Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem

Abstract

This article presents findings from a German–Swiss video-based classroom study. The research examines how three basic dimensions of instructional quality impact the development of students' understanding of the Pythagorean Theorem. The study sample comprised 19 German and 19 Swiss mathematics classes. A three-lesson introductory unit on the Pythagorean Theorem was videotaped in all classes. Multilevel analyses revealed both classroom management and cognitive activation to have positive effects on mathematics achievement. The results also provide empirical evidence that cognitive activation and a supportive climate moderate the relationship between mathematics-related interest and mathematics achievement.

Introduction

This article draws on data from a video study to examine the impact of three basic dimensions of instructional quality on students' mathematics achievement. In contrast to previous video studies (Hiebert et al., 2003) that examined lessons in different mathematical content areas, we standardised the content area of the lessons videotaped, and recorded a three-lesson introductory unit on the Pythagorean Theorem in 38 German and Swiss classes. We expected this content standardisation to provide more differentiated insights into how the quality of the learning environment impacts students' achievement in mathematics. The analyses presented in this article are part of the binational study “Quality of instruction, learning and mathematical understanding” (Hugener et al., 2009, Klieme and Reusser, 2003).

Quantity and quality of instruction are important components in models and frameworks of school effectiveness. In their influential analysis of extant empirical evidence, Wang, Haertel, and Walberg (1993) demonstrated that the effects of classroom management and quality of student–teacher interaction (especially the intensity and quality of questions and answers) are about as strong as the effects of cognitive and metacognitive abilities and family background. Moreover, recent studies emphasise that, relative to other determinants of the academic learning process, the impact of both teacher characteristics and instruction is stronger than had previously been assumed (Babu and Mendro, 2003, Lanahan et al., 2005, Scheerens and Bosker, 1997, Wayne and Youngs, 2003).

The above findings raise the question of which features of instruction are associated with stronger gains in student achievement. Although previous meta-analyses and process-product studies have produced comprehensive overviews of features of effective instruction, these lists can only be considered the first steps towards a systematic theoretical conceptualisation (Fraser et al., 1987, Scheerens and Bosker, 1997, Wang et al., 1993). Theoretical and conceptual frameworks developed in recent years provide useful structures for the interpretation and elucidation of empirical findings on how different instructional approaches influence learning processes and learning outcomes (Bolhuis, 2003, De Corte, 2004, Greeno, 2006, Hiebert and Grouws, 2007, Seidel and Shavelson, 2007). All of these frameworks emphasise the importance of students' cognitive engagement.

Recent approaches to instructional research based on constructivist perspectives do not regard learning as an information-processing activity guided by the teacher, but as an individual, self-directed and cumulative process (De Corte, 2004). This idea is reflected in complex models incorporating multiple goals, both cognitive and motivational, that focus on the learning activity of students in terms of their active construction of knowledge and acquisition of skills (Kunter, 2005, Pauli and Reusser, 2006). In the research on classroom teaching, these ideas are often connected within the concept of learning opportunities and uses of instruction that was introduced by Fend (1981) and elaborated by Helmke (2003) as a design for research on teacher effectiveness. The underlying idea is that learning processes cannot be controlled from the outside; rather, the teacher provides learning opportunities that must be perceived and utilised by the student to be effective. Researchers in mathematics education also regard the “opportunities to learn” as a key condition for student achievement. Teachers' allocation of classroom time to particular contents, the learning goals and expectations they set, and the fit between learning content and goals, on the one hand, and students' knowledge, on the other, all influence the opportunities that students have to learn (Hiebert & Grouws, 2007).

However, this conception of instruction is not specific enough to describe the interaction between student learning characteristics and students' uses of learning opportunities, or to predict which instructional features are used and how, and to what effect (Hiebert & Grouws, 2007). Moreover, instruction often varies as a function of knowledge domain or even of the context and skills to be learnt by students (Brophy, 2001, Campbell et al., 2004, De Corte, 2004, Seidel and Shavelson, 2007). More specific pedagogical–psychological theories and ideas about the teaching of mathematics are required. With respect to the promotion of conceptual understanding, which is of particular interest in this article, researchers from various backgrounds have developed approaches stressing the importance of demanding cognitive activities that prompt students to engage with the learning content (Hiebert and Grouws, 2007, Mayer, 2004, Reusser, 2006).

Various attempts have been made to specify features of mathematics instruction that are likely to offer more opportunities to learn and to promote a deeper conceptual understanding of mathematical topics. Klieme, Lipowsky, Rakoczy, and Ratzka (2006) and Kunter et al. (2007) have identified three basic dimensions of instructional quality that link teaching and students' learning outcomes in mathematics classrooms: cognitive activation, supportive climate, and classroom management. We outline these three basic dimensions below, first describing the instructional features that characterise them, and then relating them to constructs from domain-specific approaches to instruction and presenting empirical evidence concerning their effects.

