DEVELOPING PERSISTENT PROBLEM SOLVERS WITH a productive disposition toward mathematics

Good problem solvers are flexible and resourceful, demonstrating many ways to think about problems (Suh, Graham, Ferranone, Kopeinig, and Bertholet, 2011). They have “alternative approaches if they get stuck, ways of making progress when they hit roadblocks, of being efficient with (and making use of) what they know. They also have a certain kind of mathematical disposition—a willingness to pit themselves against difficult mathematical challenges under the assumption that they will be able to make progress on them, and the tenacity to keep at the task when others have given up” (Schoenfeld, 2007, p. 60).

In addition, research has linked mathematics anxiety to decreased performance on mathematics exams. Problem solvers experience a range of emotions associated with different stages in the solution process. Dweck (2006), a Stanford University psychologist with research on achievement and success, shares that there are two mindsets about learning: a growth mindset and a fixed mindset. A growth mindset holds that your basic qualities are things that you can cultivate through your efforts, whereas a fixed mindset holds that your qualities are “carved in stone” (i.e., your intelligence is something you cannot change very much).

When facing challenging problems, children who believe that effort drives intelligence tend to do better than children who believe that intelligence is a fixed quality that they cannot change. According to the research on competence and motivation (Elliot and Dweck, 2005; Weiner, 2005), students can attribute their successes and failures to ability (e.g., “I am just [good/bad] at mathematics”), effort (e.g., “I [worked/did not work] hard enough”), luck, or powerful people (e.g., “the teacher [loves/hates] me”). A student with a fixed mindset avoids challenges, gives up easily, sees effort as fruitless or worse, ignores useful negative feedback, and feels threatened by the success of others. Meanwhile, a student with a growth mindset embraces challenge, persists despite setbacks, sees effort as the path to mastery, learns from mistakes and criticisms, and finds lessons and inspiration in the success of others.

People with a growth mindset believe that they can develop their abilities through hard work, persistence, and dedication; brains and talents are merely a starting base (Dweck, 2006). Creating opportunities for success in mathematics is important, but offering students a series of easy tasks can lead to a false sense of self-efficacy and can limit access to challenging mathematics. Ironically, research indicates that students need to experience periodic challenge and even momentary failure to develop higher levels of self-efficacy and task persistence (Middleton and Spanias, 1999). Achieving a balance between opportunities for success and opportunities to solve problems that require considerable individual or group effort requires teachers to design curricular materials and instructional practices carefully (Woodward, 1999).

Dweck’s research also shows that the types of praises and validations that teachers offer have influence on the type of mindset fostered for the students. The research identifies teachers that can have a fixed mindset who tend to perceive the struggling students to be not sufficiently smart in math, whereas those that have a growth mindset would see struggling students as learners who need the appropriate guidance and the necessary feedback to improve. In other words, growth mindset teachers see this challenge working with struggling students as an opportunity to learn from their mistakes and misconceptions.

Before we can develop mathematically proficient students, we need to make efforts to develop persistent and flexible problem solvers by looking deeply at our students, the instructional practices, and the mathematical tasks that we present to them. Teaching with a belief that intellectual abilities can be fostered and cultivated through effective pedagogical practices should be integral part of helping students become successful at mathematics. As mentioned from research, we can start by taking the time to establish classroom norms that embrace mistakes as opportunities for learning and validate students multiple strategies and partial understandings so that we can help complete their understanding by connecting and or repairing their mathematical understanding.

Below are reflective prompts that the authors from that chapter developed to foster persistence, flexibility, and clear communication in their mathematics classroom (See Table 2.1). Such questions like “Did someone else solve the problem in a way you had not thought of? Explain what you learned by listening to a classmate?” can help students consider other approaches to problem solving so that they can begin to evaluate strategies based on their clarity, efficiency, and connections to other shared strategies. Setting up a classroom that values clear and respective communication and flexibility and persistence from the beginning of the school year will help students understand that these are the expectations and the norms for the classroom.

In this same way, teachers need a collaborative support network that can provide the time and space for them to inquire about their instructional practice. Professional learning communities and lesson study teams provide that supportive network for teachers to develop as reflective practitioners who persevere through challenges and work with other colleagues to examine a problem of practice from their daily teaching episodes.

Over the years, we have learned a great deal about how students learn mathematics and how teachers develop their teaching practices as they collaborate on lesson study. Lesson study is a model of professional learning that offers situated learning through collaborative planning, teaching, observing, and debriefing that afford opportunities

Table 2.1 Reflection prompts to encourage persistence and flexibility in problem solving

Clear Communication

Respectful Communication

Flexible Thinking

Persistence

What math words could help us share our thinking about this problem? Choose 2 and explain what they mean in your own words.

Did someone else solve the problem in a way you had not thought of? Explain what you learned by listening to a classmate.

What other problems or math topics does this remind you of? Explain your connection.

What did you do if you got stuck or felt frustrated?

What could you use besides words to show how to solve the problem? Explain how this representation would help someone understand.

Did you ask for help or offer to help a classmate? Explain how working together helped solve the problem.

Briefly describe at least 2 ways to solve the problem. Which is easier for you?

What helped you try your best? or What do you need to change so that you can try your best next time?

If you needed to make your work easier for someone else to understand, what would you change?

What helped you share and listen respectfully when we discussed the problem? or What do you need to change so that you can share and listen respectfully next time?

What strategies did you use that you think will be helpful again for future problems?

Do you feel more or less confident about math after trying this problem? Explain why.

for teachers to reflect individually and collectively. Teacher educators have embraced lesson study, originating from Japan, because it empowers teachers and provides a collaborative structure for eliciting reflection for critical dialogue about pedagogical content knowledge (PCK) among teachers (Lewis, 2002; Lewis, Perry, & Murata, 2006).

Another effective professional development model is called Instructional Rounds (City, Elmore, Fiarman, & Teitel, 2009), which is a practice adapted to education from the field of medicine where practitioners work together to solve common problems and improve their practice. In medicine, the clinical rounds consist of training how to care for patients, presenting the medical problems and treatment of a particular patient to doctors, residents, and medical students. In education, it is designed to help schools, districts, and state systems support high-quality teaching and learning for all students.

Instructional Rounds help teachers examine closely at what is happening in classrooms in a systematic, purposeful, and focused way. Typically, the first step in an Instructional Rounds process is determining a “problem of practice” followed by collective observation and debrief. The shared experiences designed around targeted Instructional Rounds that focused on the instructional practices such as posing rich meaningful problems, modeling using multiple representations, orchestrating mathematics “talk” and responding to students’ questions, and determining how to assess student understanding of mathematical concepts helped teachers focus on important instructional practices. One of the teachers wrote who participated in Instructional Rounds stated in her reflection,

Watching the teachers teach a lesson that was open-ended helped me alleviate some of my fears of teaching through problem solving. I am glad that the teacher allowed students to convince each other and justify their thinking. It allowed for students who approach the problem in different ways, understand a different perspective or solution strategy.

In each of our lesson study and Instructional Rounds, we focus on some of the essential research-based professional development resources that we use as high- leverage practices. These include working with teachers to unpack the mathematics, choose worthwhile tasks through cognitive demand analysis, set goals to enact some of the core teaching practices through research lessons, integrate technology to amplify the mathematics, and use formative assessment to assess their mathematical proficiency.