Andrew Otten's Math Folio
The following is Mr. Andrew Otten's Math specific portfolio. The following targets address standards of planning, instruction, assessment, evaluation, personal growth, and my field experiences during my time at Grand Valley State University. Please enjoy, and if you have any comments or questions, go to the bottom of this site's home page.
Planning
TargetsTarget P1: Effective planners exhibit a positive disposition toward learning and doing mathematics by identifying essential learning outcomes based on content and process standards and knowledge of adolescent learning, development, and behavior. [Effective planners also incorporate knowledge of learners' cultural and language diversity and select culturally relevant examples as a means to motivate and engage learners.]
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Narrative for P1: (Last updated - December 8th, 2014) In every classroom there are students who struggle. There are many different methods teachers can use to help teach the identified learning targets. One such example is a FRAME which gives students a little more structure when taking notes. By using my knowledge of my class I knew this would help them to learn about the four theorems used to prove triangles are congruent. These four theorems can easily get confused with each other, but allowing for this type of format allows students to quickly see the differences between each theorem which is part of the intended learning goal. Additionally, I have attempted to incorporate the learners' culture by selecting particularly relevant statements in the converse/inverse/contrapositive activity. This activity they need to use statements and change it into different forms, but the statements given are relevant to help motivate and engage them and show them an example where this concept could be used in their life. |
Target P2: Effective planners design focused, coherent sequences of connected lessons [that show a progression of learning over time toward proficiency and understanding].
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Narrative for P2: (Last updated - December 8th, 2014)
I selected an Interdisciplinary Unit I created for EDR 321 (Content Area Literacy) during my teacher assisting, as my artifact for target P2. This unit is mainly about the Pythagorean Theorem and is slightly connected through the story of Christopher Columbus traveling to America. This unit begins with an informal proof of the Pythagorean Theorem and an exploration into why this theorem works for only right triangles. It them moves into using this theorem to find the distances sailed by Columbus from different land locations. Then the students must use the converse of the Pythagorean Theorem to find distance of Columbus's travel on land. As you can see, the sequence of these lessons are connected by Columbus and the Pythagorean Theorem and by increasing difficulty in a common sense way using the Pythagorean Theorem. They also show a progression of learning throughout the unit of the Pythagorean Theorem and its uses. I added another unit that I used during my student teaching. The unit organizer shows at a quick glance the coherence sequence and connection between lessons as well as increasing difficulty in the topic throughout the unit allowing for learning over time toward proficiency in inductive and deductive reasoning and understanding. |
Target P3: Effective planners draw upon research-based practices to support their planning[, including print, digital, and virtual resources and collections from professional organizations].
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Narrative for P3: (Last updated - December 8th, 2014)
As an artifact for target P3 I selected a blog post that I wrote about The Teaching Gap by James Stigler and James Hiebert. This book opened my eyes to the TIMSS research found about Japanese German, and American classrooms and the systems of education that exist within each culture. This blog particularly points to my change of thought on how to orchestrate math discussions within a lesson and using the group work time to look at students work and be able to call on groups in a certain order so as to best allow the class as a whole to learn through seeing models of different strategies, and in viewing misconceptions that students may not otherwise have experienced. Additionally, I have included an activity that Samuel Otten and myself created called Selecting Systems of Equations. In this activity we use lots of different types of resources to support our planning including online sources and print. |
Instruction
Target I1: Effective teachers use questioning techniques to (a) expose and explore misconceptions and (b) engage students in productive mathematical discussions [, and they provide opportunities for students to communicate mathematics in a variety of forms and for a variety of target audiences.]
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Narrative for I1: (Last updated - December 8th, 2014)
After teaching the different methods for solving systems of equations, this activity gives the students 8 different equations and has them select which equations they will choose to solve with each of the three methods. By allowing the students to choose within their groups, a large amount of discussion ensues about which equations would be best for each method. No groups chose the same pair of equations for any one method, yet nearly all groups were able to effective defend their choices at first to an audience of their neighbors, and later to an audience with me and the entire class. Within this activity there are also chances for misconceptions to be explored during the discussions. When I asked why two equations might be better for a certain method, the students were able to identify why and elaborate that previous they were planning to simply master one method and use it to solve all problems involving system of equations, but now they understood the benefit of certain methods being better for certain types of equations. |
Target I2: Effective teachers use connections to students' prior knowledge within and outside of mathematics to help students develop conceptual understanding and procedural fluency [, and they provide opportunities for students to select and explore personally relevant problems from a mathematical perspective.]
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Narrative for I2: (Last updated - December 8th, 2014)
All students have heard of Christopher Columbus and him sailing across the Atlantic Ocean. My Pythagorean Theorem lesson plan takes part of his journey and explores it with a mathematical approach. This takes students' prior knowledge of this story which is outside mathematics and helps students develop and explore the Pythagorean Theorem. The Selecting Systems of Equations activity provides an opportunity for students to connect the previous 2-3 math lessons about the methods of solving systems. This activity also allows students to explore which method is better from a mathematical perspective and help them to understand the conceptual understanding of why we have multiple methods for solving systems and additional practice with procedural fluency of actually solving several systems of equations. |
Target I3: Effective teachers (a) provide equitable treatment of and have high expectations for all learners, and (b) they use a variety of strategies, including strategies for differentiated instruction, to build conceptual understanding and procedural fluency for all learners. [They allow multiple and varied opportunities for students to demonstrate their understanding and fluency, and they persist in helping each student reach their full potential.]
