HOPE Reflection H2 – Honor Access to Content

H2 – Honor student access to content material. Teacher candidates use multiple instructional strategies, including the principles of second language acquisition, to address student academic language ability levels and cultural and linguistic backgrounds. Many regard mathematics as a second language. Mathematica notation and use of mathematical vocabulary areLessonPlan LessonPlan 1essential to learning this topic. At the school where I work, all students are fluent english language speakers, it is unusual to encounter a language barrier in communication. Most of the language acquisition problems are through the understanding of mathematical language. To address this HOPE standard, math teachers must make the mathematical language more accessible by planning in vocabulary acquisition and teaching concepts and then naming the concepts.

The evidence I am presenting is two photocopies of some lesson plans where I am introducing two new topics. The first I am introducing students to sequences and series. In this lesson, students start with the Entry Task (ET) and are asked to complete “the list of numbers”. Since these are the teacher’s notes, the ideas are just brief notes. After the entry task, the plan is to formally put a name to “a list of numbers” which we will call a sequence. Similarly, as the lesson continues, I plan on clearly indicating and showing students the notation and the vocabulary for the notation about how to write a sequence. When I introduce combinations and permutations on the second lesson plan, I first mention “what are the possibilities of rolling two dice?” This removes technical language, the words “combination” and “permutation” and “the basic multiplication principle” are not even mentioned until the next day in class when students have acquired the conceptual understanding.

By removing the barrier of technical language students feel more comfortable with the content. The teacher will avoid the use of confusing language, but if a student uses improper language (such as the note about the difference between probability and odds in the lesson plan), the teacher will address the students misuse of language and avoid confusion of the vocabulary in the future. The HOPE standard is met because the teacher is planning for proper language acquisition and preparing students to understand content and then later naming that content when students struggle for a word to name the idea.

Over the course of my teaching internship, I have build knowledge and understanding of how to introduce new ideas to students without complicating the matter. From creating lessons that revolve around language acquisition and notation, I have learned that while I may have a deeper, technical understanding of mathematics, many students do not, and become intimidated by advanced language. By using lower level language, many of my challenged students become engaged.

Since many students are apprehensive about mathematics, this technique for introducing new ideas is helpful for students who are overwhelmed by math language and concepts. Providing instruction in an order that is helpful not to overwhelm students is important, especially in mathematics where there is a risk for pushing students away from the topic. To improve my understanding of this program standard, I will need to interact and prepare planning for more students who have different language needs, especially english language learners or those who lack significant mathematical skills. Since this HOPE standard is similar to Differentiation, I hope to address language acquisition, use of language and assisting english language learners as an improvement to my practices in differentiation.


P2 – Differentiated Instruction

P2 – Practice differentiated instruction. This means that teachers use a variety of instructional strategies or personalized instruction to help students acquire knowledge. Teachers will create opportunities for students to learn the same standards in different forms or with small modifications to fit the students’ needs.

The evidence is a series of mini-lessons presented over three days of instruction as outlined by a previous blog post found here. (LINK TO OTHER POST CLICK HERE) This post also includes some background information about the project, goals and outcomes. These lessons used student activities to help students to learn about and become familiar with vertical asymptotes, horizontal asymptotes, x- and y-intercepts and holes in a graph. Rather than providing students with direct instruction, the activities are built to facilitate student discussion around the topics and the teacher can target students with special learning needs during the activity. Each group was strategically selected to include students who brought different strengths (such as good communicator, critical thinkers in a single group). Group roles were assigned to draw out strengths or compensate for weaknesses of individual groups (quiet students were assigned as readers, critical thinkers assigned to questioner role).

