Function Approach to Algebra for Student Success

(FA2S2)

Pamela R. Hudson Bailey

George Mason University

EDCI 857: Preparation and Professional Development of Mathematics Teachers

Dr. Jennifer Suh

 

 

 

 

 

 

 

 

 

 

Function Approach to Algebra for Student Success

(FA2S2)

Introduction

            The Function Approach to Algebra for Student Success (FA2S2) is a logical next step for teachers of secondary mathematics in Spotsylvania County, Virginia.  The county has five high schools, seven middle schools, and an alternative education center that teaches algebra.  Approximately sixty middle school and high school algebra teachers will have the opportunity to be involved in the professional development sessions.  Multiple representational approaches have been encouraged for the past two years which the teachers refer to as the Rule of 5, taken from approaches used for AP Calculus instruction.  Even though only a handful of teachers have fully embraced the Rule of 5 in their classrooms there has been a concern by all regarding student growth in mathematics.  FA2S2 will be implemented through modeling sample lessons using the Rule of 5 and a focus on the function approach, looking at the graphs first for conceptual understanding.  Participants will also be implementing technology appropriately using the graphing calculator and a computer based ranger (CBR) to gather and illustrate the data.  Collaborating with their peers, participants will develop lessons to meet the standards of the state of Virginia, the process standards from NCTM (2000) and will participate  in a lesson study in the spring.

Needs Assessment

Students are struggling to meet the assessment standards in mathematics for Adequate Yearly Progress (AYP).  Our county scores increased drastically till we reached the 70 to 90 percentile, depending on the high school.  Teachers are becoming stagnated while blaming everyone but themselves (i.e. students, parents, and administration) for lack of growth.  Seventy-five percent of the students that are having difficulty passing the end-of-year assessment belong within one of the AYP subgroups with a large portion being African American.  The attitude of the students in the lower levels of mathematics in the high schools is defensive as many display a no-care demeanor.  Students routinely make negative, derogatory comments as reported by the teachers.  Success in mathematics has been very small for these students.

Professional development sessions have been delivered for the past two years covering curriculum alignment, assessment, and instruction.  The Rule of 5 (graphing, tables, symbolic, math talk, and concrete) and the 5 E’s (engagement, explanation, exploration, elaboration, and evaluation) have been modeled in county wide professional development and stressed that all need to be considered when planning.  Activities were facilitated during sessions that implemented the concepts as well as involved the teachers as students thereby having them experience the difficulties, the connections, and the joy found in a lesson.  Playing the role of a student was to hopefully let the teachers acknowledge what the students will experience as well as to put them in a state of disequilibrium so that they might see the need to step onto the path to becoming a change agent and to changing their beliefs on how students learn mathematics.  Lessons facilitated during the workshop were for teachers to further their pedagogical content knowledge in addition to being models of good instruction.  Teachers were given the opportunity during the workshop to section the curriculum maps, by each quarter, in to “big ideas”.  Follow up sessions involved additional activities facilitated around the concepts as well as different methods of differentiation.  The final assessment for the teachers was for each to develop a lesson within a unit of study determined during the summer workshop. They were to teach the lesson and reflect in writing on how the facilitation of the lesson went and how the students perceived the lesson.  Presenting the lesson to the other workshop participants along with their personal critique was encouraged to spur constructive comments from their peers.  The concern for these sessions is the buy-in from teachers and administration as very few teachers participated.  The professional development helped to make the Rule of 5 known but additional understanding, practice, and assistance with planning are needed.

Besides having a concern for the beliefs of the teachers participating, and seeing the need for change, is the implementation of the new mathematics course, Algebra, Function, and Data Analysis.  This course is based on instruction facilitated within a laboratory environment using exploration and being given or gathering data to enable students to learn algebra from a function approach.  Students have the opportunity to see and experience mathematical concepts in action that are relevant to their lives.  Of the four teachers that are currently instructing the course, only two are teaching the course with the vision in which it was intended.  Traditional instruction takes less time to prepare and teachers have the feeling of more control over what is happening within their classrooms.  Teachers of the students in the traditionally taught classes have stated that the students are not ready nor have the ability or responsibility to think on their own, to approach a problem creatively, or to see multiple concepts and their connections.  When the students do not understand concepts, teachers break it down into even smaller portions which lacks meaning to the students.  It is just a process that they have to memorize versus having meaning.