Researchers from various backgrounds have drawn similar conclusions with respect to mathematics instruction: Mathematics instruction that promotes conceptual understanding attends explicitly to concepts and specifies the connections among mathematical facts, procedures, ideas, and representations (Hiebert & Grouws, 2007). Conceptual instruction encourages students to discover and understand the meaning underlying procedures, to discuss the relationships between concepts, to compare different solution strategies, and to solve non-routine problems (Brophy, 2000). New concepts are introduced by building on students' ideas, experiences, and prior knowledge (Greeno, 2006, Reusser, 2006).

Another key feature of mathematical instruction promoting conceptual understanding is the cognitive level of students' activities. Mathematical tasks and problems that make higher cognitive demands on students—or, more generally, mathematical instruction that prompts high levels of cognitive functioning and processing—are regarded as a prerequisite for conceptual understanding (Brown, 1994, Greeno, 2006, Hiebert and Grouws, 2007, Hiebert and Wearne, 1993, Mayer, 2004, Stein and Lane, 1996).

The quality of interaction and participation in classrooms is another important factor (Greeno, 2006). According to Brophy (2000, p. 19) “effective teachers…use questions to stimulate students to process and reflect on content, recognize relationships among and implications of its key ideas, think critically about it, and use it in problem solving, decision making or other higher-order applications. The discourse is not limited to rapidly paced recitation that elicits short answers to miscellaneous questions. Instead, it features sustained and thoughtful development of key ideas. Through participation in such discourse, students construct and communicate content-related understandings”. Grouws and Cebulla (2000) stress the importance of conflict and contradiction during whole-class discussion for students' conceptual understanding.

Klieme et al. (2006) integrated these key features of mathematical instruction—challenging tasks, activation of prior knowledge, and a content-related discourse practice—within the construct of “cognitive activation”. Cognitive activation is an instructional practice that encourages students to engage in higher-level thinking and thus to develop an elaborated knowledge base. In cognitively activating instruction, the teacher stimulates the students to disclose, explain, share, and compare their thoughts, concepts, and solution methods by presenting them with challenging tasks, cognitive conflicts, and differing ideas, positions, interpretations, and solutions. The likelihood of cognitive activation increases when the teacher calls students' attention to connections between different concepts and ideas, when students reflect on their learning and the underlying ideas, and when the teacher links new content with prior knowledge. Conversely, the likelihood of cognitive activation decreases when students are requested to solve mathematical problems and tasks in a standard manner previously demonstrated by the teacher, when many of the questions set are at a low cognitive level, and when the teacher merely expects students to apply known procedures—in sum, when the teacher believes that learning mathematics means the transmission of subject-matter knowledge.

Empirical evidence for the benefits of cognitive activation from classroom studies is still weak. In a re-analysis of the TIMSS video study, Klieme, Schümer, and Knoll (2001) found a positive relationship between cognitive activation and achievement gains in mathematics at the classroom level, but they did not verify their results through multilevel analyses. The results of the British intervention programme “Cognitive Acceleration in Mathematics Education” (CAME; Shayer & Adhami, 2007), which is based on the theories of Piaget and Vygotsky, provide further empirical evidence for the positive effects of cognitively activating instruction: a significant increase in student achievement was observed in the programme classes. However, detailed analyses show that the programme's effects varied considerably across classes. It must therefore be assumed that other aspects of the programme, such as the way in which it was implemented, accounted for its success (Shayer & Adhami, 2007).

Wenglinsky (2002) also found a significant relationship between higher-order thinking skills and students' mathematics performance. However, Wenglinsky's study had some limitations. First, it was cross-sectional in design, meaning that the causal direction of the relationship remains unclear. Second, classroom practices were measured using a teacher questionnaire and not direct observation. Drawing on data from the QUASAR project, Stein and Lane (1996) showed that students benefit from challenging and meaningful mathematical tasks. Like the CAME project, QUASAR did not investigate “regular mathematics instruction”; rather, it involved a time-consuming and staff-intensive reform project and a site-based professional development program.

According to the framework of learning opportunities and uses of instruction, instructional features do not impact students' achievement in a direct manner; rather, the uptake of learning opportunities is thought to depend on various learner prerequisites. Domain-specific interest has been shown to be an important prerequisite for students' cognitive engagement. For example, Rheinberg and Vollmeyer (2000) found that the relationship between domain-specific interest and achievement is particularly strong when students feel challenged by demanding tasks. In other words, demanding tasks—a key feature of cognitively activating teaching—foster achievement gains by moderating the relationship between student interest and achievement. The authors concluded that interest influences achievement only when the situation is recognised as opportunity to increase competence, thus leading to deeper engagement with the material.

