CREATING A MINDFUL CLASSROOM:  ONE TEACHER’S BEGINNINGS

 

Ron Ritchhart

Project Zero, Harvard Graduate School of Education

Ron@PZ.harvard.edu

 

The research reported here concerns itself with two types of “beginnings.” First, there is the literal beginning associated with the start of the school year.  Within this context, the research looks at how a mindful learning community is initiated and built through the establishment of learning routines and practices.  The second type of beginning is an investigation of the psychological origin or knowledge base from which these teaching practices stem.  These two types of beginnings are explored through case-study research of a middle school mathematics teacher whose classroom exemplifies the principles of mindfulness (Langer, 1989, 199; Ritchhart & Perkins, 2000).  This particular case comes from a larger ethnographic study having four main components:  teacher selection, teacher interviews, classroom observation and videotaping, and a repertory grid methodology.

 

 

There has long been a concern in the mathematics-education community that students’ instruction go beyond the basic development of skills and knowledge to cultivate an understanding of mathematical concepts and cultivate students’ ability to reason and think mathematically (NCTM, 1980; NCTM, 1989).  The justification for this position rests on the awareness that without understanding knowledge and skills are often inert and compartmentalized, leading to inappropriate or inexpert application (Gardner, 1991).  Likewise, without the ability to think and reason, it is impossible to create new knowledge and engage in effective problem solving (Perkins & Salomon, 1987).  However, the ability to reason, think, and understand also have limitations when it comes to performance.   Ability in and of itself does not necessarily imply action.  There must be the disposition to use those abilities as well as a sensitivity to and awareness of occasions for that ability’s use (Perkins, Jay, & Tishman, 1993).  To avoid such an ability-action gap, instruction must extend beyond the cultivation of skills, knowledge, and understanding to the enculturation of a disposition toward thinking and of awareness.  In short, classrooms must strive to be mindful places in which students can become wise, not just smart.

To better understand what such environments might look like and how they are established, this study investigated how exemplary middle school teachers develop and nurture students’ disposition toward thinking and mindfulness.  While this research extended over the course of the school year with six teachers, this particular report focuses on the case of one mathematics teacher and his instruction during the first days of school  The specific questions being addressed are:  How do teachers set the stage for student mindfulness during the first days of school?  And, how does a teacher’s own thinking, values, and beliefs about thinking and the discipline play out in beginning of the year instruction? 

 

Theoretical Framework

Mindfulness.  Langer (1989) describes mindfulness as a facilitative state that promotes increased creativity, flexibility, and use of information, as well as memory and retention.  It is characterized by an increased recognition of possibilities and formation of new categories, openness to new information, and an awareness of more than one perspective.  According to the theory, mindfulness results from drawing novel distinctions, exploring new perspectives, and being sensitive to context while mindlessness is fostered through the premature formation of fixed mindsets, overgeneralizations, automaticity and acting from a single perspective. 

While experiments often focus on the promotion of mindfulness as a temporary state, Ritchhart and Perkins (2000) argue that the cultivation of mindfulness as an enduring trait is a worthwhile and achievable goal of education.  However, the accomplishment of such a goal requires educators to challenge many of the norms of schooling, including traditional conceptions of what it means to be smart and do well in school.  Rather than focusing on developing ability, education for mindful-ness is more dispositionally based.  This means that in addition to developing students’ abilities, such as the ability to consider multiple perspectives, educators must also seek to nurture students’ inclination to engage that ability and a sensitivity to occasions for the appropriate deployment of that ability.  This model of education is more about enculturation in a set of norms and patterns of thinking than the dispensing of knowledge and training of skills (Perkins et al., 1993).

Mental Models.  Just as teachers’ beliefs and conceptions about subject matter, pedagogy, and their students’ abilities have been show to affect classroom practice (Thompson, 1992), it was hypothesized that teachers’ thinking and beliefs; that is, their mental models (Johnson-Laird, 1983) of thinking, would also influence their instruction and their ability to create thoughtful classroom environments.  In particular, teachers’ thinking about thinking might influence the types of thinking they choose to develop and their ability to spot and exploit occasions for students’ thinking.  Just as good mathematics instruction proceeds from a foundation of solid content knowledge (Stodolsky, 1988), good instruction in thinking and mindfulness might likewise be dependent on the teacher’s understanding of what it means to think and be mindful.