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Narrative for I3: (Last updated - December 8th, 2014)
I expect all of my student to learn the skills I teach, and to master all of the skills for each lesson. My lesson on exponents of monomials is an example where I refuse to move on until every students is ready. This lesson uses plickers at each scaffolded step to assess if the class is ready to move on. If there is even one student not ready, then the entire class does a think-pair-share about which answer was wrong and why someone might think it was correct. My expectation is that my classes will get 100% on a plicker question before we move on. The FRAME and review Activity artifacts address multiple ways I have worked to vary strategies in my classroom. The FRAME is an alternative to traditional note taking and allows the theorems for triangle congruence to be more clearly seen and thus conceptual understanding easier to reach. The review activity blog post discusses multiple review activities used throughout the semester that allowed varied opportunities for students to demonstrate their knowledge of the unit being reviewed. Some activities worked better than others, but they all helped different students in different ways to reach their full potential. |
Target I4: Effective teachers engage students in a (a) sequence of developmentally appropriate and challenging learning activities in which they are actively building new knowledge, (b) including investigations that use math-specific technology [, and they facilitate students' ability to develop future inquiries that extend their past investigations.]
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Narrative for I4: (Last updated - December 8th, 2014)
My lesson about the exponents of monomials shows a step by step lesson that utilizes scaffolding to sequence challenging learning steps that build off of each other. The next step is not taught until the class is ready to move on. This lesson plan also works to keep the students engaged by having the majority of the math done in groups and by the students. The teacher occasionally comes in to facilitate a discussion and assess whether the class is ready to move on to the next scaffolded step. The GeogebraTube page shows an example of an investigation done in two of my classes. By showing many different triangles, students are able to see that the sum of the interior angles of any triangle is 180 degrees. After seeing this the class naturally started asking if the angles of all squares or quadrilaterals also add up to a consistent value. These type of inquires were then explored briefly, and in later lessons more indepth. |
Target I5: Effective teachers make appropriate choices regarding when to use math-specific technology and manipulatives to support deep learning, and they recognize the benefits and limitations of such tools. [They also participate in professional learning related to current and emerging technologies that support math teaching and learning.]
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Narrative for I5: (Last updated - December 8th, 2014)
To help students discuss their reasoning and show them the difference between inductive and deductive reasoning, they worked to put a comic strip into its correct order. These manipulatives allowed them to discuss why a particular slide would be before or after another slide. These manipulatives are limited since their are multiple ways the joke makes sense but the general discuss process is what was important. This activity was much more effective then just telling them what reasoning is and what it looks like. I also attended a symposium at GVSU about technology and some of its new uses in the classroom. My blog post talks about how I could use survey type apps to encourage differentiated learning in the classroom and use this technology to address specific learning targets with individual students and support ownership of students' learning. I have also heard of emerging technology such as plickers through twitter which supports teaching and learning in general. I used plickers in many lesson including my exponents of monomials lesson plan. |
Assessment
Target A1: Effective teachers identify [and design] formative assessments that can inform instruction and monitor learners' progress toward meeting essential learning outcomes.
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Narrative for A1: (Last updated - December 8th, 2014)
As an artifact for target A1, I selected a series of exit passes I designed, used, and collected during a linear relationship unit. Each exit pass was created directly from the learning targets in the textbook assigned for that lesson. With these exit passes it was easy to monitor individual learners and their progress toward the lessons, and units, learning outcomes. I also used these exit passes to see what I needed to reteach the next day before moving on to the next step in the unit. The passes proved quite useful and gave me a place to start when lesson planning day to day. |
Target A2: Effective teachers identify [and design] summative assessments that can accurately gauge students' achievements of essential learning outcomes.
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Narrative for A2: (Last updated - December 8th, 2014)
For target A2, my artifact is a Select Response Assessment that I made in ED 337 (Introduction to Learning and Assessment). This assessment has a table which shows where each question on the assessment addresses a specific learning target in the form of "I can" statements. This assessment allows both teachers and students to accurately gauge what achievements an individual student is excelling at, while also giving information about what learning targets need additional attention. This assessment accurately gauges student achievement because my ED 337 professor evaluated my work positively and a peer review session during class allowed for any additional changes to be done. |
Evaluation
Target E1: Effective teachers use timely analysis of assessment data to accurately gauge students' progress toward essential learning outcomes identified during planning [, and use this information to inform future planning, modify instruction, and increase students achievement.]