The degree to which students cognitively engage with learning content can be assumed to depend not only on the cognitive demands of the tasks but also on a supportive classroom climate (Brophy, 2000, Cornelius-White, 2007, Fraser, 1994, Pintrich et al., 1993). Turner et al. (1998) found students in classes with a more supportive climate to be more engaged and to show more involvement than students in classes with a less supportive climate. In the conceptualisation of Klieme et al. (2006), the construct of supportive classroom climate covers features of teacher–learner interaction such as supportive teacher–student relationships, positive and constructive teacher feedback, a positive approach to student errors and misconceptions, individual learner support, and caring teacher behaviour.

Studies investigating components of supportive climate often draw on self-determination theory. Self-determination theory postulates the needs for competence, autonomy, and relatedness to be crucial for motivational and cognitive learning outcomes and identifies features of the learning environment that fulfil these needs (Deci and Ryan, 1985, Ryan and Deci, 2002). Empirical studies have found that a warm and caring atmosphere and a respectful and appreciative climate, both of which foster the experience of relatedness, to be important preconditions for successful learning (Assor et al., 2002, Reeve, 2002, Ryan and Deci, 2000). Research findings on the relationship between aspects of classroom climate and achievement are mixed, partly because of differing operationalisations of the constructs “climate” and “teacher–student relationship” (Ang, 2005).

Predictions also differ as to whether climate variables can be expected to have direct or more indirect effects on student achievement. The German SCHOLASTIK study did not find any significant relationships between classroom climate and achievement gains in mathematics (Weinert & Helmke, 1997). Gruehn (2000) analysed the state of theory and research from the perspective of the classroom climate approach and concluded that empirical evidence for a direct effect of classroom climate on achievement gains is scarce. Campbell et al. (2004) found no evidence that classroom climate, as rated by observers, influences achievement gains in mathematics. Similar results were reported by Dunn and Harris (1998). In contrast, Goodenow (1993) found a significant relationship between teacher support, as rated by students, and the English grade awarded by teachers. However, Goodenow's sample was very small and the relationship was analysed only at the student level. Yet, there is empirical evidence that teacher support, teacher warmth and respect, supportive teacher feedback, being liked by the teacher, and a sense of belonging to one's classroom influence students' engagement, behaviour and pursuit of academic goals (Butler and Shibaz, 2008, Cornelius-White, 2007, Goodenow, 1993, Ryan et al., 1994, Wentzel, 1997).

It might, therefore, be hypothesised that a supportive climate has more direct effects on student motivation and student effort (see also Klieme et al., 2006) and more indirect effects on academic achievement. Specifically, it can be hypothesised that the classroom climate moderates the relationship between students' interest and academic outcomes: students with the same level of interest are likely to make more effort in classes with a supportive climate than in classes with a less supportive climate (cf. Wentzel, 1997). The relationship between students' mathematical interests and learning achievement might thus be expected to be closer in classes with a positive climate than in classes with a more negatively charged climate.

Opportunity to learn is crucially dependent on the quality of classroom management. In effectively managed classrooms, students are able to spend more time on task and therefore have more opportunity to engage with learning content than do students in less effectively managed classrooms. Providing time to learn mathematics is one of the key features of effective mathematics instruction (Grouws & Cebulla, 2000).

There are various means for developing and sustaining an orderly classroom atmosphere: preventing disruptions and minimising the likelihood of disciplinary problems, on the one hand, and dealing with misbehaviour, disruptions and conflicts, on the other (Borich, 2007). To this end, teachers must be able to establish clear rules and procedures, manage transitions between lesson segments smoothly, keep track of students' work, plan and organise their lessons well, manage minor disciplinary problems and disruptions, stop inappropriate behaviour, and keep a whole-group focus (Evertson, 1989, Kounin, 1970).

From this perspective, effective classroom management can be seen as a critical prerequisite for students' cognitive engagement, in as much as it provides them with sufficient time and an orderly atmosphere in which to engage in content-related activities. Numerous empirical studies have confirmed that effective classroom management has positive effects on student learning (Campbell et al., 2004, Doyle, 1986, Seidel and Shavelson, 2007, Wang et al., 1993, Weinert and Helmke, 1997).

The present study is part of a larger research project using video observation as research methodology. The data are based on observation of teachers and students in the classroom while a specific topic was being taught, namely the Pythagorean Theorem. Drawing on the theoretical insights and empirical evidence outlined in the previous sections, in the present study we posed the following research questions and hypotheses. (a) Which are the effects of the three dimensions of instructional quality on students' mathematics performance when relevant student characteristics and conditions within the classroom are controlled for? We expected cognitive activation and classroom management to have direct effects on students' mathematics performance. However, we did not expect supportive climate to have a direct effect on their mathematics performance (Hypothesis 1). (b) Do our data provide empirical evidence that cognitive activation and supportive climate moderate the relationship between students' interest and their mathematics performance? We expected cognitive activation and supportive climate to moderate the relationship between students' interest and their mathematics performance (Hypothesis 2).