Secondary Mathematics Teachers:

Beliefs and the Implementation of the Function Approach

Pamela R. Hudson Bailey

George Mason University

Currently student seeking PhD in Mathematics Education Leadership

 

 

 

 

 

 

 

 

Abstract

A mixed method study assessing teacher beliefs with regard to reform-based teaching and learning using a function method in secondary mathematics is assessed using two different styles of professional learning.  Method A involves teacher participation in a Math and Science Project grant while Method B is school district based.  Both methods focused on tasks that were high in cognitive demand, the Rule of 5, and the National Council of Teachers of Mathematics Process Standards.  Data was collected through interviews, observations, and pre- and post-surveys.  Method A participants attended small group monthly sessions along with monthly online sessions that are whole group.  Method B participants attended quarterly meetings based on the course.  Both groups attended summer sessions to set the foundation for the sustained follow-up meetings. 

 

 

 

 

 

 

 

 

 

 

 

 

Secondary Mathematics Teachers:

Beliefs and the Implementation of the Function Approach

Introduction

What comes first?  Professional learning focused on changing teacher beliefs about reform-based instruction OR professional learning focused on educating teachers about new strategies that when implemented with support will result in changing teacher beliefs?  What role does the success of students affect teacher beliefs?  The chicken and the egg concept repeated with teacher beliefs and content implementation strategies.  So which should come first?  Is there an answer that fits all teachers and students?  These questions are prevalent in our state as teachers are encouraged to implement reform-based approaches to teaching high school algebra.  A professional learning opportunity through the Math and Science Project (MSP) grant (Method A) on the topic was offered during the summer of 2010 and throughout the following school year with a limited number of participants from three school districts.  Method B, similar in content, was offered to all remaining algebra teachers in one of the school districts with less participant involvement hours.  This study is designed to look at teacher beliefs before and after the professional learning along with how student achievement and engagement effects beliefs.  The purpose of the study is to determine if 

1.      teacher beliefs regarding teaching and learning mathematics changed after participation in one of two different professional learning sessions and follow ups,

2.      student achievement and reactions effect teacher beliefs, and

3.      years in education and level of education of the teacher plays a role in teacher beliefs and willingness to change,   

Literature Review

Being aware that teachers need to change their beliefs about mathematics will lead to professional learning session expectations changing to meet needs of reform-based teaching and learning.  Weidemann and Humphrey (2002) posit that every professional learning session should be based on a set of goals while Cwikla (2002) states that setting small and large goals will help to acknowledge success with small incremental changes that lead to instructional improvement.  The overall goal should be to increase teacher knowledge and pedagogical content while establishing the vision for change (Weiss and Pasley, 2006). 

The professional learning sessions exposed participants to tasks that are reflective of the function (reform-based) approach to teaching and learning algebra so they might experience feelings similar to students during a lesson.  Planning, units and lessons, is centered on the  NCTM Process Standards (2000) of communication, connections, problem solving, reasoning and justification, and representations, and the Rule of 5 which includes concrete, symbolic, graphical, tabular, and math talk.  Professional learning sessions included discussions and time to practice questioning skills, creation of assessments, and developing and maintain cognitively demanding tasks.  Anticipated benefits for teachers are an increase in content knowledge, pedagogical content knowledge, and self-confidence to facilitate instruction as well as creating and developing lessons and tasks high in cognitive demand. 

In order to realize benefits of the sessions teacher beliefs needed to change or evolve but what causes or affects a teacher’s beliefs?  A teacher’s beliefs, as stated by Weiss and Pasley (2006), will take time to change.  If their beliefs do not correlate with the goals of the professional learning then the implementation will be ineffective.  Content knowledge, pedagogical knowledge, and student epistemology are all governed by an individual’s beliefs.  Guskey (1986) views change as taking time due to the cyclic nature of change as teachers apply a new concept and note the change in student results which leads to the change in beliefs.  Administrative requirements affected teacher beliefs with comments made throughout the sessions about what is expected when a teacher is being observed by the principal. Teachers are held accountable through quarterly benchmark and state assessment scores.  Change brings trepidations about student achievement on these assessments which are perceived as a direct reflection of teacher ability. 