 

Methods

This study was designed primarily as an effort to learn from best practice.  Therefore teachers who valued the promotion of student thinking and were effective at doing so were sought out through a process of community nomination (Ladson-Billings, 1994).  Prospective teachers were then screened using criteria of classroom thoughtfulness developed by Onosko and Newmann (1994).  John, an eighth and ninth grade mathematics teacher at an independent school in the West, was identified through this process.  John was a mathematics major in college with nineteen years of teaching experience at the time of the study.  As a study participant, John was interviewed six times and his classroom observed for a total of four weeks during the 1998-99 school year.  Audiotaped interviews served to elicit John’s instructional goals and values, explore his thinking on thinking, and gain his perspective on classroom events.  John’s mental model of thinking was inferred using a repertory-grid methodology (Kelly, 1955) adapted from Munby (1984). Classes were videotaped and fieldnotes taken during each observation in order to develop rich portraits of classroom instruction (see, for example, Ritchhart et al., 2000).

Analysis of John’s first-week instructional practice sought to isolate key instructional moves and features through repeated viewing of the videotapes.  These instructional moves were examined to see how they might relate to the promotion of student mindfulness and to what extent they coincided or differed with traditionally-advocated instructional practices for the first day of school (Wong & Wong, 1997).  John later reviewed and offered comments on this initial analysis.  Analysis of the repertory gird, used to infer John’s mental model of thinking, employed cluster analysis to reveal an underlying structure in the data.  John then reviewed the dendogram produced from this analysis and interpreted its meaning.  Based on this interpretation, a Venn diagram of John’s mental model of thinking was generated to visually represent the relationships he expressed.  This diagram was again reviewed by John.

 

Findings

John’s beginning-of-the-year instruction differs greatly from traditionally-advocated approaches (Wong & Wong, 1997).  Rather than focus on developing management and behavioral routines, John’s initial instruction focuses on the development of learning routines that help to set expectations for how learning will take place in his class.  During the first days of the school year, John’s instruction contains three principal moves:  looking closely, exploring different perspectives, and introducing ambiguity.  These moves clarify John’s expectations, establish an atmosphere of thinking, and help to develop students’ inclination toward thinking.

Within the first minutes of class on the first day of school, John presents his students with a problem taken from the Phantom Tollbooth.   In the book, the character Milo claims that mathematics is magical in that it can make things disappear.  Milo gives the equation, 4 + 9 – 2 x 16 + 1 ÷ 3 x 6 – 67 + 8 x 2 – 3 + 26 – 1 ÷ 34 + 3 ÷ 7 + 2 – 5 = ?, as an example.  John asks the class to work in pairs to figure out what the equation means, adding that he will also work on the equation.  Over the next four days, the class not only solves the problem but also engages in a discussion of order of operation rules, exploring the origin and application of those rules.  Below I present brief highlights from these classes to demonstrate the three principles mentioned above.

In devoting time to this problem and its discussion, John provides students with an opportunity to think beyond the surface of the problem, demonstrating the principle of looking closely.  When his students discover that order of operations rules are the key to solving the problem, John asks, “Who thought of those rules?  Why all that instead of doing it from left to right?”  When the class responds with silence, John adds that he doesn’t know the answer to that for sure, “But the real question is:  Could we do it another way?  Could we do things in a different order?”   Thus, rather than being an exercise in solving an equation by applying previously acquired rules of arithmetic, John uses the problem to explore a much bigger disciplinary issue:  Where does knowledge and truth come from in mathematics?  While the Phantom Tollbooth does not deal with this issue or even that of order of operations, John demonstrates that bigger issues and principles often lurk within problems if one takes the time to look closely.

When one student tentatively answers John’s query with, “Well, it depends,”  the opportunity arises for students to explore multiple perspectives on this problem.  Students have already been presented with the story characters’ perspective.  In addition, they have argued to justify their various answers to the problem and in doing so presented the reasons backing up their own positions.  Now, they are offered the chance to challenge the standard order of operations rules and present an entirely new perspective.  To make this a more real and plausible venture, John has students use their calculators to solve the initial problem.  In doing so, students quickly see that even their calculators, which contain different programming rules, represent a perspective.  As a homework assignment, John asks students to devise new order of operations rules and to try them out.  As the class explores these alternatives, the importance of parentheses and the problematic nature of exponents quickly emerges. As one student states, “ I think that parentheses only exist because we have order of operations, you wouldn’t need them otherwise.”