Lessons 1 through 4 use student’s prior knowledge of polynomial functions to build on new understandings. Stations which revolved around asymptotes had students use limits by completing a table of values. For horizontal asymptotes, the values approached infinity and negative infinity. For Vertical asymptotes, the values approached a fixed value of x. Structuring groups with specific roles, students were able to converse and think critically about each of the four topics. Since the conversations were NOT teacher lead, students could explain to each other concepts they were unsure of. Most importantly, I would circulate the room during the activity to check on students progress and assess needs or misunderstandings with groups of about 4 students. I would target groups that were working fast to ensure they understood the intricacies of the activity and would provide challenge or extending information to groups who were able to build on more complex ideas.

After completing this activity, I learned that station learning can be valuable but should be thought through carefully. I would reconsider several processes to make this better.

  1. Allow more time for students to complete the activities. Some groups seemed rushed and were not able to complete ideas.
  2. Debrief with groups after each activity to ensure students understood the purpose of each question.
  3. Provide a little bit of direct instruction before turning to station learning activity to motivate the learning more.
  4. Remove the unit about holes since it is not a standard, but a good to know topic.
  5. I would remove the idea of making the students physically move around the room during the activity, this wasted time.

There are some pieces of learning that I thought were beneficial to the activity.

  1. Assigning students to groups to ensure there are a variety of learners in each group of learning.
  2. Assigning group roles to draw out strengths of students to benefit others in the group.
  3. Circulating the room to provide direct instruction as needed rather than lecturing at the front of the room. The dynamic of a teacher roaming helps students by providing small group instruction AND if the teacher is unavailable, groups must work together to problem solve before asking for assistance and receiving help. The delayed gratification is more effective because students are more receptive to the learning (Meyer, 2010).

While many of the suggestions above would help students learn and are keys to improving the instruction better for next time, I can continue to improve by learning and practicing differentiated instruction and providing alternate means of learning to students when station activities are not being used, such as times when direct instruction is used more. There is more research and practice that can be learned.

References: Meyer, D. (Speaker) (2010, March 1). Math class needs a makeover. TEDxNYED. Lecture conducted from TED Conferences, LLC, New York City.

P1 – Intentional Inquiry and Planning

P1 – Practice intentional inquiry and planning for instruction. This means that classroom teacher plans instruction around a learning target and creates a lesson which encourages students to critically think about the function of the learning target. Within the realm of mathematics, students should build intuition around how mathematical processes and can therefore build from current understandings to unique situations in the future. For a teacher, building this intuition must be well planned.

Student Work Matrix POGILStudent Work Matrix POGIL2Student Work Matrix POGIL3 Student Work Matrix POGIL4

Student Work Matrix POGIL5Student Work Matrix POGIL6

In the images, I’m presenting the work of a group of students who completed a group inquiry activity that was monitored me. While POGIL (Process Oriented Guided Inquiry Learning) activities are highly structured and I lack the training to adequately design a true POGIL, this is my best attempt at guiding students through mathematical concepts using discovery as the motivator for the lesson. This POGIL is about matrix multiplication, the purpose of this activity was to build on their prior experiences of matrices, create intuition about mathematical practices, and assist students in making meaning behind matrix multiplication. Many students know the procedure, yet few understood how this process was applicable in the real world.

This classroom activity was built and designed by me, although I used a textbook to find problems with student interest in mind and adapted the questions to fit my instructional goals. Since this activity was designed by me, this demonstrates I am able to ask students good questions which lead to conceptual understanding. This also shows my ability to plan for 100 minutes of instruction and facilitate an activity, probing students for more advanced thinking.

While planning this activity, I learned about how challenging it is to create clear questions which lead students to understanding of the learning targets. Since I teach multiple sections of the same course, after each class, I revised my questions to ensure each question challenges students and leads them to more complete understanding of matrix multiplication. Students also state they enjoy the POGIL’s as a learning activity. Students get to work in groups and ask questions to their peers. Providing group based activities, students break the routine of back to back 100 minute learning segments. Additionally, this provides students the opportunity to practice new skills without the traditional “drill and kill” of many math classroom. Practicing with inquiry also helps students create meaningful understanding rather than the process of symbol manipulation alone.