A teacher’s beliefs predict how they will practice instruction in their classrooms (Cross, 2009; Harel & Lim, 2004; Nathan & Koedinger, 2000a; Nathan & Koedinger, 2000b; Sztajn, 2003).  Some teachers take new ideas of reform mathematics and adjust it to fit their own beliefs which was explained by Cross and Sztajn.  Others state that “…this is the way I’ve learned it…so I am teaching the way I’ve learned it (Harel & Lim, p. 28).”  But how can teachers practice something they have never seen or experienced was a question posed by Orrill (2006).  Professional learning sessions need to place teachers in a state of disequilibrium thereby allowing them to experience learning using reform ideas.  Sztajn disclosed that the current mathematical reform movement was expected to alter a teachers’ ideological vision or beliefs.  The reverse was the reality; teachers’ ideological beliefs influence their understanding of reform.   

Cooney, Shealy & Arvold (1998) discuss that if the belief is not based on evidence then providing evidence will not influence change.  In contrast if the belief is based on evidence then teachers can challenge the evidence and modify their belief based on outcomes.  Teachers need to believe that students can perform, apply thinking skills, and are willing and wanting to learn the material (Uworwabayeho, 2009).  Having an open mind and a willingness to change one’s beliefs will enable a teacher to learn from their students.

Professional learning sessions should involve activities that challenge teachers, causing them to doubt evidence they hold that supports their beliefs and then encourage them to reflect (Cooney, Shealy, & Arvold, 1998).  Activities need to make connections with the curriculum and world, encouraging collaboration.  Centering on the reform movement, teachers will take on the role of change agents, increasing student achievement and engagement (Cooney, Shealy, & Arvold; Cwikla, 2002).  Collaboration among peers was stressed by Lappan (2000) and Olson and Kirtley (2005) as playing an important part in change, calling on teacher’s expertise to discuss, create, and analyze lessons and assessments.  Being involved in collaborative sessions leads to instructional improvement and teachers having more confidence to learn and share with each other (Cwikla).  Given the opportunity to collaborate, teachers will garner support from each other as they grow, change, and learn from their students on how to meet their needs (Guskey, 1986; Lin, 2004). 

Weiss and Pasley (2006) posit that teacher weaknesses in content knowledge and pedagogical practices are partly to blame for poor student performance in mathematics.  Teacher’s procedural skill knowledge is not the concern; truly understanding concepts and connections is a problem.  The search for connections will lead to improved pedagogical content knowledge.  So who should be included in professional learning?  Do you only include teachers if their beliefs correlate with the new approach and goals or do you jump in, educate, and support teachers so they might see a change in their students by providing evidence that will lead to a change in beliefs?  Method A participants volunteered for the professional learning however some of Method B participants were participating due to school district requirements.  Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010) state that “It is important not to wait to provide professional learning until the entire school community is united around a common vision (p. 33).”  Teacher beliefs play a role with planning and instruction so must be considered, along with students, when developing sessions. 

Research Questions

            The two research questions addressed teacher beliefs before and after professional learning sessions from the teacher viewpoint and the influence of their students.  Question #1 is “Will sustained professional learning sessions focused on reform positively affect teacher beliefs toward approaching teaching and learning via functions?” and #2, “How do the actions and reactions of students affect teacher beliefs and teacher pedagogical content knowledge?”  

Methodology

Participants

Teachers had the option of attending the summer and school year sessions or just the summer.  A large number of participants dropped out of Method A professional learning after the summer session was completed (29 down to 15).  Some who did not continue during the school year stated that the requirements of the sessions was more than they wished to do while teaching.  Of those that completed both sessions only 11 turned in the post-survey.  Method B professional learning summer session began with 36 teachers however the school year follow up sessions averaged 12 for Algebra II and 10 for Algebra I.  Since the sessions were not required some teachers elected to participate in other professional learning offerings within the school district.  Only ten of Method B completed the post-survey. 

