The Function Approach to Teaching Algebra in Secondary Schools: Mathematics Instructional Leaders Efforts to Change Teacher Beliefs, Provide Support, and Professional Development

Pamela R. H. Bailey

George Mason University

The Function Approach to Teaching Algebra in Secondary Schools: Mathematics Instructional Leaders Efforts to Change Teacher Beliefs, Provide Support and Professional Development

Introduction and Problem

Reform mathematics has been a topic of discussion and promoted by the National Council of Teachers of Mathematics (NCTM) in the Process Standards (NCTM, 2000). More specifically, reform methods such as the function approach to teaching mathematics in high school algebra courses encourages teachers to approach instruction with student-centered methods using multiple representations, data given or gathered, and real-world situations that students can relate to and reason through. The function approach is not spreading; lecture driven, sit and get, classes are still the norm in many schools. Most teachers have never experienced student-centered instruction therefore cannot envision instruction any other way (Orrill, 2006). There is fear and uncertainty in what to do to facilitate a student-centered classroom, how to do it, and a hesitancy or lack of ability to make decisions on the spot during a lesson (Derry, Wilsman, & Hackbarth, 2007). Changing mathematics instruction requires shifting the focus from being a teacher towards being an instructional leader. Understanding the bigger picture of how mathematics instructional leaders assist secondary teachers to support and encourage the implementation of the function approach to teaching algebra will benefit teachers and students so that both are productive.

Research involving secondary mathematics teachers and/or students has centered on teacher content knowledge, or the lack therein, and pedagogical content knowledge, instruction based on conceptual understanding as viewed by NCTM, as influences that affect the implementation of reform (Becker, Pence, & Pors, 1995; Cross, 2009; Cwikla, 2002; Derry, Wilsman, & Hackbarth; Harel & Lim, 2004; Lepik & Kaljas, 2009). Professional development that endorses mathematical reform has also been studied with a teacher focus on various aspects such as content knowledge (Harel & Lim), length of the program (Becker, Pence, & Pors), goals of the professional development matching the participants (Orrill, 2006), and support (Sztajn, 2003; Wilson & Lloyd, 1995). The importance of teachers having the time to network with each other was addressed as very important to their growth as teachers and in changing beliefs to a student-centered environment (Becker, Pence, and Pors; Orrill) such as the functional approach to teaching algebra.

To date, my research on mathematical reform from the perspective of how instructional leaders can be supportive to help change teacher beliefs and encourage implementation of the functions approach has not been found. Existing research on understanding how teachers change their beliefs, and therefore their instruction, will help instructional leaders to plan and support reform implementation and to promote legitimacy for the methodology. At the same time instructional leaders need to keep in mind that meeting student’s needs is not based on one approach, but on knowing the students and meeting their needs. Overall, the focus should be on increasing a student’s mathematical conceptual knowledge and a teacher’s self-confidence and knowledge when facilitating instruction such as the function approach.

Purpose

Assisting individual teachers in their journeys of changing beliefs and practices is limiting, addressing one teacher or a group of teachers during a professional development. However implementation of the function approach or reform mathematics has been hampered by administrators and central office personnel who have power over the teachers as well as the teacher’s own inabilities. The purpose of the study will be on understanding how mathematics instructional leaders can connect with and affect more teachers to successfully implement the function approach to teaching algebra. Instructional leader’s beliefs regarding mathematical instruction will be of the utmost importance if they are to change other’s views. Their ability to work with the teachers, given the constraints of the amount of time permitted for the effort, decision or beliefs held by the school board and individual schools, will need to be addressed prior to beginning any program to transform instruction. Working with the instructional leaders of mathematics teachers so that they might support their entire team will be more beneficial and reach more teachers. At the same time, teachers will have the support of their peers as they collaborate and deal with the changes that they experience during the journey. Using a top down approach to change will add additional layers of support for teachers and administrator buy-in.

