The Affective Component of Teaching Conceptually

Challenging Mathematics in Urban Classrooms

Cecilia Arias, Roberta Y. Schorr, Lisa Warner Rutgers University

With special thanks to the rest of the MetroMath Affect Research Team

The MetroMath Center for Learning and Teaching Mathematics in Urban Schools is funded by the NSF grant # 0333753;

Additional funding comes from the NSF-funded Newark Public Schools Systemic Initiative In Mathematics grant # 0138806.

Motivation for the Study

The prevalence in the student and adult population of negative affect (including profoundly painful or humiliating experiences) in relation to school mathematics is well-known.

Large numbers of students in middle school disengage (where possible) from mathematical thinking.

In many inner-city schools, prevailing mathematical expectations may be low (Pedagogy of Poverty - Haberman, 1991).

Affective knowledge in relation to mathematical learning remains a domain where expertise, by and large, is not offered to teachers in the course of existing professional development.

Goals of the MetroMath Affect Study

To understand the constellation of affective, social, and cognitive structures that encompass the development of mathematical success in students--and how these evolve over the course of a school year;

To understand how teachers interact with students in this process.

Our Underlying Conjecture

Powerful affect, in relation to conceptually challenging mathematics, is a very important component of developing mathematical ability and achievement in students.

The Affective Domain Engagement and Motivation

– Engagement or disengagement in solving problems, and varying levels in between, etc.

Emotions– Curiosity, confusion, anticipation, frustration, annoyance,

anger, fear, threat, defensiveness, pleasure, elation, satisfaction, safety, trust, etc.

Attitudes– Interesting, dull, enjoyable, hateful, frustrating, etc.

Beliefs– What math is, how good I am at it, etc.

Values– What kind of performance or understanding do I value, etc.

Powerful Mathematical Affect

Involves patterns of emotions, attitudes, beliefs, and values that foster children’s intimate engagement, interest, concentration, persistence, and mathematical success.

An important distinction

Mathematically powerful affect (i.e., the affect that enables individuals to do mathematics powerfully) is not the same thing as positive affect.

Mathematically Powerful Affect Involves both positive feelings about mathematics

(e.g., curiosity, enjoyment, elation in relation to mathematical insight, pride, satisfaction) and ambivalent or negative feelings (e.g., annoyance, impatience, frustration, anxiety, nervousness, fear).

Negative feelings occur in safe contexts, so that students (and their teachers) are able to manage and benefit from these feelings.

Frustration with a difficult problem leads to anticipation of learning something new, and increased pride of achievement when the problem is solved (Goldin, Richardson, Schorr, & Shtelen, 2005).

An Emotionally Safe Environment

Some characteristics:– Inquiry and sense making are encouraged.– Mistakes and criticism do not lead to fear,

pain, humiliation, shame or submission.– Students’ experiences include trust,

confidence, dignity, and shared respect.

During this Session:

We will share theoretical constructs that specifically relate to to the development of powerful mathematical affect in students.

We will provide examples (through video) of students involved in heated debate about a mathematical idea.

Methods

SubjectsSubjects

DataData

Three urban middle school mathematics classrooms in New Jersey

(For the purposes of this session, we focus on two classrooms, one that was a part of the formal study, the other was part of the pilot for the formal study. Both classes were located in schools in Newark, NJ)

Videotaped classroom sessions

4 to 5 cycles – 2 consecutive days each cycle

Pre-lesson interview with the teacher

Post-lesson retrospective interviews with the teacher and 4 – 5 focus students

Student artifacts collected

Descriptive field notes compiled

Key Affective Moments

• An occasion in the context of doing or discussing mathematics where significant affect (of a student or the teacher) or a change in affect is expressed or inferred.

• Examples range from passionate argumentation over mathematical ideas to determined disengagement and withdrawal.

Sources of Data

Observations of video (including transcripts of verbal statements)

Stimulated recall interviews with students with questions posed to obtain the student’s explanation of observed events

Background information interviews with students

Interviews with teacher to get information about the student and the events.

Analysis

Informed by research that has been an integral part of the MU research seminar (taught by Schorr, Epstein & Goldin).

Informed by the multi-disciplinary experiences of the faculty and graduate students involved (mathematics education, mathematics, social psychology, cognitive science, urban studies, etc.).

Viewed through four lenses:– Mathematical (cognitive)– Affective– Teacher interventions (actions, behaviors, etc.)– Social interactions

Engagement Structures

Involve a recurring pattern, inferred from observing classrooms, and conducting interviews, that is a kind of behavioral / affective / social constellation within an individual.

Involve recurrent, “idealized” patterns of actions and reactions composed of a situational component and emotional feelings.

Can be better understood by hypothesizing a possible progression of thoughts or ideas leading toward a specific outcome (see Goldin, Epstein, & Schorr, 2007, for a more complete description).

Engagement Structures

Some contribute directly to mathematical engagement and persistence, while others may impede this.

However, they are not viewed as completely “good” or “bad”: – Most or all engagement structures seen

as present within individuals and becoming operative under given sets of circumstances.

Issues relating to “face” and “respect”

One important stimulus for many youngsters in inner-city environments is danger that can arise from an insult (tacit or explicit) by another youngster (or teacher), an act that makes one look wrong or foolish, or lose “face” (Anderson, 2000; Dance, 2002; Devine, 1996; Fine, 1991).

Losing “face”

Anderson (1999): “Life in public often features an intense

competition for scarce social goods in which ‘winners’ totally dominate ‘losers’ and in which losing face be a fate worse than death” (p. 37).