When designing lessons, it may be useful within my lesson plans to prepare questions each day which probe at the students understanding. Also, I think that creating a classroom goal everyday (and actively writing it down in the lesson plan) will help lead to a meaningful result from the lesson. With a goal like “Students can understand the meaning and operations of matrix multiplication,” I can create quality lessons, ask questions which probe for understanding, and measure the effectiveness of my lesson.

Instructional Strategies Observation

This observation compares two very different types of instruction instruction strategies between STEM related topics. The first strategy is project- and inquiry-based instruction the other is a game to demonstrate a concept. In the first class, a class titled “The Physics of Flight,” students are tasked with creating a protection system for a payload on a bottle rocket they will launch at the end of the week. Students are provided a budget, materials and a critical friend who must approve the design before the build. Students must use their knowledge of drag, friction, air pressure and mass (topics of physics) to design their payload protection system to minimize damage. Students who are careful with their design and focus on the prior knowledge built more robust systems.

The project is a long term project where students will revise their plans and rebuild their payload protection system many times as they learn more about the physics required for flying and space. What I like about this project and instructional strategy is that it is very real world. Students have to work within a budget, they need to be creative, their plans need to be approved by a critical friend and finally they can actually build and test their end product and have the opportunity to revise their original plans. I asked a student about what they would do differently, they mentioned that they would not have used such heavy material to protect their payload because the mass is difficult to slow down when the object is falling. They need a lighter protection system to be slower. I think these students are really learning about the concepts of physics in a real world environment. Some students were confident in their protection systems and the teacher didn’t challenge their thinking much after they took their mind off the task. If I were to provide feedback I would encourage this teacher to talk one on one with the students who claimed they were done and ask them about how their learning changed design elements on their product. This would re-engage these students who felt they already knew how to do the activity well. I think that mathematical modeling is one of the most useful applications of math, so I may use the project based strategy to provide a project for my students to apply their math knowledge to the real world.

The other instructional strategy that I observed was a game to unpack a scientific concept. The students were studying the carbon cycle and the teacher wanted to emphasize that particles of carbon get stuck in different areas. For instance, carbon that forms oil will be stuck in the ground for a long time until it is drilled up and then moved through the air as oil emissions. Students played a game were each student was a carbon molecule and they started evenly distributed. Students would roll dice and read a legend to determine their fate as a carbon. Some tabled became very full while others were less full because carbon stays in certain forms longer. Students recorded their fate and then at the end of the game the teacher had students discuss what happened to their molecule. I think this was beneficial since it was an activity where students could move around the classroom and see/feel what a carbon would be in the larger scheme of the carbon cycle. I especially liked that the class debriefed the activity so that those student who could not make the conclusion about the activity could be clued into what learning was supposed to take place. This type of activity could implemented in a statistics unit where randomness can be visualized.

Between these two instruction strategies, I think they were both effective because they had clear goal for the students and were well planned out. Students were able to articulate the goals of the activity and the activity was differentiated so learning could be achieved despite different learning styles. The take away from these observations was that I need to incorporate more movement into my classroom and differentiate instruction with intentional activities for students.

Bloggary #5: Writing Workshop for Math

Daniels, Zemelman and Steineke (2007) suggest teachers implement a writing workshop in classes providing critical feedback for emerging writers. Writing in a math class may seem foreign to an outsider, but writing can help students articulate their mathematical understanding. In fact, mathematical language is highly technical and best practices include direct writing with clear concise explanations and supporting graphics. The style of writing is very different from expository or novels and requires great attention to detail.

High school students taking Advanced Placement (AP) classes are exposed to a style of writing called Free Response Questions (FRQ’s). For math classes, many students are not previously exposed to FRQ writing which requires a direct style supported by evidence and facts. Educators wait until junior and senior level classes to present FRQ style writing which is unfortunate because the thought required to write such documents is invaluable. Teaching students to write FRQ’s using the Writing Workshop would provide students with tools to exceed standards on AP exams. Math writing tends to be a little bit more focused than expository writing since students are limited by their content knowledge. However, if teachers take a “toolbox” approach to the learning, students can access many previous skills. The “toolbox” approach is where skills are learned and then stored in a student’s toolbox as a collection to access when approaching challenging problems.

