Critical Thinking And Problem Solving In Math

Originally developed by Rotman’s former dean, Roger Martin, integrative thinking is a broad term to describe looking for solutions through the tensions inherent in different viewpoints. Martin noticed that effective CEOs understood that their own world view was limited, so they sought out opposing viewpoints and came to creative solutions by leveraging seemingly opposing positions. For the past seven years, a spin-off group called the I-Think Initiative has been training teachers in the Toronto area on how integrative thinking can build critical thinking in students from a young age.

LADDER OF INFERENCE 

One of the tools Jason Watt learned about in his training is called the ladder of inference. It’s a model for decision making behavior developed by Harvard professors Chris Argyris and Donald Schön. Essentially, it helps students slow down and realize which data they are taking into account when they make a decision and how the data they choose is informed by their past experiences. Assumptions are often made in a split second decision because the brain is wired to prioritize data that confirms the model a person already holds. The ladder of inference is a way to check those assumptions.


Watt first used the ladder in a very basic way; he showed his grade two students an image of a soccer player lying on the ground, one leg up, holding his head. The image was intentionally a little vague. At first Watt’s students concluded that the man had fallen. But as they worked their way up the ladder of inference they began to notice different aspects of the image and add those to their “data pool.”

“Students started to realize there was a lot more going on in the picture just in terms of data than what they first said,” Watt said. For example, students would say the man was hurt. That’s not a data point, it’s an inference. Watt could tease out from them that they thought the man was hurt because he was on the ground, holding his head and had a pained look on his face. “I started getting much deeper, more thoughtful answers from students,” Watt said.

As students practiced using the ladder of inference in various content areas they also started to use it on their own when dealing with social problems. When there is a disagreement, students now use the ladder of inference to back up and think through the data they chose and the assumptions that stemmed from that data. Watt says now students solve problems on their own or ask a friend to help them make their ladders.

“We’ve learned that there’s nothing wrong with questioning, so the kids have become much more willing and accepting of criticism because it’s not really criticism anymore,” Watt said. He feels the integrative thinking tools have naturally encouraged his students to build a growth mindset about all aspects of life because multiple viewpoints or ways to solve a problem are a core part of why integrative thinking works. Difference is the strength of the model.

PRO/PRO

Another integrative thinking tool called the pro/pro chart offers some good examples of how students are learning to think flexibly. Most people are familiar with pro/con charts, but in a pro/pro chart the group thinks through the positives of two different ideas. Rather than deciding between two choices, this tool helps students identify the positive traits of different viewpoints, and then create a third option by merging the good qualities of both.

Watt asked his students to brainstorm ideas for the worst restaurant of all time. When they had a good list of terrible ideas, Watt then asked groups of students to each take one idea and explain why it was the best restaurant of all time. One group had initially proposed a restaurant with no seating would be the worst; they reframed that to say if everyone was standing up they would move through the restaurant faster and turn more of a profit. A second group had said a restaurant in the woods would be terrible; they reframed that as dining under the stars.

“They were coming up with these really good ideas out of a terrible idea,” Watt said. “It helps kids see that they are capable and switches those mindsets.” Watt built on the activity, asking the groups to pitch their ideas in a Shark Tank or Dragon’s Den style contest. Students came up with hilarious slogans and designs for their restaurants and what started as a silly, fun activity became a rich interdisciplinary project with written and oral communication, presentation skills, media literacy, and of course, the process skills that enable them.

“The students now are no longer afraid to think,” Watt said. “They’re being more creative thinkers.” He even uses integrative thinking in math instruction, asking students to use the ladder of inference to determine information in a word problem, or asking them to do Pro/Pro charts for different multiplication strategies and then letting them come up with their own third way. His students’ math scores started skyrocketing, and even better, they no longer felt they weren’t “math people.”

PROVOKING SELF REFLECTION

Jennifer Warren became curious about integrative thinking through her daughter who kept coming home from her grade six classroom saying things like, “we had the most interesting discussion today.” That piqued Warren’s interest.

“The way she was talking about her own thinking developing, I was kind of thinking I didn’t think my students were saying the same kind of things,” Warren said. She wanted to be sure she was provoking the same response from her high school English students at Dundas Valley Secondary School in Hamilton. So when her board of education decided to fund the I-Think training she signed up.

The integrative thinking tools gave Warren a solution to a problem she and many other teachers have struggled with for a long time: how to deepen student thinking. Until then, Warren had tried to do this by modeling what deep thinking looks like. She was confident she could help any student become a strong writer. But the integrative thinking training forced her to ask some hard questions about her instruction and prompted her realization that her students were recreating her example, not creating in on their own.

“It completely flipped what mattered to me in an English classroom,” Warren said. She used to be mostly concerned with the product. Now, “instead of defending a stance, I’m so much more interested in having students reflect on their stance and shift and explain why they shifted. That metacognitive piece is more interesting to me now.”

CAUSAL MODELS

Warren starts the first semester by asking students to do a causal model -- another core integrative thinking tool -- of their values. She asks them to pick three to five things they value, anything from profound qualities like independence or kindness, to passions like music or hockey. They then having to dive deeply into why they value those qualities, what caused that? Often this requires them to have conversations with family about values taught to them from a young age.

She then asks them to make visual representations of their causal models and present them to one another. “I like that because they realize people don’t value the same things that they do,” Warren said. Those causal models go up on the wall as a reminder that everyone in the class is different and that the diversity of values, perspectives and opinions makes them better problem solvers.

