Reflect on each lesson. Within the context of your reflection consider the following questions.

 Rocket Man

• If you taught this lesson again what changes would you make to improve the effectiveness of your teaching?

During the 'hook' I would have given students more time to form the problem in front of them. Instead of giving the students 'think time' I often gave them the problem and solution. As mentioned before,  during the 'hook', I would have asked students to make an estimate and placed those on the board. This gives all students a level playing field. Even students that wouldn't normally participate in everyday math activities have an instinctual idea of the size of everyday objects. This starts the lesson off on a good foot and the students are self-starters.

In order to make the topic relevant to students I would have them dilate their own names. The concept of dilation didn't really take hold with concrete learners and days or weeks later they would confuse it with other transformations. I got this idea from a Dan Meyers lesson (slide 28-36) where the students blow up their names. After this dilation exercise the students would have a greater understanding of dilations and I would then move onto triangles in the coordinate plane. 

• How did your knowledge and skills in your content area enable your students to meet the standards or Grade Expectations?

My coursework in physical science along with math allows me to create lessons that combine multiple disciplines to make sense of real world applications (MHS:9). My strong content knowledge also helps me be proactive about typical misconceptions in geometry. While starting a new concept such as dilations I can reiterate the connection of our eyes dilating. This will help students learn the vocabulary and not confuse dilations with other transformations such as slides where the students glide objects as opposed to enlarge them.

Using a real world example of scale (MHS: 13) and having the students find the scale factor numerous times enabled my students to exceed the Grade Expectations. We found the scale factor together in class by calculating the dilation from the photo to Rick's true height. The students also completed a problem solving activity (student work) where they found the scale factor of objects at school in relation to their height. The students went outside with a mirror and tape, and using similar triangles found the height of trees, goalposts, and parts of the building (add lesson plan). 

• What are your strengths and challenges in providing opportunities for all students to meet or exceed the standards or grade equivalents?

In TAP seminars our teachers modeled many formative assessments including: whiteboards, yes/no cards, thumbs up/down, RAFT, popsicle sticks, graphic organizers, prezi, vocabulary games, jeopardy, PRS clickers, entrance cards, exit cards, preassessments, choral response, and info gap.

Variation in activities throughout the class help provide opportunities for all learners. I used different strategies such as cold calling, whiteboards, no-opt out, HW checks, think time, direct instruction, guided note-taking, and multimedia: pictures and wikepedia.  The rocket photo as well as Wikipedia multimedia appealed to both visual and concrete learners. Going outside and measuring trees, in the following class, appealed to naturalistic learners. These techniques were commented on when I surveyed my Geometry classes in a final evaluation. They most enjoyed going outside, games & activities,  whiteboards, and the way I explained new concepts! There were a number of students (Evaluation 1, 2) that suggested I slow down, speak clearly, and listen. I was so grateful to have so much constructive feedback. These comments really emphasized in my practice the value of varied and coherent instruction.

In Geometry, I always used guided note-taking which was highlighted in Marzano's Classroom Instruction that Works. "Teacher-prepared notes are one of the most straightforward uses of notes. [They] provide students with a clear picture of what the teacher considers important, and they provide students with a model of how notes might be taken (Marzano, p. 46).

It is a challenge to meet the readiness of all students in a heterogeneous classroom. I have learned different techniques for meeting many readiness levels. I tiered the hook by asking students to measure the rockets while some went further out of curiosity or skill level. Those ready students measured whether there was a difference between the horizontal and vertical scale factor. 

There were many behavior challenges 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' and didn't master until well into my solo period.  100% is waiting for all students' undivided attention before speaking. I also set up clear expectations and consequences when starting my solo period to address electronics use.

Shadow Man

• If you taught this lesson again what changes would you make to improve the effectiveness of your teaching?

Similar to the previous lesson, Rocket Man, I would have been more patient with the students. I didn't give them enough time to formulate the problem in front of them. I told them there were similar triangles in front of them. The purpose of Dan Meyer's technique is to create initiative within the students. When the students develop the problem they have more interest in solving it.  I think if the images were set up next to each other like on Dan Meyer's blog, the concrete learners would have picked up on the similar triangles faster. It takes a lot of preparation to think of the series of questions to scaffold the problem for the students.

Many students had trouble setting up proportions as it was a review of algebra.  In the future I would use a hands on approach when reviewing proportions. The golden ratio of our body parts is a great combination of anatomy and math! Students could discover the many ratios in their bodies. One ratio is when you have your arms outstretched and measure the distance between your fingertips and compare this to your height, the two measurements are the same. The class could report different measurements and find common proportions (NCTM example).