Significance of the study

Virginia mathematics Standards of Learning (SOL) have changed and will be implemented this year with the expectations that students will justify their thinking and processes, make sense of the mathematics, and make connections between representations and to the real-world (Virginia Department of Education, 2009). Additionally concepts such as statistics are included in the standards for algebra courses in high school which pose further concerns for teachers and their lack of knowledge of the concept. Past standards allowed for some repetition of concepts with each successive course which is not the case with the new standards where overlapping concepts have been omitted. Changes in the standards will add to a teacher’s frustration as they are used to repeating concepts and approaching the material procedurally with a lot of drill and practice. The amount of material that must be taught is overwhelming without teachers having to repeat past concepts.

Teachers tend to teach to the SOL test, procedurally, which will be changing to have some questions that are interactive and require students to think and reason using their understanding of the concepts. Teachers, and students, have been able to be successful in the past on the SOL tests by using substitution to determine which of the choices is correct. Many teachers have remarked that students do not really need to know the math as long as they know the vocabulary and can substitute values to determine answers. The interactive questions will be worded so that students will need to apply their knowledge which will lead to teachers needing to change their method of instruction to promote retention and conceptual understanding but with the remaining concern and fear that students need to pass the SOL test.

Teachers have expressed their concerns with student’s mathematical achievement and retention of concepts as they progress through the various courses. Besides needing to review past concepts for most of the first marking period, teachers must deal with students that are not academically ready for the mathematics of the course. The result has had drastic effects on advanced level mathematics courses. Those that have attempted to implement the function approach within the limitations of the existing county curriculum maps and standards of learning have stated how the students retain the material and are more engaged in the learning process. Wilson and Lloyd (1995) posit that teachers are more concerned that students will not be able to do the mathematics, make connections between the concepts and to real-world applications, and move fluently between small and large groups. It was also revealed that they felt that they had to take charge and lead the class in order for the students to learn. Teachers need to have faith that their students can think, discover, apply, and even struggle with the mathematical concepts.

Change is not only effected by teacher beliefs but also by leaders, administrators, central office, and the community’s beliefs. Instructional leaders are expected to assist teachers with changes in the standards and the related assessments therefore a better understanding of the instructional leader’s efforts, beliefs, and constraints as they try to change instruction will benefit research and understanding of the beliefs and change involving teachers. Constraints include how the public views the new approach to instruction. They will be either an asset to change or a hindrance. Educating the parents will be necessary so that they will understand the rationale behind the changes and that it is not just another fad that a teacher or school district is implementing. Legitimacy to this initiative is gained by the support given by the professional association with the NCTM Process Standards (2000) and the Virginia Standards of Learning (Virginia Department of Education, 2009). Marion (2002) discusses how organizations are a product of the culture. Instructional leaders need to address the changes in standards, assessment, and instruction on the school district level instead of putting that pressure on the teachers to justify their actions and instruction.

Viewpoints of the instructional leaders on how mathematics should be taught will benefit from research conducted with teachers on changing beliefs (Becker, Pence, & Pors, 1995; Breyfogle, 2005; Cooney, Shealy, Arvold, 1998; Cross, 2009; Derry, Wilsman, & Hackbarth, 2007; Guskey, 2002; Harel & Lim, 2004; Lepik & Kaljas, 2009; Sztajn, 2003; Wilson & Lloyd, 1995) and on how students learn mathematics (Nathan & Koedinger, 2000; Wilson & Lloyd, 1995). Guskey’s model of teacher change begins with professional development that leads to alteration of classroom practices. Changes in the classroom instructional practices will lead to improvements in student achievement. The teacher’s beliefs and attitude will transform if the modifications in students are positive. Professional development will lead to change but there are still the other influences that affect teacher’s choice of instructional methods as mentioned before. In addition to those in power, leaders need to be able to help teachers acknowledge what influences their instruction on how students learn. One of the main influences that reinforce teacher beliefs is textbooks. Word problems and the problem solving questions are placed at the end of a section, after the drill and practice (Nathan & Koedinger, 2000). This tells the teacher that drill and practice should be done first to gain understanding and, if there is time and the students are capable, the harder problems can be tackled.

