Introduction

The lower mathematics achievement of students of color has received extensive attention (Martin, 2009). Scholars have also documented that mathematics instruction for African American and Latino students often focuses on taught strategies, fragmented procedures, vocabulary out of context, and assessment based on following steps (Ladson-Billings, 1997). However, the variance in instructional quality does not completely explain lower achievement (Shechtman et al., 2010). We understand little about other mechanisms that affect student outcomes (Lubienski, 2002). One mechanism that has not received attention in mathematics is relational interactions between students and teachers (Author, under review).

Theoretical Framework

Research on teacher-child relationships has found them to be predictive of achievement (Pianta et al., 2002). Studies have established that teachers rate relationships with students of color less accurately (Murray et al., 2008) and more conflictual than for white students (Jerome et al., 2009), and that conflict better predicts mathematics achievement than closeness (Pianta & Stuhlman, 2004). Moreover, evidence suggests that these teacher-student interactions can draw on broad stereotypes causing disengagement, misbehavior, or dropping out (Feagin et al., 2001). We need to better understand why teachers perceive interactions negatively, how relationships relate to mathematics instruction, and to what degree this influences achievement.

However, this previous work does not measure specific interactions or use direct observation, and is not mathematics-specific. Given these limitations, this study analyzes relational interactions through direct observation, specific to mathematics. Relational interactions are episodes that mediate instruction, occurring through verbal and nonverbal behavior (Author, under review). In conceptualizing this in mathematics, we detail five mediational modes from prior work: addressing behavior, framing mathematics ability, acknowledging student contributions, attending to culture and language, and relating to parents and families. These serve as a framework to operationalize teachers' interactional behaviors in classrooms. This study explored two research questions:

1.     What are the types, frequency, and intensity of relational interactions in mathematics classrooms?

2.     How do these relational interactions relate to the quality of instruction?

Methods

Participants

Four 4^th and 5^th grade classrooms of approximately 25 students participated in the research (averaged 4 years of teaching experience). The classrooms were located in a K-5 elementary school of 1300 students (85% Latino, 15% African American). The school is  in an urban district in the southwest US. Over 50% of students were designated ELLs and 86% received free or reduced lunch. On the 4^th grade state mathematics test 37% of students achieved proficient or higher (state average 66%).

Data Sources

Field notes: detailed interactions between teachers and students.

Video: was collected in June on one lesson per teacher. The camera captured the entire classroom. Lessons lasted 30-60 minutes and all classes worked on the handshake problem (see below), making the mathematics comparable. However, teachers adapted the mathematics for their own students.

Twenty people are at a party. If each person shakes everybody else's hand once, how many handshakes take place at the party? How many handshakes will take place for 21 people? How does the number of handshakes grow every time someone new arrives at the party?

The lessons were transcribed verbatim. The video was supplemented with field notes when student or teacher talk was not captured.

Analysis

Teaching for Understanding Instrument (TUI): assessed the quality of instruction. We coded dimensions: cognitive depth, explanation & justification, problem solving, and mathematical discourse. Each dimension is scored on a five-point scale (1-low) and has high reliability and validity (Stecher et al., 2005).

Relational Interactions: were coded from video in five layers: 1) identify relational interactions, 2) classify (positive/negative), 3) determine mediational mode (one additional mode was added, "Setting the tone"), 4) code emphasis: extension, repetition, stress in syllabication, word choice, physical gesture, facial expression, omission, and posture, and 5) code intensity. Low (1) intensity interactions had no emphasis. Interactions with at least one emphasis were classified medium (2) or high (3).

Results

            Teachers varied considerably in instructional quality (see Table 1). Mrs. Brown's instruction focused on answers and procedures. Mr. Dylan's instruction was similar. In contrast, Mr. Gray allowed for student generation of strategies and detailed explanations. Mr. Lucia was in the middle, engaging in some depth, but lower in discourse.

Table 1: Mathematics Instruction

Dimension	Teacher
	Brown	Dylan	Gray	Lucia
Depth	1	1	3	3
Explanation	1	2	4	3
Problem Solving	3	3	5	3
Discourse	1	2	4	2

Teachers generated 90 relational interactions, mostly through addressing behavior and acknowledging contributions (see Table 2). There were more negative interactions and they had slightly higher intensity. Gray only engaged positively across dimensions. In contrast, Lucia engaged in negative interactions with lower intensity. Brown's interactions are decidedly negative with high intensity. Dylan's interactions were more moderate.

