Critical Points for Critical Thinking

1. Critical Points for Critical Thinking Peter Appelbaum Arcadia University

2. Is it hard to teach critical thinking? • Are we always disappointed? • “There has probably never been a time in the history of American education when the development of critical and reflective thought was not recognized as a desirable outcome of ... school. Within recent years, however, this outcome has assumed increasing importance and has had a far-reaching effect on the nature of the curriculum.” Harold Fawcett, 1938

3. Our Students: Select the significant words and phrases in any statement that is important, and ask that they be carefully defined. Require evidence supporting conclusions they are pressed to accept. Analyze that evidence and distinguish fact from assumption. So after all these years we should be able to say we’ve figured out Fawcett’s goals

4. Our Students: Recognize stated and unstated assumptions essential to the conclusion. Evaluate these assumptions, accepting some and rejecting others. Evaluate the argument, accepting or rejecting the conclusion. So after all these years we should be able to say we’ve figured out Fawcett’s goals

5. Our Students: Constantly reexamine the assumptions which are behind their beliefs and actions. So after all these years we should be able to say we’ve figured out Fawcett’s goals Can we say this now? Should we? Should we reconsider our assumptions?

6. Can we re-conceptualize critical thinking? “A climate should be established in the classroom that places critical thinking at the heart of instruction ... to give students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervades all mathematical activity.” -- NCTM Standards Challenge:

7. Can we re-conceptualize critical thinking? “Teachers can easily correct the products, but there is no direct access to the individual (internal) processes of constructing. Thus, on the surface of the official classroom communication, everything can be said and presented acceptably, but the hidden strategies of constructing may lead the child astray in other strategies or in the face of even minor variations.” –Heinrich Bauersfeld Challenge:

8. Can we re-conceptualize critical thinking? “… there is no help from mathematics itself, that is, through rational thinking or logical constraints, as teachers often assume. Mathematics does not have self-explaining power, nor does it have compelling inference; for the learner there are only conventions.” –Heinrich Bauersfeld Challenge:

9. More research will help us figure out what to do in the classroom? Erna Yackle: “It is necessary for teachers to understand that students’ activity is reflexively related to their individual contexts, and that the teacher contributes, as do the children, to the interactive constitution of the immediate situation as a social event.” What’s the question?

10. Can research help us figure out what to do in the classroom? Alfie Kohn: “a brief smile and nod are just as controlling as a dollar bill -- more so, perhaps, since social rewards may have a more enduring effect than tangible rewards.” What’s the question?

11. Peter: Historical haphazard association of mathematics with any sort of rationality or clarity of thought Idea of Cultural Construction (what people think mathematics “is”) If I am steering my students toward an objective, then even critical thinking is not “critical thinking”! What’s the question?

12. Peter: Accept students as critical thinkers. (Don’t think, “I teach critical thinking skills.) Think: Students enrich their abilities and deepen their conception of themselves as critical thinkers through classroom experiences. What’s the question?

13. How do we enact critical thinking in the classroom?? • Ask: In what ways can we…? • Recognize the critical thinking that our students are capable of. • Generate classroom activities that allow or enable acceptance of critical thinking. • Choose from options for classroom activities.

14. Critical Points for Critical Thinking • Imagine a lesson as a path through space. There are particular moments of critical change at which the flow has a sudden shift in acceleration towards a critical thinking classroom.

15. Critical Points for Critical Thinking • Imagine a lesson as a path through space. There are particular moments of critical change at which the flow has a sudden shift in acceleration towards a critical thinking classroom. • Treat Mathematical Actors as Mathematical Critics • Make a Choice, Pursue It, & Consider the Consequences • Obsess About Functional Relationships

16. Critical Points for Critical Thinking • Imagine a lesson as a path through space. There are particular moments of critical change at which the flow has a sudden shift in acceleration towards a critical thinking classroom. • Problematize the “Answer” • Problematize the Pedagogy • Understand Mathematics as Rhetoric

17. Critical Points for Critical Thinking • Imagine a lesson as a path through space. There are particular moments of critical change at which the flow has a sudden shift in acceleration towards a critical thinking classroom. • Realize Apprentice Mathematicians and Citizens as Objects of Mathematics • Perform Celebratory Archaeology

18. Treat Mathematical Actors as Mathematical Critics • Old idea: • Students invent their own procedures and algorithms, search for another way to do a problem. • Shift to: • Everyone has to use another person’s strategy or procedure, and participate in discussions of the strengths and weaknesses of each. • Identify situations where each strategy is optimal.