Leaders need to address mathematical discourse as they lead teachers to change instructional methods to implement reform. To open a discussion and encourage students to ask questions and explore the mathematics may lead to them probing into areas that are uncomfortable for teachers. As the leader of the class, mathematics teachers expect to be the all knowing ones and feel that student questions challenge their knowledge and authority. Derry, Wilsman, and Hackbarth (2007) revealed that discourse improved through professional development when it was planned but interaction on the go remained very much the same. A teacher that was videotaped as he interacted with his students showed progress in the type of discourse initiated in the classroom by reflecting on his actions which led to more thought provoking statements in the class (Breyfogle, 2005). Discourse plays a large role in the NCTM Process Standards (2000) and in the function approach so mathematics leaders need to be cognizant of this concern and take steps to counteract teachers as the giver of information with little or no input from students. All of these concerns and constraints need to be foremost in an instructional leaders mind as they plan, support, and attempt to bring about change in instruction with respect to the function approach to teaching algebra.

Additional Questions

By recording the common themes in the research gathered to date on changing teachers beliefs in an effort to implement mathematical reform, research questions listed below have been developed that concentrate more from the viewpoint of the mathematics instructional leaders. The questions include:

· What forces will influence and/or effect changing mathematics instruction to implementing more of the function approach to teaching algebra?

· How do the instructional leader’s beliefs effect the beliefs of teachers, administrators, and central office?

· How is the school board willing to support innovative approaches, such as the function approach, in order to maintain legitimacy?

· What kinds of support can an instructional leader give teachers and administrators to assist in their efforts to change?

· How can an instructional leader use the research results of teacher change with respect to mathematics reform to benefit change on the district level?

· How might the instructional leader address teachers reflecting on instructional practices, content knowledge, pedagogical content knowledge, support to the teachers, administrators supporting the teachers, and collaboration in a professional development setting?

· What do the instructional leaders know about the function approach, classroom discourse, and assessment?

· How can we use knowledge from research on how students learn mathematics to support teaching and learning mathematics using the function approach?

· How will the loose coupling of school systems help or hinder change in mathematics instruction?

These questions will lead to searches for additional studies on topics centered on the instructional leaders, professional development, function approach to teaching algebra, and change leadership. Constructs that assess change, leadership, beliefs, professional development, and influences on change will also be sought.

Conclusion

Changing mathematics instruction mainly affects teachers but needs to begin at a higher level in a school division. Assistance by mathematics instructional leaders can help by supporting and encouraging the implementation of the function approach to teaching algebra with their secondary teachers. Implementing the function approach in algebra classes affect the outcome and learning that goes on in those classes but also leads to future success in more advanced mathematics courses. If students are successful then so are the teachers. If the teachers are successful so are the schools and the school districts. Ultimately, we are working to creating a society of individuals with thinking and problem solving skills.

Research found thus far has focused on changing teacher beliefs, professional development to change mathematical instruction to implementing the reform movement, and various types of knowledge. We need to step back and look at change from the higher level in school districts. Transforming mathematical instruction is more than a fad but instead is an expectation as stated by NCTM Process Standards (2000) and the Virginia Department of Education Standards of Learning (2009). In addition to the school system there are other factors that influence mathematical change and growth.

The other factors and constraints such as political and social interest will affect how the implementation of the function approach will be accepted. Besides individuals impinging on the change in instruction, textbook companies and other resources used by the teachers will also be effected when the teachers begin gathering their own data and looking for interactive and relevant problems. Resources for the function approach are slowing being created which will be a positive influence and affect teacher reactions to change. Change is not just based or centered on teachers, other factors must also be considered in order for school systems to eventually help to produce productive citizens that are capable to think through and make informed decisions using available information.

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