Implementation of a writing workshop would not be challenging for an instructor. FRQ’s are already written with grading materials easily accessible for teachers and provided by the College Board (the testing agency for AP tests). Since all previous tests are published and accessible online, students will have no problem selecting a topic of their choice to write a full response. Teachers can select specific criteria for grading the writing aspect while the published solutions can be used for the accuracy of the mathematics. Features math teachers may look for include; precision of language, supported claims (words, pictures or other) and organization of thought. In my classroom, I envision a workshop where each student works on up to five FRQ’s at one time and uses partner grading or teacher conferencing to work through many technical writing issues. Students would be presented with a model and brief writing instruction and their work packet would be graded for specific writing elements to help students articulate their argument and write clearly.

Since teaching a topic is a true indicator of understanding, the project could be extended by having students write their own FRQ’s and create a grading rubric for future students. Students could set goals for what they hope to gain from their writing, possibly understanding a challenging concept more deeply. Students need not be limited to the domain of calculus, but rather they could use their creative license to write a question for students in lower classes. This intense focus and peer review helps students engage in the material for a longer period of time which increases the chances of comprehending the content (Borich, 2014). Since the AP exam expects content mastery, the writing workshop technique can help students engage in challenging material for longer.


Borich, G. D. (2014). Effective Teaching Methods: Research-Based Practice (8th ed.). Upper Saddle River, NJ.: Pearson Education, Inc.

Daniels, H., Zemelman, S., & Steineke, N. (2007). Content-Area Writing: Every Teachers Guide ( ed.). Portsmouth, NH: Heinemann.

M4 Reflection: Cognitive Development to Assist Student Learning

H1 – Honor student diversity and development. To me, this standard means that teachers should be studying current and relevant research articles which pertain to educational development and work to apply the findings within the classroom. Modern educators have the duty to use cognitive research surrounding student learning in their classrooms. After reading “Brain Rules,” (Medina, 2008) chapters about short and long term memory, researches have clarified many tools educators can use to help student retain information. Medina states that sadly “People usually forget 90 percent of what they learn in a class within 30 days”(p. 100, 2008). Many classmates have suggested tools for helping students remember, two stands out in particular. First, using cumulative tests and quizzes encourages students to revisit old ideas. According to Medina (2008, pp.147), “The way to make long-term memories more reliable is to incorporate new information gradually and repeat it in timed intervals.” Other class suggestions for repeating information was incorporating games in class as review at the end of a unit (Benton, 2014).

Another cognitive development strategy which can foster learning in the classroom is constructivism teaching. With constructivism, teachers build upon older ideas, this too requires students to rehearse old concepts for long term memory storage. Particularly in math, where content continually builds onto itself, constructivism provides teachers the opportunity to remind students about how their old knowledge can apply to a new situation. Prompting and coaching can be assistive for this teaching strategy in which students develop critical thinking skills consistent with their own perceptions of information. This does not imply teacher provide answers, but rather give students resources to direct their learning. Students can learn to adapt their current schematic understanding of math concepts and adjust their thinking of new information when teachers construct learning to build upon itself.

Constructivism Evidence

Teaching methods assist educators in implementing these cognitive skills into the classroom. Students can review concepts through words-, quote-, person-of the day, suggested by Antje Brewer (2014) when supported by rich class discussions. Additionally, providing students with opportunities to explore, though class projects can encourage curious learners similar to that of Medina’s explorations through his childhood (2008, pp. 272). Using many strategies to encourage cognitive development helps educators teach their students through proven, research based discoveries of cognitive development.



Benton, Alex. (2014, July 17). Teaching Strategies to Help Encoding & Consolidation. BlackBoard Discussion Post. Module 3. (Web)

Brewer, Antje. (2014, July 16). Teaching Strategies to Help Encoding & Consolidation. BlackBoard Discussion Post. Module 4. (Web)

Medina, J. (2008). Brain rules: 12 principles for surviving and thriving at work, home, and school. Seattle, WA: Pear Press.

