For each of the two lessons, analyze the effectiveness of your teaching and the level of student learning. Use the evidence from the video, transcription, or detailed supervisor’s notes and your analysis of the students’ work when considering the following questions:

 Rocket Man

• How appropriate were the standards/grade equivalents you selected for each lesson?

Two standards from the VT Framework of Standards and Learning Opportunities were very appropriate; MHS:13: Applies concepts of similarity to solve problems involving scale, and MHS:9 Models situations geometrically to solve problems connecting to other areas of math or to other disciplines (i.e. transformations). We used the similarity of the scaled down photo and the actual size of both Rick and the rocket engine. We modeled the concept of dilation (a transformation that produces an image that is the same shape as the original but is a different size).  The photo was a dilation that shrunk the original figure.   

'Application of transformation' was a standard from  VT Department of Education's Professional Knowledge Standards. Another appropriate standard was 'Real world application of mathematical ideas and interconnection with physical sciences' from VT Department of Education's Professional Performance Standards.  These professional standards were well suited to this lesson given the real world application of dilation.

• How did your goals, assessments, activities and teaching materials align with the standards/grade equivalents?

The 'Rocket Man' activity aligned very well with geometry high school standards 9 and 13. Whiteboards helped the students apply their new concept of dilation. Other formative assessments included homework, classwork, and a HW check. There were several concept tests as a summative assessment. Students grasped the concept of dilations quickly.  The average results of the first, second and third concept tests were: 89, 89, and 60. 5% of the students took a fourth concept re-test. After all testing was final, 80% of the class had mastered the concept or gotten a perfect four twice. 95% of the class had gotten one four, a 4/5, or 80%. 

This lesson and the connected problem solving activity (student work) were very appropriate for the real world application standard. The performance standard is one which designs mathematical activities that enable students to investigate, explore, and discover structures and relationships; create and use mathematical models; formulate and solve problems collaboratively; and justify and communicate their conclusions orally and in writing. For the indirect measurement problem solving activitiy, the students gathered data in groups and formulated the measurement problem collaboratively. In their write up the students needed to communicate their findings in proportions and graphically. 

• What information did your assessments provide about student learning? 

After three rounds of concept tests, the students had a 80% success rate on the concept of dilations. The concept test assessment was invented by Dan Meyers.  Dan Meyers visited CHS last year for professional development. Since his visit the math department has completely realigned its testing procedure for Algebra I, Geometry, Algebra II, and Algebra II/Trigonometry.  Each course is broken up into 30-40 discrete concepts with separate records on students' performance on each skill, and students are tested three times always counting their highest scores. What I found attractive about this assessment system was the idea that with goals broken down into such small, discrete pieces, students were empowered and motivated and take the initiative to learn what they needed to make the grade.

Multiple measurement is something Marzano mentions in Classroom Assessment and Grading that Work: " from a mathematical perspective, at least three scores are required to estimate the power law score" (Marzano, p. 114). The students also kept track of their progress on a concept checklist. "One of the most powerful and straightforward ways a teacher can provide feedback that encourages learning is to have students keep track of their own progress" (Marzano, pg. 89).

We had a HW check on dilations so both the students and I got feedback on their progress.  Roughly 70% of the class understood the steps needed to dilate x,y coordinates when we used the whiteboards. Bloom's taxonomy of verbs analyzes the different questions teachers ask of their students.

I had the students apply their newly attained dilation knowledge in a whiteboard review. Application is the act of solving problems in new situations by applying acquired knowledge, facts, techniques and rules in a different way.  The students manipulated a graphical dilation by showing me its algebraic and verbal representation.  According to my supervisor, Ellen's Observations there was a student in the front that got none of the dilation questions correct. I helped him with the concept during the following day's homework check.

• In what ways were your teaching and management strategies effective or not effective in helping students meet or exceed the expectations for each lesson? 

There were a few behavior issues during this lesson. According to my supervisor's notes, there were many points throughout this lesson when I was talking over the students.  There were also a couple students listening to music when they were doing their practice sheet. I have learned not to rush through my lesson in an effort to get 100% from the students. A technique I learned from 'Teach like a Champion'

• How did you create a positive learning environment?

I was always positive when leading a class. Classroom environment is called Domain 2 in Danielson's Framework for Teaching. This positive environment  is something I observed in Marijke Reilly's biology class. She established a culture for learning by taking immense pride in student work. I also observed an incredible amount of respect and rapport when I observed Heather Baron's earth science class. Heather's students were presenting that day so she spend a great amount of time reinforcing a climate of respect and support.

I congratulate students for contributing answers and for volunteering their thoughts. I use no opt out, a technique I learned in Teach like a Champion when picking on students randomly with popsicle sticks. This technique sets a high expectation for the classroom. I am not satisfied with "I don't know". I help students who are struggling with the material by giving them steps in order to solve the problem or I will ask their classmates to volunteer with some help.