Virginia is also in the process of implementing new mathematical standards.  Teachers are concerned with the quantity of concepts that will need to be taught and how they will complete the curriculum.  A lack of seeing how teaching using big ideas and multiple concepts with connections will enable them to successfully do so has not been accepted by many.   

Teachers want their students to be actively engaged but are lacking skills and confidence to do so.  The few teachers that have begun the journey to having students engaged in the discovery of mathematical concepts are a wonderful influence for those who have not, but more teachers are needed to jump on the boat.  Developing a plan that will be endorsed and promoted by the central office with the administrators backing is needed.  Ed Laughbaum’s workshop on the function approach to teaching and learning algebra has been attended by two of our teachers.  The comment that each has made upon their return is that the experience has changed their outlook and approach to teaching algebra.  One teacher has been very instrumental in changing her approach to instruction by having her students gather data, explore the mathematics, and with a willingness to let them take the time to do so.  The pay offs have been enormous.  Student’s participation and attendance have gone up, discipline issues have gone down, and attitudes of the teacher and students have positively increased.

Research

            The reform movement in mathematics.

            Students need to understand and apply mathematical concepts to real-world applications.  The Principles and Standards for School Mathematics (NCTM, 2000) gives teachers guidance on how to approach student learning for understanding with the process standards.  Going beyond rote drill to students engaging in problem solving, reasoning why they are doing the process or justifying their thinking, communicating their thoughts in writing and verbally, using multiple representations as an aid in the discovery process or as a mode to illustrate their findings, and making connections between the mathematical concepts or between real-world situations (NCTM).  Being encouraged to explore, explain, and elaborate on the mathematics will allow the students to solve authentic problems that they might encounter (NCTM).  Student motivation is also a concern for teachers.  The instructional method that the teacher selects will influence their students to want to learn more (White-Clark, DiCarlo & Gilchriest, 2008).  Prince and Felder (2006) posit that students need to know how they learn mathematics in order for there to be a transfer of knowledge to long term memory.  Once this transfer is made then it becomes readily available for recall and application.  Weiss and Pasley (2006) state that the goal of professional development should be to increase a teacher’s knowledge however there are many types of knowledge.  Harel and Lim (2004) reveal three types of knowledge that should be addressed in professional development, the knowledge of content, the knowledge of student epistemology, and the knowledge of pedagogy.  Knowledge held by the teacher effects what they teach along with how they teach (Cross, 2009; Cwikla, 2002). Knowing the content is important but being able to facilitate the instruction so others might understand is even more important according (Lappan, 2000).  All three types of knowledge will be addressed as the facilitators of the sessions model lessons, lead teachers in discussions of articles, and as teachers reflect on lessons.  Keeping foremost in the planning and implementation of instruction should be the students (Lappan).  In order for teachers to implement a reform-movement approach to teaching mathematics their beliefs about mathematics need to change. 

Teacher beliefs effect growth.

What happens in the classroom is a reflection of a teacher’s beliefs (Cross, 2009; Harel & Lim, 2004; Nathan & Koedinger, 2000a; Nathan & Koedinger, 2000b; Sztajn, 2003).  Many teachers believe that they are following the NCTM process standards and implementing the ideas of reform mathematics however the ideas are modified to fit in to their specific beliefs (Cross; Sztajn).  The level of education of the teacher will also effect their beliefs according to Weidemann and Humphrey (2002).  As the teacher’s level of education increases some may develop the false ceiling effect, believing that they have obtained all necessary knowledge needed in order to be successful.  Engaging teachers in challenging activities that result in disequilibrium may result in teachers reflecting on their knowledge, the problem solving process, and their motivation to complete an activity (Cross; Harel & Lim).