19. Make a Choice, Pursue It, & Consider the Consequences • Old idea: • Structure activity to include critical thinking skills: comparing, contrasting, conjecturing, inducing, generalizing, specializing, classifying, categorizing, deducing, visualizing, sequencing, ordering, predicting, validating, proving, relating, analyzing, evaluating, patterning, ... • Shift to: • Provide open-ended situations, and facilitate the students’ development ofquestions and investigations.

20. Obsess About Functional Relationships • Old idea: • Collect data in explorations and support students’ identification of patterns in the data, search for more than one pattern, articulate more than one rule. • Discuss strategies for winning games. • Shift to: • How do changes in one or more categories of data relate to changes in other categories? • Explore changes in strategy games.

21. Problematize the “Answer” • Old idea: • Students shared different ways they obtained an answer, or used their explanation to justify their answer; they expressed how they got their answer without using any numbers or shape names. • Shift to: • Students offer several possible answers based on their calculations. • Do things that complexify the choice of “answer” based on computations. • Start with the “answer.” Come up with questions.

22. Problematize the Pedagogy • Old idea: • Offer different options for learning a particular topic or skill. (Centers, jigsaw, etc.) • Shift to: • Present the teaching of a topic or skill as a controversy among educators: at least three experiences. Have students evaluate the approaches. • For themselves • For mathematics • For connections to everyday life

23. Understand Mathematics as Rhetoric • Old idea: • Collect examples of the use of mathematics in newspapers and on television. Use them to prompt investigations. • Shift to: • Study how the mathematics is represented. • Consider together why the author chose a mathematical representation over other possibilities. • Examine the role of mathematics in persuasion.

24. Realize Apprentice Mathematicians and Citizen Mathematicians as Objects of Mathematics • Old idea: • Students create original mathematics; they act like apprentice mathematicians learning a craft. • Students behave like discriminating members of a democratic community. • Shift to: • Report on investigations from the perspective of the object of study. • Study the ways in which mathematics is used to turn ourselves into objects of study.

25. Perform Celebratory Archaeology • Old idea: • Assess student understanding and performance with a variety of assessment strategies (tests, performance tasks & rubrics, portfolios, journals …). • Shift to: • Crucial for students to see for themselves that they have learned particular skills and concepts, and that they can apply these in new contexts. • Look back over a period of work and collect important learnings. • Critical Point: design further, interesting investigations.

26. Skills We Need to Achieve Our Goals • New vision of critical thinking as something other than skills to be learned. • Recognize potential critical points. • Must be able to: • treat out students as critical agents. • Make knowledge problematic. • Utilize critical and affirmative dialogue. • Make the case that it is worth trying to make a difference.

27. “Critical Insight” • “In part this suggests taking seriously the need to give students an active voice in their learning experiences. It also means developing a critical vernacular that is attentive to problems experienced at the level of everyday life, particularly as they are related to pedagogical experiences connected to classroom practice.” • Svi Shapiro

28. “Critical Insight” • “An awareness that pervades the ideology of surface description (in which our world is named in particular and distorting ways.)” • Svi Shapiro

29. Are We Willing to? • Avoid searching for the perfect critical-thinking lessons. Re-think current lessons and units in terms of the critical points of practice. • Stop figuring out how to teach critical thinking. Find out how our students are thinking. Document for ourselves the ways in which our students exhibit critical insight in mathematical ways.

30. Opportunity! • When we drop restrictions like certainty or timelessness it is like breaking out of the restrictions of the real number line in algebra:

31. Opportunity! • “Dropping the insistence on certainty and indubitability is like moving off the line into the complex plane … We don’t throw away all sound sense. The guiding principles remain: intelligibility, consistency with experience, compatability with philosophy of science and general philosophy…” • Reuben Hersh