Three of the teachers, Bethany from Method A, Carl and Donna from Method B sessions agreed to be interviewed and observed.  Bethany, a six year veteran is teaching Algebra II and Advanced Algebra II.  Donna, with eleven years of experience, is currently teaching Advanced Algebra II classes.  Carl is teaching Algebra II and has been in education for about four years.  The makeup of students in the classrooms of the interviewees is very similar so all teachers being interviewed are working with comparable levels of students (Table 1).  Math 8 and Algebra I Standards of Learning assessment scores were compared for these teachers to determine the percentage of students passing and those passing advanced.      

TABLE 1

Student Abilities in Interviewee Classrooms

                        Percent Passing           Percent Passing     Percent Passing     Percent Passing

                        Math 8 SOL                Advanced                 Alg 1 SOL            Advanced

                                                              Math 8                                                      Alg 1

Bethany              

    Algebra II        56.40                        38.50                           92.70                      7.30

    Adv. Alg II     23.70                        76.30                           51.30                    48.70

Donna                 

    Algebra II        ------                          ------                             ------                     ------

    Adv. Alg II     28.90                        71.10                           51.10                    48.90

Carl                     

    Algebra II        56.60                        32.10                           89.30                    11.00

    Adv. Alg II     ------                          ------                             ------                     ------

Data Sources

Each type of professional learning began with participants completing the Conceptions of Mathematics Inventory (CMI), a teacher/faculty survey from the National Science Foundation provided by Online Evaluation Resource Library (OERL, 1994).  A Likert scale was used on the survey with value choices of 1 to 6 with 6 representing beliefs that reflected the function standards-based approach to a 1 representing traditional beliefs.  For the purpose of reporting a score of 1 or 2 will be considered traditional, 3 or 4 as wavering, and 5 or 6 as reform.  Star and Hoffman (2005) used the same survey with students to determine their mathematical beliefs in the categories of composition, structure, status, doing, validating, learning, and usefulness.  The Star and Hoffman categories are defined in Table 2 and aligned with categories given by OERL as shown in the last column, faculty/staff categories.  Titles of the categories for teachers are content, classroom management, academic profession, teaching profession, student understanding, methods, and practical value are aligned with the categories by Star and Hoffman based on these definitions. 

TABLE 2

Alignment and Definition of Categories

Star & Hoffman                                                                                                       Faculty/Staff

Student Categories            Traditional                             Reform-Based                   Categories

Composition                Facts, formulas, and                Concepts, principles, and        Content

                                    algorithms                               generalizations

Structure                     Collection of isolated              A coherent system                  Classroom

                                    pieces                                                                                      Management

Status                          Static entity                             Dynamic field                         Academic

                                                                                                                                    Profession

Doing                          Process of obtaining                Process of sense-making         Teaching

                                    results                                                                                      Profession

Validating                   Via mandating from an           Through logical thought          Student

                                    outside authority                                                                     Understanding

Learning                      A process of                            A process of constructing       Methods

                                    memorizing intact                   and understanding     

                                    knowledge.    

The interviews explored teacher’s experiences in the professional learning sessions, their beliefs about teaching and learning mathematics, and perceived effects of change on students.  Interviewees were observed twice in the classroom setting to assess if actions matched words and survey results.  Archived data of the interviewee’s current students included math 8 SOL scores (2008) and Algebra I SOL scores (2009).  In January 2011 those that had taken the pre-survey were requested to take a post-survey to determine if there were changes in their beliefs. 

Data Analysis

            For completing participants, mean values for each of the categories, pre- and post, was determined.  A Paired-Samples t-test was performed to determine significance of the difference between the mean values for pre- and post-survey results.  Observations and interviews, along with the survey results, helped to give the total picture of how the interviewees perceived learning and teaching using the function approach.  Talking or assessing students was not permitted in the school district so discussions were limited to the teachers.

Professional Learning Sessions

            Method A, a four day summer session offered through a local university grant, was followed by monthly follow-up sessions online as a whole group and monthly small group meetings among the participants.  Method B, a two day summer session, had quarterly follow ups held throughout the school year for the teachers of Algebra I and of Algebra II, separately, to better meet their needs.  This group is from the same school district which encourages collaboration among teachers during common planning. 