How a teacher learned mathematics, the courses they took, and their mathematical background effects their beliefs (Cross, 2009; Cwikla, 2002).  Nathan and Koedinger (2000a; 2000b) investigated how teachers rated various types of problems, arithmetic and algebraic, ranging from symbolic to verbal, with and without context.  The results revealed that the teachers believed that verbal problems were the most difficult and that the symbolic would be the easiest for students to complete successfully.  Students were also assessed with a different result, verbal was the easiest with no difference found between problems in context or not and the symbolic problems were the most difficult for them to complete successfully.  Textbooks also influence and confirm teacher’s beliefs in the approach they use when presenting material, symbolic to verbal (Nathan & Koedinger).  This conclusion effects how teachers plan for instruction.  Nathan and Koedinger posit that teacher’s beliefs on what is more difficult for students to complete need to correlate with the students they instruct.  Engaging in problem solving activities will allow teachers to experience what students encounter during an activity so that their beliefs might change with the experience (Derry, Wilsman, & Hackbarth, 2007). 

  Beliefs take time to change. Teachers need to learn how to develop investigative, open-ended problems as many believe they are difficult to develop (Lin, 2004).  Curriculum alignment and flow needs to make sense to the teachers and the students so that connections can be made and investigated (Cwikla, 2002; Lappan, 2000).   Centering the curriculum on the reform movement encourages teachers in their efforts to change and increase student achievement (Cwikla).  Support is also needed.  Collaboration between teachers will help to provide support for the questions that arise in planning, during instruction, and when evaluating student growth (Lin).  

Goal Description

            The goals of the professional development will be:

·         75% - 80% of all Algebra I and Algebra II teachers will participate.

·         Administrators will support summer and school year teacher participation with encouragement and knowledge of the goals of the professional development.

·         To practice and learn pedagogical content knowledge applicable to the implementation of the goals.

·         Four day summer workshop will focus on participants

o   experiencing lessons from a function approach,

o   incorporating technology appropriately from an investigative approach,

o   acknowledging the importance of mathematical discourse,

o   making connections between concepts using the Rule of 5 and to their daily lives,

o   collaborating to solve real-world problems and to develop lessons,

o   the role that assessment should play in guiding instruction, and

o   understanding the lesson study concept and implementation.

·         School year and follow up sessions will focus on teachers:

o   incorporating concepts learned during the summer workshop in to their daily lesson plans,

o   collaborating with peers within their specific school with a focus on students conceptually learning mathematics,

o   going through one lesson study per semester in subject area groups within each school,

o   presenting the lesson developed for the lesson study with their team,

o   saving all lessons to SCORE, a county-wide program that will allow teachers to share ideas and plans,

o    participating in follow up sessions for all teachers so that they might    

      continue learning pedagogy, content, and technology, and

o   participating in follow up sessions to continue discussions on mathematical discourse, assessment, and the function approach.

·         The overall goal is to encourage teachers to be agents of change by transforming their attitudes and beliefs about implementing student-centered instruction from the function approach with knowledge and support.

Activities

The four day summer session will have a combination of presenter/facilitators.  The main presenter will be Ed Laughbaum.  His role is to present the function approach to all the teachers along with making connections and learning technology.  The participants will be divided in to two groups, Algebra I and Algebra II, for a better focus on their needs.  Three expert teachers within the county will also assist in the summer session by facilitating sessions on curriculum and unit/lesson planning, writing key questions for units, and ideas for assessment.  Each day will be split into two parts, one with Ed Laughbaum and one with the expert teachers.  The summer session will occur during the week prior to the teachers coming back to school.  This will enable them to know what they will be teaching and have a focus during the session on their needs for the school year.

