FinalReport

 Lindsay Tucker
 * Geometric Proofs ISD Report **

Executive Summary (Mandatory) 10 points For my instructional systems design project, I decided to take something that I have taught before and improve on it. I chose the unit on Geometric Proofs because it is always a difficult topic for students to understand. Students are scared to give them a try before they even enter my classroom doors on the first day of school. I wanted to use this opportunity to create a product that would help motivate my students to give proofs a chance.To create this book, I followed the ADDIE process by Analyzing, Designing, Developing, Implementing, and Evaluating my product and instruction.

Analysis 20 points
 * Objectives **

The main purpose for my instruction and the use of a BookBuilder lesson as my final product was to help motivate students to learn and understand geometric proofs. Geometry is a tough subject for many students and motivating them to learn difficult topics within the subject is one of my greatest challenges as a teacher. I decided to take this opportunity to introduce the topic of geometric proofs by creating a BookBuilder lesson to show students when they will use deductive reasoning (the reasoning used to support each statement with a known or proven fact) in real life.

The main objectives of the unit on geometric proofs are for students to be able to:
 * use deductive reasoning to logically write a two-column proof, and
 * to make a connection to how this type of logic is used outside of the classroom.

My goal was to have student effectively communicate, with 80% proficiency, mathematically by writing geometric proofs using deductive reasoning. I evaluated their success with two different types of summative assessments.
 * Process used for this analysis **

Although it took some time to decide on a topic, I knew I wanted to focus on the Geometry courses that I teach. I have been teaching Geometry for six years now, and each year I try incorporating something new to improve my instruction. Geometry is a tough subject for most students. One reason is due to the fact that adolescent brains are beginning to develop to think analytically (“Cognitive Development,” n.d.). Another is that students have to consistently work hard to understand many of the concepts taught in Geometry and therefore need some additional motivation to work at the level required. One of the most difficult concepts to teach in Geometry is how to think logically and how to write Geometric proofs. I decided this would be a great topic for my instructional systems design project for a couple of reasons, (1) it is always a tough topic to motivate students to attempt, and (2) it was something that I would be teaching during the time of this project.

Teaching geometric proofs require direct instruction with examples to show how to complete a geometric proof. Repeated practice is a necessity. In addition, students require motivation, a positive attitude, and determination in order to be successful. For this project I decided to focus on the introduction of my unit on proofs because I wanted to change my method of approach to see if motivating students in a different way might possibly have a positive effect on student's overall performance. Students need to feel confident in themselves as they make their way through the understanding of the logical process (deductive reasoning) used to write geometric proofs. One way to motivate students is to let them know how they will use this knowledge in the real world. I thought introducing the topic using book builder would be a different experience for my students. As a small group they were able to discuss and respond to the provided questions. If students were not sure how to answer a question, the coaches were there to assist them. Having the coaches able to assist allowed for me to be able to move about the classroom and listen to the conversations transpiring (and to make sure everyone maintained focus).
 * Needs analysis **

Geometry or math in general, is not a subject that most people can read and comprehend on their own. It takes a lot of practice and different modes of instruction for most learners to understand the material. Geometric proofs are no exception. In my teaching experience proofs are probably the most difficult concept in geometry for my students to grasp. Students spend a few days "in the dark" before I begin to see some light bulbs turning on. It is important to meet students where they are and offer support through the darkness in order to help them find a path to the light.

There is a definite need to teach students how to understand geometric proofs in a way where they are confident not only in their approach, but also confident that they have reached a logical conclusion. I believe that the main problem students have with completing proofs is due to a lack of knowledge as well as a poor attitude towards proofs. For the knowledge aspect, proofs are brand new. No previous math teacher has ever required such a difficult thought process. For the most part math up to this point has required following a specific process...there was no need to "think outside the box." Also up to this point in mathematics there is typically one right answer or approach to solving a problem. Now, we have no numbers, a few symbols, and a lot of WORDS! In order to successful complete a proof students must also activate prior knowledge which can often be a difficult task. As for the attitude, it comes from friends or siblings who have taken geometry previously and been unsuccessful in mastering proofs. Students sometimes have their minds made up before they even get started that proofs are impossible and they aren't even going to waste their time trying.

My challenge and need for instruction comes in helping students to overcome their fear and attitudes and instead instilling confidence in their mathematical abilities. I am also required to help them activate their prior knowledge by reminding them of what we have previously learned which will help them logically come to a valid conclusion. I found it most difficult to instill confidence in my students.
 * Content analysis **

Originally, I had six major steps in my task, or content, analysis. The first step was to address the question, “Why do we need to use Geometric Proofs?” This is where I spent my time developing my final product using BookBuilder. From there I briefly mapped out the overall unit. After the introductory lesson we began with algebraic proofs where students supported each step in solving an equation with a reason. Then we moved onto an interactive lesson that included examples and guided practice problems. Once students felt a little more comfortable with proofs, they worked in groups to complete several proofs. Then they complete proofs on their own before an individual summative assessment.