 During our TAP seminar 7 we did a gallery walk that highlighted what you hear and see in a classroom filled with respect, rapport and a culture for learning. I always tried to greet every student by their first name when they entered the classroom. I was always positive and enthusiastic.  In my student teaching evaluation I surveyed 32 students and got the highest rating for creating a 'positive classroom climate.'

• How did your planning and preparation impact your teaching and student participation? 

I did a considerable amount of research on this rocket engine in order to demonstrate knowledge of resources (Danielson's Domain 1d). I opened the lesson with many open ended questions after some background on the rocket. Students participation was surprisingly low because they don't often come up with the problems. After reading more about Dan Meyers I realized that I needed to give the students some incentive for estimating the height of the engine. In the future I would place student's guesses on the board before having them make measurements. This minor contest gives the lesson some buy-in which is what we're trying to access with the Hook method. I got the 'Hook' technique from Teach like a Champion, which is especially important at the onset of a new unit. I know now that student's love to have their guesses on the board and I will plan future lessons around this knowledge of students (Danielson's Domain 1b).

• How did this lesson demonstrate strength in your endorsement area?

I've earned a Bachelor's of Arts in Mathematics in which I took Foundations of Geometry, passed the Praxis II exam in Mathematics: Content Knowledge and was exempted from the Praxis I exam due to my SAT scores. Dilation meets the Professional Knowledge Standard for Geometry: Core concepts and principles of geometry in the plane... and applications to transformations and problem solving.

Dilation was an entirely new concept for the Geometry students to grab onto.  There were many misconceptions that I was looking for when circulating the room such as the confusion of setting up proportions from Algebra, and which units to use and to keep those units consistent.  Anticipates, elicits, and corrects typical errors and misconceptions in mathematical reasoning through the planning and execution of lessons is a VT Performance Standard. Many students needed a refresher on how to set up and solve proportions.  My strong content knowledge enables me to guide students towards the right path. The 'rocket man' hook conveyed the real world applications of mathematical ideas. Scale factor is a great application that we learned in this lesson and students found again in their indirect measurement Problem Solving Activity (student work). Our DSRP model allowed students to discover structures and relationships present in transformations.

Lesson Two

• How appropriate were the standards/grade equivalents you selected for each lesson?

Two standards from the VT Framework of Standards and Learning Opportunities were very appropriate; MHS:13:Applies concepts of similarity to solve problems involving scale, and MHS:9:Models situations geometrically to solve problems connecting to other areas of math or to other disciplines (i.e. transformations). We used similar triangles to solve the goal post problem. The shadows of the goal post and Wayne formed two similar triangles. Similar triangles are proven through Angle- Angle. One similar angle was formed by the ground and object. The other angle was formed by the top of of our objects by the ray of sun. Therefore the heights of our objects over the size of their shadows are proportional or similar.

• How did your goals, assessments, activities and teaching materials align with the standards/grade equivalents? 

The 'shadow man' activity aligned very well with geometry high school standards 9 and 13. Using multimedia was a wonderful way to helped students visualize similar triangles.  Other teaching materials included: handouts, rulers, popsicle sticks,the problem of the day, calculators, smartboard, and colored paper.  The 'shadow man' hook conveyed the real world applications of mathematical ideas. Summative assessments included HW and a HW check.  The students practiced their similar triangles for HW (student work) that night. This lesson was assessed in the next class through a similar triangle HW check (student work).

The summative assessment included a series of three concept tests. Students had a harder time grasping the concept of similarity.  The average results of the first, second and third concept tests were: 92, 46, and 52. They had not retained the similarity concept.  25% of the students took a fourth concept re-test. 10% of students took a fifth concept re-test. After all testing was final, 60% of the class had mastered the concept or gotten a perfect four twice. 85% of the class had gotten one four, a 4/5, or 80%.  I think next time they need to draw their own conclusions about similarity and the summative assessment should take concrete learners into consideration.

This lesson and the connected problem solving activity (student work) were very appropriate for the real world application standard. The performance standard is one which designs mathematical activities that enable students to investigate, explore, and discover structures and relationships; create and use mathematical models; formulate and solve problems collaboratively; and justify and communicate their conclusions orally and in writing. For the indirect measurement problem solving activity, the students gathered data in groups and formulated the measurement problem collaboratively. In their write up the students needed to communicate their findings in proportions and graphically.  Teaching materials used for the indirect measurement activity included: mirrors, handouts, and measuring tapes.

• What information did your assessments provide about student learning?

 The students had a difficult time grasping similar triangles. Our concept testing system is set up almost as a slow formative assessment. There were many students who continued to retest after seeing the concept on three classroom tests. On average, 25% of student took a voluntary, fourth similar triangles re-test. They had a hard time visualizing similar sides and often got them confused when setting up proportions. The problem solving activity helped the class understand how to set up their proportions to solve for the objects' heights. But the triangles were all set up the same way. The tricky part for most students was when the triangles got flipped around or were nested on the concept test.

