It is a principle of educational psychology that a moderate level of challenge enhances student learning and motivation. Researchers Julianne C. Turner, University of Notre Dame, and Debra K. Meyer, Elmhurst College, reviewed the research surrounding challenge as a motivator and its application in mathematics teaching. They argue that a combination of challenging instruction and positive emotional support is necessary to promote motivation in mathematics classrooms.

Previous research suggests that the standard for moderate challenge may be more modest in real-life math classes than in experimental studies because academic success is a key factor in many students’ self-worth, and failure in school has negative personal and social consequences. In the classroom, activities are compulsory and performed in public. Few students are likely to be motivated in situations in which they believe they have only a 50%chance of success. The target for optimal challenge may be only slightly above the level of their current development. These researchers describe the complex relationship between teaching, learning and motivation in mathematics.

### How teachers present challenge

Turner and Meyer report that students and teachers frequently seem willing to trade the benefits of challenge seeking (competence, pride and enjoyment) for the safety of avoiding mistakes and appearing competent. Students who prefer challenge, however, tend to persist longer at difficult tasks, show greater interest in the subject, and have higher achievement than their peers. These researchers sought to describe the kinds of classroom environments that are likely to support challenge seeking and learning in mathematics. They found that the general classroom climate and specific instructional interactions are related to the way in which challenge is integrated into learning activities and to how it is perceived by students.

The ways in which teachers present challenge can affect understanding and motivation to learn. The National Council of Teachers of Mathematics (NCTM) states that “teachers’ actions are what encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions. The teacher is responsible for creating an intellectual environment where serious mathematical thinking is the norm. The classroom environment communicates subtle messages about what is valued in learning and doing mathematics.”

### Classroom discourse

The two central features of a classroom’s culture are its learning activities and its discourse. Research has shown that when the instructional goal is more oriented to task completion than to conceptual understanding, and when teacher talk dominates classroom discourse, students avoid taking risks. Accepting challenge, using errors diagnostically, and seeking understanding with the help of others are norms that facilitate taking risks to learn difficult material.

Students are more likely to take academic risks when their classroom’s culture is one in which effort more than ability, and admitting misunderstanding more than demonstrating correct answers, are seen as competent mathematical behaviors. But it’s a difficult task to create a culture in which challenging learning is the norm, and mathematical understanding and a positive attitude toward mathematics are goals.

### Instructional practices that increase motivation

Teachers implement challenging instruction by pressing students to be accountable for their thinking. Teachers who press for understanding give feedback rather than answers, ask open-ended questions, and provide cognitive and affective support.

Four characteristics of rigorous instruction include:

- pressing students to give reasons for what they did;
- asking them to examine the similarities and differences among multiple strategies;
- using errors as opportunities to rethink problems rather than just accepting them as “normal”;
- and stressing that collaboration involves individual accountability and reaching a consensus, not simply working together or following the lead of a skilled peer.

Students in such classrooms report a more positive attitude toward mathematics in addition to increased conceptual understanding.

Studies reveal that the affective environment was the most powerful predictor of students’ motivation. Students take risks associated with making mistakes, constructing understanding and justifying solutions when teachers provide trust and support. Several researchers have demonstrated that teachers’ caring for students as learners and people is a central component of a motivational classroom.

Teachers increased student involvement by modeling mathematical thinking, emphasizing important concepts and adjusting instruction to meet student needs. They treated errors as informational, provided feedback on progress, and helped students manage frustration. And they always demonstrated their own enthusiasm for mathematics. Successful teachers were able to balance the demands for accountability, encouragement and support.

A supportive emotional environment appears essential for helping students build the confidence needed to meet conceptual challenges such as justifying solutions. The most positively motivated students had teachers who devoted significant amounts of time to helping students understand why they pressed them and held them accountable for learning.

### One teacher’s middle school classroom

Turner and Meyer describe one middle-school math teacher’s classroom as an example of how teachers can negotiate the delicate balance of helping students understand, holding them accountable for learning, and offering sufficient affective support for challenging mathematics. Both her average and pre-algebra classes reported the highest levels of academic press and teacher support. She begins class with motivational statements, calling the activity an “experiment” and using the analogy of a game, asking students to follow the clues. She represents the problem as a game in which they are expected to take risks because they have to experiment with solutions.

The teacher makes it very clear that the problem is challenging and that students will have to concentrate and put forth effort to solve it: “If you struggle with something, that is very good.” Rather than simply answering students’ questions, she responds with questions designed to suggest ways they can think about the problem. She is willing to change plans to accommodate their interest in the problem. She acknowledges the value of the problem when she changes the homework assignment to allow students to continue working on the problem at home. She orients all activities toward the students’ learning. During homework check, she asks for other ways to solve the same problems, demonstrating multiple solutions.

When a student is confused, she often asks other students to explain solution strategies and asks if that made sense to the other students. She demonstrates that learning and understanding are more important than either curriculum coverage or correct answers, and that students are capable of helping each other achieve this understanding. Also, when students make mistakes or reveal misconceptions, she does not correct them, but instead asks the student to explain. “I’m not sure I understand — come here and show me how you did that. I think you have a good idea, I just don’t know what you did.” In this way, mistakes become an opportunity for learning and provide opportunities for a child to demonstrate her ability to self evaluate and then solve her own misunderstanding.

A norm is established throughout the lessons that challenging problems should be viewed as enjoyable. The climate of the class, the activities and what the teacher says to students, all emphasize challenge, effort and student accountability. But she shows enthusiasm and creates a positive environment in which students are willing to take risks with challenging material.

Turner and Meyer’s research shows that moderate challenge can be highly motivating in mathematics classrooms under certain conditions. Such challenge depends not only on a student’s chance of success but also on the social context of the classroom — on the support that the student can expect from his teacher and peers.

*“A Classroom Perspective on the Principle of Moderate Challenge in Mathematics”, The Journal of Educational Research, Volume 97, Number 6, August 2004, pp. 311-318.*

**Published in ERN September 2004 Volume 17 Number 6**