• How did your knowledge and skills in your content area enable your students to meet the standards or Grade Expectations?

 My coursework in natural science along with math allows me to create lessons that combine multiple disciplines to make sense of real world applications (MHS:9). My strong content knowledge also helps me be proactive about common misconceptions in geometry. When starting a new concept such as similarity I can reiterate the concept of similar objects such as similar body parts, similar cars, similar shapes. This will help students learn the vocabulary and give them a real world, concrete context.

Using a real world example of similarity (MHS: 13) and having the students find similar triangles numerous times enabled my students to exceed the Grade Expectations. We found similar triangles through the shadow man activity, the problem solving activity  (student work) and practiced with (add word problems).

• What are your strengths and challenges in providing opportunities for all students to meet or exceed the standards or grade equivalents?

 I have found great strength in using a varied approach to teaching math. The multimedia approach has helped bring the a real world approach into my classroom  (MHS:9). All students, both concrete and abstract thinkers, have an idea of how big a football post is. There is a general sense that its way bigger than we are, but how much bigger?  It was fun to go through this process with the kids and then highlight the actual height. 

Our homework and homework check policy created a positive feedback system. As long as students put effort into these assignments they were awarded full credit. Reinforcing effort and providing recognition was highlighted in Marzano's Classroom Instruction that Works.  One study found that "students who were taught about the relationship between effort and achievement increased their achievement more than students who were taught techniques for time management and comprehension of new material" (Marzano, p. 51).

There is a challenge for some concrete thinkers to meet expectations which are measured through abstract diagrams. Here is a sample of a NECAP similarity question, number 13.  These questions are Type II items which "typically address more complex mental procedures .. they involve embedded  procedures for gathering information" (Marzano, p. 71).

Reflect on the two lessons together. Within the context of your reflection address the following questions:

• What changes did you make in your teaching between the two lessons and what impact did those changes have on student learning?

 Between the two lessons I improved on my 100% technique that I gained from 'Teach like a Champion'. "[If any less than 100% of students follow directions] your authority is subject to interpretation, situation, and motivation" (Lemov, p. 168).  I would wait for everyone's attention before speaking. 

During the first lesson I would spoon feed the students too quickly. During the second lesson I would give them greater 'think time' to figure out the problem.  As a new teacher, it is tempting to not let a question linger for too long.  I have become more comfortable with having the students struggle through not only finding the answer but also finding the problem. I also worked on setting a high standard of correctness by using the 'right is right' technique that I read in 'Teach like a Champion'.  Instead of jumping ahead to the solution of a problem I worked with students to develop their problem solving steps.

• What are your strengths and challenges in providing opportunities for all students to meet or exceed the standards or grade equivalents?

Bringing students real world, multimedia problems was the great strength of these two lessons. It was so nice to start a new concept with a 'hook' featuring multimedia that grabs the students attention. Oftentimes, the abstract math student gets along just fine and the concrete learner gets left behind. When orchestrated well, these two lessons have the potential to give both concrete and abstract learners even footing.

Instruction is the third domain in Danielson's Framework for Professional Practice. Danielson highlights engaging students in learning, providing feedback, communicating clearly, demonstrating flexibility, and using questioning. I believe both lessons had strong 'hooks' that started with questioning to engage all students in learning. The students got instant feedback when using whiteboards as well as the following class during HW checks.  I communicated each lesson clearly through a posted objective and agenda. I demonstrated flexibility and responsiveness by letting the students direct the problem in front of them.

The problem solving activity brought together the idea of similar triangles and dilation. The students exceeded the standards when asked to figure out the algebraic scale factor of their object's dilation in relation to their own height.  The majority of students met the expectation for standard MHS 9 and MHS 12 which we communicated through the PSA rubric. A minority of students exceeded the standards by successfully solving the scale factor.

It was a challenge to meet the needs of concrete learners in our formative assessment. The assessment was strictly abstract and many student struggled to master the similar triangles concept. 

Classroom management was a major challenge during these teaching episodes. My supervisor noted that students talked, listened to music, played iPod games, and went to the bathroom during instruction.   I learned that my expectations must be clearly stated and followed through with defined consequences. 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. If electronics were out the consequence was to put it in the 'parking lot' or the student and their iphone would be sent to the planning room.