The overall unit took ten days to complete the six steps. For the most part I followed my original task list; however, I did also include a group summative assessment as well as the individual. I like for students to have an opportunity to participate in a group and receive a grade because with their pooled knowledge, students tend to score higher than they would individually. This group assessment was actually a proof that they ended up writing on a poster as a two-column and a flow chart proof. They were graded using a rubric both on the accuracy and creativity of the poster.

Overall, I think my analysis was a complete inventory task list for the items to be developed for the overall unit. It helped being a subject matter expert completing the analysis. I can imagine that this would be a much more difficult task without knowing the content. When I originally completed the analysis I wish I had analyzed my first task, the BookBuilder lesson, more in depth. At this point in the process I should have put a little more thought into exactly what my final product would look like and the types of questions my book would ask. I instead spent more time analyzing what the overall instruction for the unit would be instead of just the product, which was the new part of my instruction. I also wish that I had done a better job at “viewing the content from the learner’s perspective” as Morrison, Ross, & Kemp describe as an important step of any task analysis (as referenced by Brown & Green, 2011).
 * Learner analysis **

My learners are in the 10th grade (with a few in 11th) taking a college prep course in Geometry. All students have a background in Algebra I where some were more successful than others, but all have passed an Algebra I course. I have two classes of 28 students each, my fourth period has 15 girls and 13 are boys and my sixth period has 14 girls and boys. Overall most students are of average ability. I have two ESOL students; they both speak English very well but struggle with comprehension sometimes. I have three students who are learning disabled, two require extra time on assessments and another has slow motor skills and is unable to copy notes or complete assignments at the pace of other students, but is very intelligent. Attendance has proven to be the biggest challenge among 5 or 6 of my students. Some students have been out due to illness and others for a lack of motivation/desire to attend school. Also, in the middle of this unit, three of my students were transferred to the Alternative School for behavior and/or attendance problems.

In my college prep course, my students all have a diverse background, both in their education as well as in their home lives. Although these students have all passed an Algebra I course, some of them passed it the first time, some have taken and passed the course in summer school, and some have taken the course split up over two years. In any case, this provides a diverse learning environment where some students catch on very easily and for others it is a struggle. Also, home life has a huge impact on school performance, and that also must be taken into consideration when teaching and placing students in groups to work together.

Universal Design for Learning (2011) explains the necessity to reach all learners through multiple means of representation, action and expression, and engagement. In a 50 minute period, it is sometimes difficult to address each one of these; however, over the course of a unit it is possible. I knew for teaching geometric proofs it was definitely necessary to include multiple means of representation which include my book builder lesson, interactive lesson, guided practice, group practice, and individual practice. I found engagement to be the most difficult because students are so quick to give up on something they find difficult. It took a lot of work on my part to make sure that students stayed engaged throughout the unit and to encourage them to not get frustrated with the mathematics.
 * Context analysis **

This instruction is intended to take place in a classroom environment for 10th and 11th grade students. In my classroom, desks are arranged in rows, which will need to be changed during the times where students are required to work in groups. I have access to a computer with a projector as well as 10 netbooks that students will be able to use to access my BookBuilder lesson.

Geometric proofs are taught typically during the second quarter because it is at this point in the year where students are learning about congruent triangles and they have enough background geometric knowledge (postulate, definitions, and theorems) to use deductive reasoning to write proofs.

Design and Development 20 points
 * Description of the Instruction **

I teach at North Augusta High School in Aiken County, South Carolina. My lesson was taught in my college prep Geometry course to 10th and 11th graders. Student ages range from 15 to 17. My classroom is set up with 6 rows of desks with 5 desks in each row. I have 28 students in both of my geometry classes. When the students worked in groups, they moved the desks from rows into groups. The sequence of events for the 10 day unit on geometric proofs follows.
 * Setting, Activities, and Sequence **
 * 1) Address the question, “Why do we need to use proofs?” to provide motivation for the unit using my BookBuilder lesson. Students worked in groups of 3 to 4 to read the book on netbooks and answer the questions provided on screen. They submitted the responses when finished. (1.5 days)
 * 2) We then reviewed prior knowledge on how to write a basic algebraic proof where students have a supporting reason for each step in solving an algebraic equation. (.5 day)
 * 3) I had an interactive lesson on Geometric Proofs which includes examples and guided practice problems with self-check (SMART Response) features. (2 days)
 * 4) Students then worked in groups of 3-4 to complete geometric proofs. (2 days)
 * 5) Then students had a group summative assessment on two-column and flow chart proofs. The final product is a poster which included a picture, the given and prove information, a two-column proof, and a flow chart proof. (2 days)
 * 6) Students then completed proofs individually which took place both in the classroom as well as for homework. (1 day, plus additional time outside of class)
 * 7) Students then completed an individual summative assessment (1 day)
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Development Process Supporting the Instructional Approach **