Throughout his initial instruction, John has taken arithmetic and tried to make it problematic for students in an almost Socratic way.  Rather than present mathematics as straightforward, he has introduced a degree of ambiguity into situations to make students more mindful of what is going on.  An example of this can be seen on the fourth day of school when John asks the class to interpret the different meanings of x2, (x)2, –x2, and –(x)2 when x = –2.  Immediately, the class erupts in opinions as students argue about the meaning of x2  when x = –2.  When the majority of the class is convinced that the answer is –4, John comments, “Where we’re getting bogged down is that we’re trying to remember a rule rather than think about what is going on.  I need you to think about what is going on here. Let’s go back to something that was brought up in the discussion.  What does x2 mean?” The point that a variable has to be treated as an entity just as an expression in parenthesis is treated is then made.  Exasperated, a girl in the second row asks, “Why didn’t you just put the parenthesis in the problem then?”  John turns the question back, “Why didn’t I?’  With a sigh the girl responds, “To make us think?”  John responds and concludes, “Yes, that’s the main reason. This isn’t something just to memorize.  I need you to understand it.”

John’s mental model of thinking (Figure 1) reveals a rich and nuanced conception of thinking.  As part of the repertory-gird process, John generated thirty-two synonyms for thinking, which he placed in six major categories.  The importance of playing with ideas, taking risks, and considering different perspectives emerge in the “open thinking” cluster.  Thinking associated with looking closely can be found in the “keying in” category.  While it is difficult to say what an all-

 

Figure 1.  John ’s mental model of thinking

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


inclusive or complete mental model of thinking might be, John’s model does tend to capture a

variety of types and facets of thinking useful in problem solving, developing understanding, creating new ideas, and decision making.  While one cannot say that John’s instruction is dependent on his mental model of thinking, the richness present in both certainly suggests that, in at least John’s case, his teaching is relatively compatible and in line with his mental model.

 

References

Gardner, H. (1991). The Unschooled Mind. New York: Basic Books.

 

Johnson-Laird, P. N. (1983). Mental Models. Cambridge, MA: Harvard University Press.

 

Kelly, G. A. (1955). The psychology of personal constructs. New York: Norton.

 

Ladson-Billings, G. (1994). The dreamkeepers. San Francisco: Jossey-Bass.

 

Langer, E. (1989). Mindfulness. Reading, MA: Addison-Wesley.

 

Munby, H. (1984). A qualitative approach to the study of a teacher's beliefs. Journal of Research in Science Teaching, 21(1), 27-38.

 

NCTM. (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: National Council of Teachers of Mathematics.

 

NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

 

Onosko, J. J., & Newman, F. M. (1994). Creating more thoughtful learning environments. In J. N. Mangieri & C. C. Block (Eds.), Creating Powerful thinking in teachers and students:  Diverse perspectives (pp. 27-50). Fort Worth: Holt, Rinehart and Winston Inc.

 

Perkins, D. N., Jay, E., & Tishman, S. (1993). Beyond abilities:  A dispositional theory of thinking. Merrill-Palmer Quarterly, 39(1), 1-21.

 

Perkins, D. N., & Salomon, G. (1987). Transfer and teaching thinking. In D. N. Perkins, J. Lockheed, & J. Bishop (Eds.), Thinking:  The second international conference . Hillsdale, NJ: Erlbaum.

 

Ritchhart, R., & Perkins, D. N. (2000). Life in Mindful Classrooms:  Nurturing the Disposition of Mindfulness. Journal of Social Issues, 56(1), 27-47.

 

Stodolsky, S. (1988). The subject matters:  Classroom activity in math and social studies. Chicago: University of Chicago Press.

 

Thompson, A. G. (1992). Teachers' beliefs and conceptions:  A synthesis of research. In D. A. Grouws (Ed.), handbook of research on mathematics teaching and Learning (pp. 127-146). New York: MacMillan Publishing Company.

 

Wong, H. K., & Wong, R. T. (1997). The First Days of School:  How to be an Effective Teacher.