The objectives for Method A participants is to experience and develop lessons based on the functions approach, learn about and create cognitively demanding rich tasks, make connections between topics, and collaborate with peers.  Focusing on the Rule of 5 and the NCTM Process Standards, teachers planned, facilitated, and reflected on lessons with students the center of their self-analysis.  Presentations were made to the whole group about high cognitive demanding tasks, processes used to solve problems, and justification of their solutions and procedures.  Connections were continually made to real-world situations and to additional mathematical concepts.  Manipulatives are a challenge for many secondary teachers so were incorporated into sessions, concentrating on when and how they might be implemented.  Planned discussions during the sessions included grouping methods to encourage collaboration, how to maintain a high cognitive demanding lesson, and the importance of higher level questions being determined during the planning stages. 

Monthly follow up sessions for Method A continued to center on new mathematical concepts needed to facilitate revised state standards.  Teachers worked in groups online and presented their approach and solution to the other participants using technology.  Monthly face-to-face small group meetings, a requirement of the grant, was a time for teachers to create stations for students to review concepts prior to the state assessment, discuss topics from the online sessions, and reflect and plan lessons.  Small groups went through the lesson study process of researching and developing a lesson to be presented in the spring 2011.  Total time participants were engaged in Method A professional learning was 72 hours of summer and school year face-to-face and online meetings.  

The goals are the same for Method B but opportunities to experience lessons using the function approach were less due to shortened involvement time.  Summer sessions focused on experiencing lessons presented using the function approach, understanding and creating cognitively demanding rich tasks, and questioning techniques to maintain the demand.  Quarterly follow-up sessions centered on participants experiencing additional lessons that focused on new standards using the function approach and discussions about the following quarter concepts.  Separate sessions were planned for Algebra I and Algebra II teachers so they had a chance to collaborate to discuss pacing of material, connections they could make, as well as concerns and successes.  In addition to the follow-up sessions, teachers were to meet with their peers at their individual school’s to continue planning.  Total time outside of the school day for those participating in Method B was 20 hours of summer and follow-up meetings.  A case study of each of the interviewees is shared below, relating their interview, observations, and response survey mean scores.

Findings

Case Study 1: Bethany, Reluctance to Enthusiasm

            A few years ago Bethany was reluctant to receive any help or guidance with lessons.  Through professional learning on the Rule of 5 and questioning she began to show a willingness to change by volunteering to participate.  During the interview Bethany posited that the mathematics had not changed but students are learning the concepts differently.  Her view of teaching and learning mathematics evolved from reluctance to enthusiasm as she expressed that the function approach is hands-on and investigative with students figuring everything out.  The time has passed where the teacher is the giver of all information.  Bethany also stated that “mathematics is an exercise of the mind” where students need to learn how to think and problem solve, knowing the constraints, the conditions, and the rules.  Growing more toward reform, Bethany’s pre-survey score of 4.25 (wavering level) increased to 4.88 on academic profession which is on the edge of becoming reform.  She recognized prior to the professional learning sessions that mathematics instruction was changing but her beliefs held her back.  She states that “I also didn’t know what I really wanted them to get out of the lesson…” referring to implementing the reform-based instruction of which she had not experienced

The professional learning session was eye opening for Bethany.  As she worked on problems the expectation was to approach using different methods and manipulatives.  Bethany found both difficult.  Elementary school teachers are well versed on how to use manipulatives correctly however high school teachers, who tend to be more procedural, do not take advantage of manipulatives (Olson & Kirtley, 2005).  Looking at questions and problems with a new lens, Bethany began applying her knowledge about cognitive demand and open ended problems which was also challenging as she struggled to understand the connections.  Her students were working on open-ended problems in one of the lesson observed.  Given a situation groups of students had to problem solve, decide how to represent their thinking and solution, present to their classmates, and be able to answer questions.  Most of the groups applied the Rule of 5 to illustrate and explain their thinking.  Due to not having a collaborative group when developing the lesson, Bethany learns while doing.  While students were working I had the opportunity to talk with Bethany about her expectations of the lesson.  Connections arose during this discussion about the characteristics of the representations that she could prompt students to expound upon such as domain, range, discrete and continuous data.  Instead of giving questions from the text for homework, Bethany had the student’s select three presentations to compare and contrast.  Her results on the pre- and post-survey for teaching profession increased from 4.50 to 4.88 as shown in her increased confidence in her own teaching ability as well as the confidence she has in her students to perform.  Bethany’s average for content in the survey dropped from a pre-survey score of 4.63 to a post-survey score of 4.00, contradicting her actions which are more reform-based.  When questioned about the decline she stated that she didn’t know what she didn’t know, the connections, the big ideas, and the levels of cognitive demand.  The gained knowledge gave her a different view of problems.  She expects more of herself and her students. 

