assistant teacher, brings Barry, a large bat puppet, out of his cave to help them solve math problems. Sitting on Ms. Winter’s knee, Barry tells the children that he needs their help to fix a hole in his cave (a decorated cardboard and papier mâché creation in the corner of the room). The children inspect the cave and find the hole. Ms. Hunter suggests that perhaps rocks will work to make the repair.

Ms. Hunter shows the children a large tray of rocks fairly equal in size and two different-size empty boxes. Barry asks the children questions and reviews vocabulary they might need to talk about the problem: “How will I use the rocks to repair the cave’s hole?,” “How many rocks do you think I need?” Ms. Winter writes the numbers the children predict on a piece of chart paper. “Which is the larger box? Which box is smaller?” Ms. Hunter shows the children how one box fits into the other, and she asks the children to predict which box they think will hold more rocks and which box will hold fewer.

The children take turns placing rocks in the boxes until they observe that both boxes are full. Then the children and Ms. Hunter join Barry in counting the rocks in each box to find out which box holds more. Ms. Hunter writes the two numbers on the chart paper. The children compare the numbers and decide that the larger box holds more rocks.

Barry takes the larger box of rocks and flies off to his cave, with the children yelling advice about cave repairs as he goes.

The problem-solving and critical thinking skills the children use to help Barry fix the cave are an important part of early mathematics development. Young children have plenty of curiosity about the world around them, and they like to figure things out about it (Clements & Sarama 2009, 2012). Mathematics includes important tools for doing just that. Continue reading

About the Authors

Gretchen Butera,​ PhD. is associate professor of special education at Indiana University, in Bloomington, and co-investigator for the Children’s School Success Plus project. As an experienced preschool teacher, Gretchen has an ongoing interest in effective instruction for preschool children with disabilities. gbutera@indiana.edu

Amber Friesen, PhD. is an assistant professor in early childhood special education at San Francisco State University in California. Amber teaches graduate courses and works on research related to promoting learning in all young children, including those with disabilities. afriesen@sfsu.edu

Susan B. Palmer,​ PhD. is research professor at the Beach Center on Disability and the Center for Developmental Disabilities at the University of Kansas in Lawrence. Susan engages in research to support access to the general education curriculum and self-determination for children with disabilities. spalmer@ku.edu

Joan Leiber, PhD. is professor in the Special Education Program at the University of Maryland in College Park. Joan has worked with her colleagues to develop and implement CSS+ (Children’s School Success Plus), an integrated preschool curriculum, in Head Start programs. jlieber@umd.edu

Eva M. Horn,​ PhD. is a professor in the School of Education at the University of Kansas. Eva coordinates the Early Childhood Unified Education teacher preparation program and conducts research on the provision of quality early learning opportunities for all children. evahorn@ku.edu

Marci J. Hanson,​ PhD. is a professor of early childhood special education at San Francisco State University (SFSU). She also directs the SFSU Joint Doctoral Program in Special Education with the University of California, Berkeley.

Carol Czaja,​ PhD. has most recently worked as a field coordinator, coach, and early literacy specialist in research on preschool children at risk of school failure, conducted at Purdue University, Department of Human Development and Family Studies, in West Lafayette, Indiana. Carol previously worked as a special educator.

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