Table 2: Relational Interactions

Meditational Mode	Teacher				All
	Brown	Dylan	Gray	Lucia
Behavior
Positive	2(1.0a)	0	0	2(1.0)	4(1.0)
Negative	1(1.0)	3(2.7)	0	23(1.8)	27(1.9)
Ability
Positive	0	0	6(2.5)	2(1.0)	8(2.1)
Negative	2(3.0)	2(2.0)	0	0	4(2.5)
Student Contributions
Positive	5(1.6)	6(2.0)	8(2.1)	3(1.7)	22(1.9)
Negative	11(2.5)	3(2.3)	0	2(2.0)	16(2.4)
Culture
Positive	0	0	4(2.3)	1(2.0)	5(2.2)
Negative	0	0	0	1(1.0)	1(1.0)
Families
Positive	0	0	0	0	0
Negative	0	0	0	0	0
Tone
Positive	0	0	3(1.7)	0	3(1.7)
Negative	0	0	0	0	0
Total
Positive	7(1.4)	6(2.0)	21(2.2)	8(1.4)	42(1.9)
Negative	14(2.6)	8(2.4)	0	26(1.8)	48(2.1)

^a Average intensity

Comparing the classrooms instructionally and relationally, we find no clear link between the two. Mr. Gray engaged in substantive mathematics with very positive interactions. Mrs. Brown's class displayed low quality of instruction and more intense negative interactions. Mr. Lucia's instruction was of average quality, but drew considerably on negative behavioral interactions. Finally, Mr. Dylan's instruction was low in quality and his interactions were marginally more negative.

Discussion

We can imagine differing affects of relational interactions on achievement. Does negative behavioral management lower instructional quality to levels of Dylan or Brown? Does the degree of acknowledging student thinking spur or hinder learning? While beyond the scope of this study, the results merit further research on mediating effects of relational interactions on mathematics achievement.

References

Author (under review)

Feagin, J. R., Vera, H., & Imani, N. (2001). The agony of education: Black students at White colleges and universities (2nd ed.). New York: Routledge.

Jerome, E. M., Hamre, B. K., & Pianta, R. C. (2009). Teacher–child relationships from kindergarten to sixth grade: Early childhood predictors of teacher-perceived conflict and closeness. Social Development, 18, 915–945.

Ladson-Billings, G. (1997). It doesn't add up: African American students' mathematics achievement. Journal for Research in Mathematics Education, 25(6), 697-708.

Lubienski, S. T. (2002). A closer look at Black-White mathematics gaps: Intersections of race and SES in NAEP achievement and instructional practices data. The Journal of Negro Education, 71(4), 269 - 287.

Martin, D. B. (2009). Researching race in mathematics education. Teachers College Record, 111(2), 295-338.

Murray, C., Waas, G. A. and Murray, K. M. (2008), Child race and gender as moderators of the association between teacher–child relationships and school adjustment. Psychology in the Schools, 45, 562–578.

Pianta, R. C., La Paro, K. M., Payne, C., Cox, M. J., & Bradley, R. (2002). The relation of kindergarten classroom environment to teacher, family, and school characteristics and child outcomes. The Elementary School Journal, 102, 225–240.

Pianta, R. C., & Stuhlman, M. W. (2004). Teacher-child relationships and childrenʼs success in the first years of school. School Psychology Review, 33(3), 444-458.

Shechtman, N., Roschelle, J., Haertel, G., & Knudsen, J. (2010). Investigating links from teacher knowledge, to classroom practice, to student learning in the instructional system of the middle-school mathematics classroom. Cognition & Instruction, 28(3), 317–359.

Stecher, B. M., Wood, A. C., Gilbert, M. L., Borko, H., Kuffner, K. L., Arnold, S. C., & Dorman, E. H. (2005). Using classroom artifacts to measure instructional practices in middle school mathematics: A two-state field test (CSE Report 662). LA, CA: National Center for Research on Evaluation, Standards, and Student Testing (CRESST), Center for the Study of Evaluation (CSE), & The Regents of the University of California.