Bethany explained that when planning a lesson she first decides on the content and then an activity.  Trying to anticipate student questions helps Bethany feel more confident but there are some questions that come up during the lesson that she did not anticipate.  She shared that determining the big picture, the big questions, and making connections is a lot to do for one class.  This is probably due to not being a part of a collaborative team.  Collaborative discussions provide teachers an opportunity to discuss lesson topics and questions.  “I miss a lot of opportunities to make connections and don’t even realize it,” states Bethany. 

She believes her students do like math class and that the class is becoming a learning community, stating “…the good kids will keep the annoying kids on task.”  When presenting problems the students love explaining concepts, feeling empowered by their knowledge as they jump up to be “the teacher”.  She divulged during the interview that she “…never thought [she] would get to the point where [she] could just walk around … and not talk.”  Personally, Bethany loves teaching algebra this way and enjoys seeing the students discover concepts.  At the end of the year Bethany wants her students to “not be afraid of math…”

The category of methods increased drastically from 4.75 (wavering) to 5.38 (reform) with Bethany response to mathematics being more than remembering content going from a 4 to 6 as well as recognizing that memorizing formulas may not be helpful when problem solving.   Student understanding also rose from 5.00 to 5.63, where her answer choice of having students reason and problem solve with less teacher input increasing from 3 to 6.  Practical value, while still in the reform level did drop slightly from a pre-survey score of 5.25 to a post-survey score of 5.13 as Bethany struggles with the practicality of the mathematics.  She rated mathematics applicability after high school as 2, dropping her choice from a 4.  She questions whether students will actually need to know the quadratic formula for their work as an adult.   Classroom management, also in the reform level, rose slightly from 5.50 to 5.75.  Bethany increased her response from 4 to 5 revealing that mathematics consisted of many unrelated topics.  This is a reflection of her concerns for not recognizing connections.  Bethany progressed from being reluctant to showing enthusiasm for her own learning and the learning of her students.     

Case Study 2: Donna, Traditionalist to Awareness

            Donna admits that she is very traditional.  Understanding mathematics came when Donna began teaching the concepts and include understanding the procedures, recognizing context clues, and being fluent with numbers.  Donna stated that she was not ready for the professional learning and went into the sessions with her arms crossed and a negative attitude.  She expressed that she is “…a little stubborn so I like to do things, I guess, at my own pace,” feeling forced to change when students are already weak and lacking number sense.  After the majority of sessions had been completed, Donna stated that mathematics is “A bunch of numbers, variables, generally when I think of math I think of algebra because I believe everything starts with algebra.”  She went on to define the function approach as the ability to make connections to multiple concepts in one lesson by “…keeping everything involved with graphing”.  These beliefs correlate with pre-survey and post-survey results in academic profession from 3.75 (traditional) to 4.00 (wavering).    