In Marzano's Classroom Assessment and Grading that Work we learned the three levels of assessment. In TAP seminar 11 we created a guided note-taking sheet to follow chapters 3,4,&5.  Type I items are basic questions that would accompany a concept addressing basic details. A Type II question is a little bit more in depth and Type III questions have students infer or make their own conclusions. The nested triangle tasks in this concept were Type II questions. "Type II items and tasks typically address more complex mental procedures .. they involve embedded  procedures for gathering information" (Marzano, p. 71). The embedded procedure was the act of figuring out which sides of the triangles were corresponding.

Bloom's taxonomy is a hierarchy of words to signify the difficulty/depth of questioning/activity.  The students applied their newly attained similar triangle knowledge in the indirect measurement Problem Solving Activity (student work). Application is the act of solving problems in new situations by applying acquired knowledge, facts, techniques and rules in a different way.  Students found the heights of three different objects using their knowledge of proportions and similar triangles

• In what ways were your teaching and management strategies effective or not effective in helping students meet or exceed the expectations for each lesson?  

In Domain 2 of Danielson's Framework for Teaching, Component 2d is Managing Student Behavior.   My supervisor's observation (add day 22 file) is evidence that students talked, listened to mp3s, played iPod games, and went to the bathroom during instruction. My supervisor, mentor and I brainstormed ways for me to "take over" the class and establish expectations and rules for behavior. To manage behavior, we developed three non-negotiable expectations during my solo unit: I would not talk over students,  no electronics (ipods, cellphones), and bathroom use was reserved for the beginning and end of the class.

 Other components in the Framework include: establishing a culture for learning, managing classroom procedures, and organizing physical space.  I set high expectations by following through on student's answers. 'I don't know' was not an acceptable response in my classroom. Management of supplies and materials was smooth as each student had their own folder and supplies were labeled and readily available.

• How did you create a positive learning environment? 

TAP seminar 7 we did a gallery walk that highlighted Danielson's second domain. The classroom environment includes: creating an environment of respect and rapport, establishing a culture of learning, managing classroom procedures, organizing physical space, and managing student behavior.I was always positive and enthusiastic.  In my student teaching evaluation I surveyed 32 students and got the highest rating for creating a 'positive classroom climate.' My mentor and I set up strong classroom procedures including our daily problem of the day, a folder system, and a  posted objective/agenda.

• How did your planning and preparation impact your teaching and student participation? 

Domain 1 of Danielson's Framework for Teaching is planning and preparation.  Component 1e is designing coherent instruction which includes learning activities, instructional materials, groups, and lesson structure.  Given more time to digest the concept, research lesson activities, and reflect I realize that my planning for this lesson was basic. My mentor helped me create this lesson but after doing it several times I can see where it needs improvement. It takes more time and experience to craft a wonderful lesson or you can model them after amazing educators such as Dan Meyers .

The teaching strategies in this lesson were not totally effective because it took a while for many students to grasp the concept. In an ideal world I would always have the sun shining for this day so the kids could create their own similar triangles using shadows. Another learning activity would have students build their own similar triangles. There could be a set of three different colored straws to represent similar triangles.  The concept of similar triangles needs to be presented for concrete thinkers and assessed through concrete ways.

My student participation would have been greater if I had more buy-in. Component 1b is demonstrating knowledge of students which includes their interests and varied approaches to learning.  Waging guesses on the height of the football post would have accessed many more minds in the classroom that day. 

• How did this lesson demonstrate strength in your endorsement area?

 I've earned a Bachelor's of Arts in Mathematics in which I took Foundations of Geometry, passed the Praxis II exam in Mathematics: Content Knowledge and was exempted from the Praxis I exam due to my SAT scores. Similar Triangles meets the Professional Knowledge Standard for Geometry: Core concepts and principles of geometry in the plane... and applications to transformations and problem solving.

Similar triangles was a tough concept for the Geometry students to grab onto.  There were many misconceptions that I was continually looking for when teaching this concept.  Anticipates, elicits, and corrects typical errors and misconceptions in mathematical reasoning through the planning and execution of lessons is a VT Performance Standard. Students would confuse the corresponding sides of the triangle when setting up their proportions. It would help when I would distinguish between the sides with different shapes which many of the students grabbed onto. A triangle, square or circle may represent one corresponding side of similar triangles. 

Many students needed a refresher on how to set up and solve proportions.  My strong content knowledge enables me to guide students towards the right path. The 'shadow man' hook conveyed the real world applications of mathematical ideas. Students applied similar triangles again in their real world  indirect measurement Problem Solving Activity (student work).