Throughout the development of my product, I kept John Keller’s ARCS Motivational Design in mind. This model seamlessly fit into my plan for instruction which was to grab students’ Attention, make the learning Relevant, build their Confidence, and Satisfy their hard work with meaningful assessments.

Before I planned my instruction, I began with my Geometry textbook as well as the Common Core state standards. After aligning my objectives and goals with the standards, I could move forward with my design. I focused this project on the introductory lesson where I incorporated the BookBuilder lesson to help motivate students to learn about geometric proofs by learning about jobs and situations where deductive reasoning is used in the real world. The diagram below describes the process I used to create my book builder product.

Created using gliffy.com
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Major components **


 * 1) Testing and Evaluation Plans: Students were evaluated at the end of the unit with a group assessment where the final product was a poster. Students were also evaluated individually using a basic paper and pencil test where students had to fill in the blanks for some proofs and also fully write their own.
 * 2) Grading Rubrics: I graded the group poster assessment using a rubric which was based on meeting the requirements, visual appearance, and logically and accurately completing the proof
 * 3) Practice Activities: Students had many different opportunities to practice writing proofs. They had guided practice, practice as a group, and practice individually.
 * 4) Feedback Mechanisms: Feedback was provided verbally to students as they worked both in groups and individually to complete proofs. Written feedback was provided on the rubric for the group assessment and on the final individual summative assessment.
 * 5) Introductory Presentation of Instruction: The BookBuilder lesson was used to introduce the unit.
 * 6) Motivational Strategies: The introductory lesson was used to help motivate students by understanding real world situations and jobs where deductive reasoning is used.
 * 7) List of Materials:
 * Introduction: Netbooks (1 per group of students) and BookBuilder product
 * Core Lesson: projector, computer, and geometric proof notes
 * Group Work: Worksheet with practice proofs
 * Group Assessment: Poster board, pencil, markers, rulers, worksheet with proof to be written
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 1.5;">Individual Assessment: Test

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 140%;">Evaluation 40 points
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Key Development Decisions and Justification **

// Effective // for Learners: // Easy // for Students (and Instructors) to Use:
 * 1) Begin a lesson with motivating students to learn. Students are always curious as to why they have to learn something. I decided to use the BookBuilder lesson to create some motivation to show students jobs where deductive reasoning is used. If they see the purpose, and how they will benefit in the long run, then they are more willing to give something a try.
 * 2) Questions were provided on each page for students to ponder and answer as a group to discuss situations where deductive reasoning could be applied in a decision that a doctor, lawyer, or politician has to make.
 * 3) Three coaches, Emma, Hali, and Alex were provided to help students if they got stumped on a question. Emma encouraged the students to think for themselves, Hali provided some helpful hints, and Alex provided a sample answer. This still allows for students to learn and come up with ideas on their own, but also gives them a little push in the right direction if needed.
 * 4) For the remainder of the unit students were able to work in groups to complete proofs before they had to do it on their own. When teaching difficult topics, proofs especially, it is beneficial to have the support and knowledge of classmates to help in the beginning and then eventually students build up the confidence to be able to complete some on their own.
 * 5) Two different types of graded assessments were provided. The first was a group activity where students created a poster of two different types of proofs. The second was an individual summative assessment. Some students are intimidated by test taking and it can negatively affect their grade, the group project provides an alternate assessment opportunity for students to hopefully help boost their grade.
 * 1) BookBuilder is easy for students to click through and helps to maintain focus. It also provides a location for students to respond to the questions right on the screen so they can type in their answers instead of having to write them on paper. This is also easy for teachers to review because all of the answers are neat and legible!
 * 2) Although technology is not always reliable, it is easy for students to use and they connect to technology a lot better than a lecture-style lesson.
 * 3) This is also easy for instructors because it allows for the teacher to walk around the classroom to monitor the classroom instead of asking the questions and having to repeat questions for those students who were not paying attention. This reciprocal teaching style allows for the students to learn from one another instead of the “all knowing” teacher.
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Evaluation **