            Still using worksheets, Donna has progressed to those that pull in multiple concepts and multiple representations.  When asked about how she would teach a specific idea Donna’s response was that she would pull in the characteristics of the graph and of the situation but also stated that she would present the concepts first and then try to make connections.  Content knowledge has been gained by doing the above because she had to recognize the connections herself.  Her pre- and post-survey scores, 3.63 and 3.75 (both wavering) upholds this belief in content; an increase in understanding but at the same time holding on to beliefs about procedural skills being very important.  She discussed not letting her students have calculators during lessons on transforming functions as they needed to understand the formula.  Also the teaching profession, 4.13 to 3.63, methods, 3.88 to 3.88, and student understanding, 4.38 to 3.88, categories are also at the wavering level and either remained constant or fell closer to tradition.  She acknowledges the empowerment and growth that can occur when students are given the opportunity to investigate and discover, stating that “…it gets them more involved so engagement is definitely better as far as everyone needs to know what is going on, before I might have been losing students.”  Donna beliefs are still very strong as she explains how she approached planning a lesson.  “Well we are on quadratics now so when I did quadratics I did not talk about everything [in the past], I didn’t talk about domain and range, I didn’t’ talk about increasing and decreasing intervals but now I am.”  Donna’s response was about what she would do during the lesson, remaining the giver of information, with students in a sit and get mode of learning.  She is trying to pull more information into the lesson so students can see a broader picture.  While not in groups, students are permitted to talk across the aisles but the class culture has not led to a sense of community where students take on the responsibility of learning and helping others to understand.  Classroom management survey results rose slightly from 4.00 to 4.13, still on the wavering level, as Donna still maintains quite a bit of control of the classroom.  She disclosed that real world scenarios and data are very difficult for her to put in a lesson, too many unknowns could happen and time could be wasted.  While admitting that she is still a traditionalist Donna is now aware that the changes have had a drastic affect on the student’s attitude about mathematics.  They are participating, asking, and answering questions.  

Case Study 3: Carl, Open-minded But Cautious

            Carl spoke about how mathematics had changed since he was in high school.  He felt that he was good in mathematics and could perform any procedure presented.  Now he realizes that he can do the procedures but understanding and making connections in another story.  Carl states “I can set up and start to do a table and that would be the only way I would probably ever get an equation” referring to an activity where participants were challenged to find the pattern using pictures or manipulatives.  He goes on to state “I was never really presented with stuff like that….professional development is the first time I have seen things like that or have been prompted to think about things like that.”  Carl’s attitude toward teaching mathematics evolved during the sessions to view the function approach as the opportunity to analyze concepts from different angles, stating “I think definitely [it’s] a positive giving them alternate opportunities or alternate methods to be successful.”  

He asserted that manipulating numbers along with problem solving is what mathematics is all about.  Carl’s survey results parallel with his words as his teaching profession scores fell from 5.25 to 4.88, reform down to wavering (understanding student responses score fell from 6 to 4).  Content survey scores remained constant at 4.38, wavering, as Carl believes that students need to be able to do the computations, memorize the formulas, and perform the procedures as students performed drills during class to repeat procedures in order to “learn”.  Drill was stressed over understanding even though he states “…if they are getting it with one or two examples…”    

            Carl believes that students will be more comfortable “doing” math than he is since he does not like the unexpected.  Survey scores on methods, 5.38 falling to 5.13, was due to Carl selecting a unit lower for the importance of being shown how to work a problem and remembering information.  Student understanding, falling from 5.63 to 4.88, was mainly due to a drastic drop from a 5 to 2 on the importance of a student acknowledging mathematical statements as being true.  Carl changed his choice from 6 to 3 on whether finding solutions to one problem being helpful in other situations resulting in classroom management falling from 5.25 to 4.88.  This contradicts classroom actions when he encourages students to look at other problems on the worksheet to help them decide on a procedure.  A radical change in Carl’s response to the need of mathematics for a student’s future work and that mathematics plays a role in their current lives led to practical value increasing from 4.25 to 5.00

            The professional learning sessions have encouraged the collaborative planning times to be more productive with Carl’s group working together to decide how to present concepts.  By bringing math to life for students, Carl watched students learn mathematics despite their strengths, weaknesses, and beliefs about their own abilities.  He also made the statement that students need to “determine the method they choose to do and what they look at.”  This is a major difference from being told how to do a process to knowing more about themselves and their own thinking abilities.  The difficulty for Carl is planning.  Even with the collaborative group meetings, planning takes time with more creating and less get, sit, and do x number of problems in the text.  Finally Carl hopes that students will “…keep an open mind and to be willing to be challenged.”        

Overall Findings

Both types of professional learning sessions together resulted in changes in teacher beliefs toward reform-based teaching and learning.  All categories except practical value (-0.07993), disclosed a positive average growth ranging from 0.0408 (content) to 0.1726 (teaching profession).  Method A and Method B revealed an average positive change of all seven categories of 0.0669 and 0.0948, respectively. 