Warren teaches a course for students who failed the Ontario literacy exam, a graduation requirement. The kids in this class often don’t have a lot of self confidence and are often missing some key literacy skills, like the ability to elaborate on a topic in writing. The ladder of inference has been an incredible tool to help Warren walk students through their thinking, modeling the tool step by step, climbing up or down the ladder as students offer insights from the text.

“It was such a simple and elegant way to allow someone who couldn’t wrap their head around inferring to do it well,” Warren said. She thinks the visual of a ladder helped these struggling students pin their thoughts to different steps and make connections.

She’s also found the tool to be helpful when she has disagreements with students. She’ll use the language of the tools to describe to students what data she’s using to make conclusions about their work ethic, their attendance, their behavior. But she always asks, “What am I missing.”

“It changes the conversation,” Warren said. It gives her a voice to express her disappointment to students in a way that is transparent and uses the shared language of their critical thinking tools. And because integrative thinking is based on the fact that one’s understanding of something is always incomplete, constantly shifting, there is room for students to be participants in the conversation.

TRUE COLLABORATION

“I’m completely and utterly blown away whenever I use one of these tools with my kids,” said Kristen Slinger, a grade two teacher at Norseman Junior Middle School. Before learning about integrative thinking, Slinger would have said she has been doing collaboration in the classroom for the past ten years. But she’s shifted her definition of collaboration and now sees what she was doing before as merely asking kids to write on the same piece of paper.

“When you use these tools [students] realize that they hit a roadblock when not everyone is participating,” Slinger said. The natural need for every students’ voice in order to solve the problem creates genuine collaboration.

Slinger remembers one boy who came from a Montessori background. He was used to a small school and small classes and was overwhelmed when he joined her class of 20 and the broader school of close to 700 students. Slinger said he was selectively mute until Christmas, an issue she raised with his mother. The news came as a surprise to his mom who said he was very chatty at home. Slinger kept the boy in a consistent group so he could develop trust with a few peers and slowly he realized that they really wanted to hear his opinion.


“It would have taken me probably months longer to get him to that point, but it was that idea that his peers valued what he had to say,” Slinger said. He went from never talking in class to volunteering to be the student who went around to other classes polling students on their favorite lemonade for a project.

Slinger said before she learned about integrative thinking she would get interesting responses from students, but she wouldn’t know how they got to their conclusions. The integrative thinking tools help make student thinking visible. “It’s the thinking that’s been put into the responses and the way it’s been broken down,” Slinger said. When she can see the steps of their thinking she has more ways to push them to go even further.

“I haven’t take a course in a very long time that has reshaped my entire program,” she said.

GETTING STARTED

“The safest way in was by using fiction stories,” Slinger said of her own attempts to use integrative thinking. “Find that story that maybe has that emotional clincher that may have different endings and then stop there and use the ladder of inference to come up with what they think might happen at the end.”

Jason Watt suggests starting with an activity that’s part of the curriculum every year. That way a teacher new to the practice can compare the kind of thinking students demonstrate when using an integrative thinking tool with their previous lesson plan.

One important element of success is choosing a topic that’s engaging to kids, that has multiple entry points and solutions, and that has a real stakeholder. “One of the biggest mistakes is when you give the tension without the problem to be solved from a particular perspective,” said Nogah Kornberg, Associate Director of the I-Think Initiative at the Rotman School of Management.

For example, a grade one teacher offered her students a challenge from the school’s janitor. In the summer the trash is stored outside and becomes infested with bees. In the winter the trash is stored inside and smells bad. What might be a better solution? Giving students the challenge from the perspective of the stakeholder helps them solve the problem for him. If it is just presented as an A or a B solution, they don’t know who to solve for.

Kornberg was a high school teacher herself before becoming part of the I-Think Initiative. She sees the program as offering two things: critical thinking skills and building better citizens.

“We’re seeing quite young students learning how to play the game of school and this is about how to become good thinkers and good questioners of our thinking,” she said. Getting started on this metacognition piece can’t start too young in her opinion. She also sees the tool as a way to empower young people. “Because it’s rooted in problem solving it’s about saying things are the way they are, but we can make them better and I have a responsibility to make them better.”

Rahim Essabhai wholeheartedly agrees with Kornberg; he’s seen the shift in his students. He teaches a class called Business and Cooperative Education for seniors at John Polanyi Collegiate Institute that asks students to work on what big problem for an outside organization over the course of the school year.

“When I have my kids coming back to visit me and they say that this course has gotten them ready for the next stage more than any course they took in high school, I don’t take that lightly,” Essabhai said. And since students are coming up with interesting solutions to problems real businesses and organizations have, they see that their thinking has value.

And he knows students are using the tools beyond his course as well. In a final reflection for his class, one student described how she constantly found herself having to choose between hanging out with her friends and spending time with her little sister. When she did either she felt bad, so she came up with a third option. Once a month she hosted a gathering for all her friends and their little sisters to spend time together.

“They’re not being a passenger in their own life,” Essabhai said. “Nothing is too messy or too tough.” Growing students who feel that way about tough challenges should be an essential function of education.

Here's a challenge for your students to tackle:


Critical Thinking


What is 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.

Why Teach Critical Thinking?

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.

References & Resources

  • 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.

Other Reading

  • 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.

On the Internet

  • 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
  • Sheridan, Marcia (No date). Internet education topics hotlink page. [On-line], April 4, 1997. Available HTTP: http://sun1.iusb.edu/~msherida/topics/critical.html

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