I created a survey for my peers to use to evaluate my BookBuilder product. On the survey I asked questions such as did the book provide a reason to learn about geometric proofs and deductive reasoning? I also asked my peers to evaluate the pictures that I chose, the coaches, and the student response questions. Finally, I provided an opportunity for my peers to offer suggestions for improvement. All of my peers felt that the book provided a reason or motivation for students to learn about geometric proofs. They also felt that the pictures were appropriate and that the coaches and questions were helpful. Some of the suggestions for improvement that were offered included changing the “/” to a comma because the coaches read “slash” every time it is used which was a distraction. Also, one of the pictures is a little big for the page. Another suggestion was to come up with a scenario for a student who might not want to go into one of the professional professions that I included. Before I implement this lesson again, I will make the improvements and come up with a situation, such as begin a parent, which would also connect to my students.

I formatively evaluated my students by walking around the room while they were on the netbooks using the BookBuilder lesson. I noticed that they were engaged in answering the questions and discussing and creating scenarios. For the most part, my students enjoyed this book builder activity. This book served as an introduction to Geometric proofs and my students worked really well together to come up with some great situations where <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">deductive reasoning could be used in the real world. It was especially entertaining to read and listen to student's perspectives on what kinds of decisions lawyers and politicians make on a daily basis. This book and introduction helped to engage my students and they gave proofs a chance. For at least a few days they showed some motivation and gave a good effort towards producing Geometric Proofs on their own. This topic is always challenging because it stretches student's minds and requires a lot of prior knowledge to be activated. This is easier for some students than others, but I can honestly say they were more motivated with this introduction than I have ever had in the past.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Since my Book was only a small portion to my overall unit on Geometric Proofs, it is hard to link my summative assessment results (a unit test) directly back to the introduction. Based on my student's performance this year vs. last year on the individual assessment, the overall average increased from an 81.5% to an 83.7%. Fortunately the scores increased, but it is impossible to tie the final assessment results to the introductory lesson. For the group assessment, the groups did a good job writing their proofs with their pooled knowledge and skills. Ten out of the fourteen groups made an A on their proof posters. The four remaining groups made either a B or C.
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">Expected Maintenance and Distribution Requirements **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">I do not anticipate any major issues for maintaining my product for its use in my instruction. Geometric proofs will always have a place in a Geometry course and therefore it will always need to be taught. I do not anticipate students as a whole to ever think that writing proofs is easy and because of the analytic thinking that is involved, this will remain a challenge to teach. The only issue that I foresee happening is if the internet is down when I am trying to introduce this unit. As a teacher I am flexible and constantly needing to monitor and adjust, so if this is ever the case, I would simply do the same! It might be a good idea to have the student response questions written or typed somewhere else so that if the internet is down we can still discuss and create the scenarios for each profession.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">I will always have access to my BookBuilder lesson as it is linked from my wiki site as well as in the CAST BookBuilder Library online. I also have all of my supplemental documents saved via Dropbox, so I will always be able to access that material as well.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">None of my lessons are ever “finished.” Each year I try to improve on my lessons and this one is no exception. As teachers we are constantly adjusting to the students that are sitting in front of us each year and something that motivates my students this year might not interest next year’s group at all. My book is definitely usable as it is though, and my students this year were motivated to learn about proof. I would definitely make a few adjustments based on my peer’s feedback before I use it again though.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 140%;">References 10 points


 * 1) Brown, A., & Green, T. D. (2011). // The Essentials of Instructional Design // . Boston, MA: Pearson Education, Inc.
 * 2) Burger, E.B., Chard, D.J., Kennedy, P.A., Leinwand, S.J., Renfro, F.L., Roby, T.W., …, Waits, B.K. (2012). // Holt McDougal // // Geometry Common Core Edition // . Houghton Mifflin Harcourt Publishing Company.
 * 3) Cognitive Development. (n.d.). Retrieved from [|http://www.lpch.org/DiseaseHealthInfo/HealthLibrary/adolescent/ cogdev.html]
 * 4) Common Core State Standards Initiative, (2012). Retrieved from []
 * 5) Gliffy, (2013). Retrieved from []
 * 6) Tucker, L. (2013). // Instructional Systems Design Project: Geometric Proofs // . Retrieved fromhttp://ltucker722.wikispaces.com/Home
 * 7) Universal Design for Learning, (2011). Retrieved from []
 * 8) What is Motivational Design?, (1988), Retrieved from <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">[|http://www.arcsmodel.com/home.htm#!motivational-design/c2275]

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 140%; line-height: 1.5;">Written Report Points (Total points = 100)