Table 3 reveals the mean values per category of the pre- and post assessments completed by the participants for Method A and Method B respectively as well as the amount of change.  A positive change occurred in all areas except practical value for participants in both methods.  The table also discloses the change in beliefs for each of the two methods separately.  Method A did not show growth toward reform teaching and learning in academic profession nor practical value; Method B did not show growth in practical value and classroom management.  Pre- and post-assessment scores were not affected by the teacher’s years of education, level of education, or the professional development method attended.

TABLE 3

Pre- and Post-Survey Results

                                                        Pre-Survey Mean       Post-Survey Mean          Post-Survey –

                                                                                                                              Pre-Survey Mean

Content    Method A                              4.5357                      4.1563                         -0.3794

     Method B                              4.1750                      4.2125                         0.0375

Classroom management                        

     Method A                              5.0938                      5.0938                         0.0000

     Method B                              4.9375                      4.8750                         -0.0625

Academic profession                                                

     Method A                              4.4688                      4.2500                         -0.2188 Method B      4.2250                                                4.4071                      0.1821

Teaching profession                                                  

     Method A                              4.8750                      4.8125                         -0.0625

     Method B                              4.6000                      4.6750                         0.0750

Student understanding                                             

     Method A                              4.0000                      4.0000                         0.0000

     Method B                              4.0500                      3.8861                         -0.1639

Methods  Method A                              3.7813                      3.8750                         0.0937

     Method B                              3.8250                      3.7750                         -0.0500

Practical value                                                            

     Method A                              3.7813                      3.5446                         -0.2367

Method B                              3.9125                      3.9500                         0.0375

Discussion and Conclusion

Some common themes emerged in the observations and interviews: the need to have confidence in their actions, being frustrated with their own learning, and the need for collaboration among peers.  The teachers have confidence in their procedural skills which correlates with how they learned.  During both methods of professional learning, teachers acknowledged being aware of the NCTM Process Standards (2000) as shown by the increase in survey responses toward more reform-based teaching and learning.  Students taking the lead and communicating their knowledge was addressed by all three interviewees.  Donna talks about student engagement in group work, Carl speaks about students discussing what they have learned in a task, and Bethany’s revelation about being able to be quiet and let the students take the lead reveals their growth in students communicating mathematics.  All three also discussed how they were taught mathematics using procedures and working many problems which in turn is how they approached teaching.  Donna and Carl stated that their collaborative groups were an essential ingredient as a support on the road to reform, giving them the confidence to try.  Assurance and the camaraderie of the collaborative team helped Donna overcome negative feelings of being “forced to change” and Carl opening his thoughts to other approaches to a topic.  Bethany didn’t have the support of a collaborative team but became more willing to let those who could help from the professional learning sessions to do so.

            Frustration was mentioned by the interviewees.  Frustration working with manipulatives, the amount of time involved when planning, and the amount of time a lesson may take were mentioned.  Carl stated that looking back over the year that he probably spent too much time on some lessons but experience will remedy the pacing concerns.  Bethany stated that she was frustrated because she did not know what questions or concerns the students may have as well as a fear of not being able to answer.     

Confidence and frustration leads to a common theme of collaboration.  Donna and Carl collaborate weekly with their Algebra II teams where all members in their teams are involved in the school district professional learning sessions.  Bethany was the anomaly of the interviewees of not having collaborative team support but she did take advantage of the support available through professional learning.  Carl shared that his collaborative team evolved during the time frame of the professional learning sessions to focus on how to present concepts, what connections could be made, and what are the big questions thereby increasing teacher confidence.  Cross states that as a teacher’s self-confidence increases so does their subject matter knowledge (2009). 

Teacher beliefs progressed toward reform as they experienced positive reactions and results to their efforts.   The evidence of success reinforced Bethany’s confidence in herself and her students with each productive lesson.  Donna disclosed that “before I might have been losing students here and there for whatever reason but now they are working doing more group work.”    Both methods of professional learning centered on exposing teachers to tasks that reflected the reform movement so they might have some of the same experiences as students.  Some of the participants became immediately “sold” on the methodology, loving the challenge and the connections being made, others expressing concern about not knowing or seeing the patterns in the data, and not being able to make all the connections.  A few teachers wondered that if they struggled with tasks what would the students do.