An important aspect of this “code” is to not appear weak and/or perceived as being a “loser.”

The Mathematical Task

Farmer Joe has a cow named Bessie. He bought 100 feet of fencing. He needs you to help him create a rectangular fenced-in space with the maximum area for Bessie to graze.

• Draw a diagram with the length and the width to show the maximum area.

• Explain how you found the maximum area.• How many poles would you have for this area if

you need 1 pole every 5 feet?

“Don’t Disrespect Me”:

Involves person’s experience of a perceived challenge or threat to his or her well-being, status, dignity, or safety.

Resistance to challenge raises the conflict to a level above that of original mathematical task.

Need to maintain “face” supersedes the mathematical issues.

Figure and Ground (Rubin, 2001)

Primary focus of attention That which is in the ‘background’

At times, the mathematics is figure and the social is ground; conversely, the social may be figure and the mathematics is ground.

Engagement Structure 1: “Don’t Disrespect Me”

I aggressively defend my idea

and am unwilling to consider other

ideas

A classmatedisagrees with

my mathematical idea

Note: At this point, both students are engaged withthe task. The focus starts out as a mathematical focus. Math is figure. One is proposing a mathematical idea andthe other is disagreeing with that mathematical idea. Atthis stage, the engagement is directed at the mathematicswhich is figure while social issues are ground.

Focus has changed from the mathematical to the social (social is now figure and math is now ground)

Student perceives the disagreement as a social challenge

“This person thinks my idea is wrong & this could make me look weak, foolish, lose ‘face’”

Therefore threatened

“Being seen as weak is potentially very harmful to me”

Therefore tension (and fear).

“How dare he ‘diss’ my ideas”

Therefore anger

“I can’t let him get away with that”

Therefore aggression.

“What does he know about math anyway”

Therefore contempt and an unwillingness to consider the ideas.

Structure 1: “Don’t Disrespect Me”

I aggressively defend my idea

and am unwilling to consider other

Ideas.

A classmatedisagrees with

my mathematical idea.

Shay Dana

Dana & Shay…a Disagreement

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Dana’s Retrospective Interview

Interviewer: So what do you think? What did you understand from that day?

Dana: That, maybe I was wrong. I don't know whether I was wrong or right. I was just, that day. He was…I couldn't say nothing to him cause he…I was mad at him.

Interviewer: Were you feeling comfortable that day?

Dana: No.Interviewer: Why?Dana: Cause he was trying to prove me wrong.

Data Leading Us to Propose a “Don’t Disrespect Me” Structure1. Dana was initially (prior to public fight) very

involved working at the mathematics of the task (video [V], interview [I], descriptive notes [N]).

2. Dana takes on the leadership of her group:A. Prompts other to work [V+N]B. Dissatisfied if others don’t care about task [V+N+I]

3. Dana very interested in understanding Shay’s group’s solution prior to fight:

A. Lingers at table looking at Shay’s group’s work [V+N]B. Tells own group members that Shay’s group is correct

[V+N]C. Tells interviewer (during stimulated recall interview) that

perhaps Shay was right and she was wrong

More of Dana’s Interview…

Interviewer: So you are uncomfortable often then?

Dana: Not all the time. But when I'm right, I'm not uncomfortable. But when I'm wrong, when they try to prove me wrong, I'm uncomfortable.

Interviewer: Is there anything else that makes you mad?

Dana: Uh.. yeah, when people try to prove me wrong too.

A Second Case to Examine

Nastashia Jamal EfrainTayshawn

The Mathematical Task

Billy's family goat liked to eat grass in their neighbor's yard. Billy's dad told his children that he would like to have a contest to see which child could solve this problem. He purchased some fencing, and asked the children to design a pen that would hold the goat in their yard. Billy's father purchased a total length of 64 yards of fencing, and wants to be sure that all of it is used. Please help Billy win the contest by answering the following questions:

Describe the shape of the pen that would allow the goat to have the smallest area in which to graze.

Describe the shape of the pen that would allow the goat to have the biggest area in which to graze.

Which type of pen do you think is best for the goat? Describe why you feel that way.

(Revised Math Exemplars II, Grs. 6-8)

A Second Case of Powerful Mathematical Affect

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A Different Way of Interacting

What are some of the similarities and differences between the two interactions?– Dana & Shay– Nastashia, Tayshawn, Jamal, Efrain

Additional Structures Involving Engagement or Disengagement

Stay Out of TroubleCheck This Out (…this is interesting)I’m Really Into This (see

Csikszentmihalyi, 1990)Get The Job DonePseudo engagement

Implications

By defining engagement structures we can better understand how to help teachers…

– Recognize, anticipate, and head off potentially unproductive situations;

– Maximize positive interactions;– Change the atmosphere from latent

resentment or hostility to a feeling of justice, fairness, and productivity;

– Keep the intellectual aspects of the task at center stage (figure).

ImplicationsFor teachers to be able to guide their students on pathways of mathematical engagement, we hypothesize that they need:

– a type of knowledge about affect, motivation and engagement, different from knowledge that has previously been delivered to middle school mathematics teachers seeking to teach conceptually challenging mathematics;

– to be able to understand, recognize ahead of time, and handle difficult mathematical situations in which feelings of being challenged may give way to those of anger, fear, contempt, hostility, or humiliation;

– to understand, recognize ahead of time, and make good use of mathematical opportunities for students to feel that a period of frustration has been rewarding, and to experience encouragement, elation, pride, satisfaction, and the maintenance and enhancement of respect.