Bloggary #3: Writing for the Next Generation

Writing to communicate with others has been around for centuries, originally starting with cave drawings and paintings and then developing into more formal writing systems. Some educators claim that in recent years students have become less able to communicate using effective writing. However, I believe that students are no worse, the writing style has evolved with new technology. Writing experts Daniels, Zemelman and Steineke (2007) mention students distaste in writing, “Teachers often sat that kids hate writing. But maybe what they hate is the kind of writing we make them to do” (p. 3). New technology encourages writing, Facebook and text messages are the main form of communication for the Millennial generation (those born into the technology revolution). The Baby Boomer generation (those born within 20 years of WWII) and Generation X (those born between the Baby Boomers and Millennials) have quickly adopted email as an effective form of replacing snail mail. Hence, people enjoy writing, just maybe not the type of writing that is valued in academia. Recognizing the differences between generations can help understand how writing has changed throughout the years.

John Seely Brown (a Baby Boomer) is credited with creating the first prototype for the modern spell checker (Krishnamurthy, 2005). Many improvements have been added throughout the years, including grammar correction (the dreaded little green line) in computer word processors. These innovations have assisted in the so called “writing crisis” facing public education. However, if we can harness the current writing style, educators can use student writing as a learning tool. Too often writing is limited to english and history classes, however, if educators use writing to help students reflect on their work, explain their ideas and learn how to articulate step by step instructions, writing has a valuable place in the education system.

Daniels et al. (2007) suggest exit slips and reflective journaling to use writing as a learning tool. Exit slips are a form of reflective journalling for students to complete after a lesson. Given a prompt, this tool can help students think about their learning and identify improvement areas, or it could provide vital feedback for teachers (p. 35-39). Future lessons can be constructed using student feedback to identify areas of struggle or class improvement suggestions. Exit slips are just as valuable for the student as they are for the teacher. Additionally, the reflective aspect helps students review concepts before moving to their next class. Reviewing material supports the cognitive process of storing short term memories into long term memories (Medina, 2008).

Within math and science topics, students are frequently overwhelmed with information in the content they are reading. By adopting a less formal writing techniques, teachers can use drawings as a form of writing knowledge obtained from a passage (Daniels et al., 2007). Encouraging students to write down all of the information in a complicated problem helps them articulate clear understanding of their reading. In drawing form, students can use math and science skills to learn within that domain. After, writing is important to explain the steps and procedures for solving them problem. Another approach to drawing is to use clusters to connect central concepts (Daniels et al., 2007). In a math classroom, making a cluster map can visually represent central ideas together and make connections between the mathematical concepts.

Finally, Daniels et al. (2007) claim, students remember “50 percent of what they see and hear, 70 percent of what they say and write and 90 percent of what they say as they do a thing” (p. 26). While visiting a mentor teacher, I observed a project where students created study guides and learning tools for future classes. They were encouraged to engage with the material as they use peer to peer writing techniques to explain the math skills they used throughout the year. According to the instructor, peer teaching demonstrates a clear understanding of the topic.

So, writing is being used in creative ways that are less formal than essays or analysis papers, it is being used to converse and effectively communicate with teachers and other students. Writing in high schools is not in crisis mode, rather the culture of formal writing is slowly changing to meet modern needs for effective writing. Teachers have invented and adopted methods of written communication that will help lead to effective workplace communicators.



Daniels, H., Zemelman, S., & Steineke, N. (2007). Content-Area Writing: Every Teachers Guide ( ed.). Portsmouth, NH: Heinemann.

Krishnamurthy, S. (2005, October 19). A-List overdue on campus. The Michigan Daily.

Medina, J. (2008). Brain rules: 12 principles for surviving and thriving at work, home, and school. Seattle, WA: Pear Press.