Discrepancies between actions, words, and the survey are believed to be due to the teacher’s beliefs and/or lack of knowledge or understanding about reform-based teaching and learning.  The small changes in the mean values between pre- and post-survey results may be due to participant involvement in previous professional learning sessions in the school district over the past few years.  Past sessions focused on questioning, big ideas, rule of 5, and the NCTM Process Standards (2000).  The participants in the study were not only knowledgeable about reform ideas but had already begun attempting non-traditional lesson presentations.  They were ready for more help and knowledge. 

The study reveals that collaboration and support are essential ingredients for teacher growth towards reform.  Teachers may believe they are addressing the NCTM Process Standards but knowledge and confidence comes with experience and evidence of results.  In addition, the amount of time spent in the professional learning sessions provided teachers with time to collaborate and learn as they were expected to put in to action what was discussed during the sessions and then reflect on those actions.  Future research on the connections and influence school administration has on teacher growth toward mathematics reform, teacher beliefs, and student achievement for secondary mathematics is needed.

References

Cooney, T. J., Shealy, B. E., & Arvold, B.  (1998).  Conceptualizing belief structures of preservice secondary mathematics teachers.  Journal for Research in Mathematics Education, 29(3), 306-334.

Cross, D. I.  (2009).  Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices.  Journal of Mathematics Teacher Education, 12, 325-346.

Cwikla, J.  (2002).  An interview analysis of teachers' reactions to mathematics reform professional learning.  Presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

Guskey, T. R.  (1986).  Staff development and the process of teacher change.  Educational Researcher, 15(5).

Harel, G., & Lim, K. H.  (2004).  Mathematics teachers’ knowledge base: Preliminary results. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, 25-32. Norway: PME.

Lappan, G.  (2000).  A vision of learning to teach for the 21st century.  School Science and Mathematics, 100(6), 319-326.

Lin, P.  (2004).  Supporting teachers on designing problem-posing tasks as a tool of assessment to understand students’ mathematical learning.  Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, 257-264.  Norway: PME.

Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W.  (2010).  Designing professional learning for teachers of science and mathematics (3rd ed.).  CA: Corwin.

Nathan, M. J., & Koedinger, K. R.  (2000a).  An investigation of teachers’ beliefs of students’ algebra development.  Cognition and Instruction, 18(2), 209-237.

Nathan, M. J., & Koedinger, K. R.  (2000b).  Teachers’ and researchers’ beliefs about the development of algebraic reasoning.  Journal for Research in Mathematics Education, 31(2), 168-191.

National Council of Teachers of Mathematics.  (2000).  Principles and standards for school

mathematics. Reston, VA: Author.

Olson, J. C., & Kirtley, K.  (2005).  The transition of a secondary mathematics teacher: From a reform listener to a believer.  In Chick, H.L. & Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4. Melbourne: PME.

Online Evaluation Resource Library.  (1994).  Conceptions of mathematics inventory.  National Science Foundation.  Retrieved from http://oerl.sri.com/instruments/te/teachsurv/instr55.html

Orrill, C. H.  (2006).  What learner-centered professional learning looks like: The pilot studies of the InterMath Professional learning Project.  The Mathematics Educator, 16(1), 4-13.

Star, J. R. & Hoffmann, A. J.  (2005).  Assessing the impact of standards-based curricula: Investigating students’ epistemological conceptions of mathematics.  The Mathematics Educator, 15(2), 25-34.

Sztajn, P.  (2003).  Adapting reform ideas in different mathematics classrooms: Beliefs beyond mathematics.  Journal of Mathematics Teacher Education, 6, 53-75.

Uworwabayeho, A.  (2009).  Teachers’ innovative change with countywide reform: A case study in Rwanda.  Journal of Mathematics Teacher Education, 12, 315-324.

Weidemann, W., & Humphrey, M. B.  (2002).  Building a network to empower teachers for school reform.  School Science and Mathematics, 102(2), 88-93.

Weiss, I. R., & Pasley, J. D.  (2006).  Scaling up instructional improvement through teacher professional learning: Insights from the local systemic change initiative [Policy Briefs].  Consortium for Policy Research in Education, 44.