Teachers in each school will determine weekly collaboration times for each of the two subject areas.  The lead investigator will attend as many sessions as possible to support and guide the flow of instruction.  Planning unit and daily lesson plans along with the lesson they will be using for the lesson study will be the objective for the collaborative sessions.  High school administrators have been attempting to have teachers collaborate in each of the subject areas but this will bring a higher level of focus to these sessions.  The lead investigator, the mathematics supervisor for the county, and the team of teachers will attend the presentation of the lesson study.  All lessons are to be submitted to SCORE with good explanations for the delivery of the lesson and the vision of how it is intended to be presented.  Follow up sessions will occur once a quarter with the expert teachers leading the sessions in discussions, activities, and sharing of ideas.

The following summer teachers will reconvene for two days to debrief.  Discussions are to include positive and negative student results, additional teacher needs, and any adjustments for the improvement of the implementation of the function approach to teaching algebra.  Curriculum maps will be amended as needed to fit student needs and flow of concepts.  

Timeline

August

Four week summer workshop

Participants will take attitude – belief assessment and content knowledge assessment.

September – November

Weekly collaborations within each school for Algebra I and for Algebra II teachers.

Last week of the month will be follow up session.

December  - January

Weekly collaborations within each school for Algebra I and for Algebra II teachers.

No follow up sessions to be scheduled.

Schedule times for presentations of the lesson study project.

February – May

Weekly collaborations within each school for Algebra I and for Algebra II teachers.

Last week of February and March will be follow up sessions.

Schedule and present lesson study project in April or May.

Participants will take attitude – belief assessment and content knowledge assessment.

June

Two day summer follow up to adjust curriculum maps and discuss the implementation of the function approach.

 

Budget

Item

 

Cost

Ed Laughbaum - 4 Day Session

 

$ 3,000.00

Ed Laughbaum – travel expensive

 

$ 1,200.00   

Substitutes for Lesson Study

     60 teachers needing substitute ½ day @

     $70.00 per day

 

$ 2,100.00

Food

*Four Day Workshop

     Breakfast  60 @ $7.00

     Lunch  60@ $10.00

*Follow Up Session 

*Reconvening – Following Summer

     Breakfast  60 @ $7.00

     Lunch  60@ $10.00

 

 

420.00

600.00

150.00

 

420.00

600.00

 

 

 

 

 

 

 

$2190.00

Expert Teacher Compensation

     3 teachers for 4 days

     2 teachers for 4 follow ups

     Planning for 3 teachers

 

2400.00

400.00

300.00

 

 

 

$3100.00

Materials for workshop

 

 

$910.00

TOTAL

 

$12500.00

 

Evaluation and Accountability

            Teachers in the professional development will be requested to complete a pre- and post-assessment in the form of a questionnaire.  The completion of the questionnaire will be voluntary and consent will be acknowledged by turning in the finished document.  The instrument, Local Systemic Change through Teacher Enhancement Mathematics 6-12 Teacher Questionnaire, was developed by the National Science Foundation (NSF), a federal agency.  Data collected will include information on teaching practices, preparation when planning, opinions on mathematics teaching and learning, and the impact of any professional development, past and current, experiences. 

Outcome

            The overall goal is to have happier and more productive teachers that understand how student learn mathematics.  Engaged students that are discovering and becoming the owners of their mathematical knowledge will lead to deeper levels of instruction and more mathematics courses being taken.  Students will be able to acknowledge why they are learning the concepts and its relevancy to their daily lives.  Teacher’s self-confidence will improve and they will be more willing to step out of their comfort zones with sustained support provided by the professional development sessions, their peers, and the administration.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References

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 development. Presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

Derry, S., Wilsman, M., & Hackbarth, A. (2007). Using contrasting case activities to deepen teacher understanding of algebraic thinking and teaching. Mathematical Thinking and Learning, 9(3), 305-329.

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.

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.

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

Prince, M. J., & Felder, R. M. (2006). Inductive teaching and learning methods: Definitions,

comparisons, and research bases. Journal of Engineering Education, 95(2), 123-138.

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

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

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

White-Clark, R., DiCarlo, M. & Gilchriest, N. (2008). “Guide on the side”: An instructional

approach to meet mathematics standards. The High School Journal, 91(4), 40-44.