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Narrative for E1: (Last updated - December 8th, 2014)
For target E1, I selected three exit passes used during three consecutive lessons in my unit on linear relationships. These exit passes show data collected at the end of each lesson and assessed/evaluated to gauge students' progress toward the learning outcomes for each lesson. I often used these to guide my planning throughout my teacher assisting semester. Each pair of exit passes includes one showing a student meeting the learning target (the exit pass on top) and an example of a student who needs to be retaught the next day (exit pass on bottom). As you can see in the exit passes, the question asked was directly related to learning targets and while the next days exit pass moved on, it was still used to plan for the next days opening warm ups and what to talk about. I also used plickers, such as in my exponents of monomials lesson, to collected data of my students. This form of data collection was done nearly immediately since the analysis is done by my smartphone. I could then instantaneous use this data to see how my students are progressing and modify my instruction during the lesson. This allowed me to have the students stop and discuss a misconception before moving on. The plickers increased student achievement since misconceptions were identified during the lesson rather than between lessons like exit passes. |
Target E2: Effective teachers use summative assessments to accurately gauge students' achievement of the essential learning outcomes identified during planning [, and they determine the extent to which individual students' mathematical proficiencies have increase as a result of instruction.]
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Narrative for E2: (Last updated - December 8th, 2014)
In two geometry classes that I taught, the final exam has fifty questions and each question is aligned with a specific chapter that was taught throughout the trimester. This alignment helps me to use this final exam to gauge student achievement and specifically which previously identified learning targets were met and which are not yet mastered. In this fifty question exam, the first twenty questions are similar to a pre-test taken by the class during the first day of the trimester. The answer of the final exam and the pre-test are compared to each other and help determine how much students' grew mathematically as a result of my instruction. In the artifact we see how many questions were correct on the pre-test and how many were answered correctly on the first 20 questions of the final exam. This comparison is evidence of my effectiveness as a teacher and the increase of individual student proficiencies as a result of instruction. |
Personal Growth
Target G1: Effective teachers actively seek, engage, reflect, and share personally-relevant collaborative learning experiences [, and they apply what they have learned to enhance mathematical learning opportunities of their students.]
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Narrative for G1: (Last updated - December 8th 2014)
Throughout most of the 2013 calendar year I worked with another Grand Valley State University student to create a math competition at Grand Valley State University called Math-Team-Matics. Together we sought out for two GVSU professors and eventually landed on a four person team with the additions of Professors John Golden and Karen Novotny. The four of us engaged collaboratively for many months to make Math-Team-Matics a reality. After the competition, we all shared our learning experiences during a session at the convention Math in Action on February 22nd, 2014. At Math in Action we discussed all the steps and biggest challenges in creating and implementing the competition; including making the competition align with the Common Core State Standards. Using these standards to guide our problem building allowed us to ensure that the mathematical learning opportunities of the participates would be as enriching as possible. This collaborative group of four (two students and two professors) created and successfully ran a second competition of Math-Team-Matics in November 2014. We are now working to enhance and improve the competition and learning opportunities of the attending teams for the next competition. |
Field Experiences
Target F1: Effective teachers have observed [and implemented] a range of approaches to orchestrating the mathematics learning environment (e.g. task selection, discourse, and assessment systems), and they reflect on how those approaches may have been influenced by one's beliefs about the nature of mathematics and how students learn mathematics.
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Narrative for F1: (Last updated - December 8th, 2014 )
During my teacher assisting field placement I was in a culturally diverse middle school and observed my coordinating teacher's approach as a math teacher and the approaches of many other teachers in math and in other subjects. I often concentrated on how each teacher orchestrated the learning environment in their subjects. My CT's approach intrigued me the most because it was the most different from my own approach. Mr. Stevens felt that his students learn mathematics through a strict structure and that control should be in the hand of the teacher. After finding that these were his beliefs, I created and implemented a lesson based on his style of teaching. The task selection and discourse was very teacher-centered and my lesson as a whole was deemed, by both Mr. Stevens and my observing professor, Professor Hasenbank, to be similar to Mr. Stevens approach to orchestrating the mathematics learning environment. I also included an artifact of my blog post reflecting on this lesson and the observations from Mr. Stevens, Professor Hasenbank, and myself. My reflection includes how Mr. Stevens' beliefs about mathematics was effected by the way he was taught and how much he likes having control of the class. |
Target F2: Effective secondary teachers have demonstrated knowledge, skills, and professional behaviors in both middle and high school settings [and have communicated to other educators what they have learned from those experiences].
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Narrative for F2: (Last updated - December 8th, 2014)
I selected three artifacts for target F2. Two are my Coordinating Teachers' narratives from my teacher assisting semester at Riverside Middle School to address the middle school setting of this target while another is the narrative from my Coordinating Teacher at Rockford High School which addresses the high school setting of this target. The third artifact shows my communication of what I have learned from my experiences in both of these settings. My Coordinating teacher's narrative says I planned lessons that targeted needs of all students and did my job of teacher assisting with respect to students and colleagues. This shows my skills and professional behavior in a middle school setting. During student teaching, my CT writes about my good relationships with colleagues, parents, and students as well as my mathematical knowledge and skill in teaching high school geometry and algebra. My blog consists of over 19 posts about my thoughts and actions taken during these two semesters. Those this communication I have allowed other educators to learn from my experiences and have connected to them and learned from them. |