There are a number of methods that I use to determine if my instructional design is responsive to the needs of each student as they access the Common Core State Standards. First, I hold **daily class meetings** to check in with the students. I have been holding these meetings since the beginning of the year and have found them very successful. Second, I hold **one-on-one meetings** with each of the students every two weeks. This allows me time to ask questions and learn more about the student’s level of understanding and level of interest. Third, I host **online office hours using Google Docs** to give students the option of asking questions or expressing their concerns through text rather than through voice. Fourth, I have students maintain **progress journals**, where they reflect on their learning. Since these are their journals, I allow them to fill them out how they choose. This also allows me to see how they are processing their learning.

# Teaching Strategy

# Hands-on Critical Thinking: Equilateral Triangles

About a month ago, I was working with my Geometry class on **equilateral triangles**. Using a ruler and a compass, I demonstrated how to construct an equilateral triangle. In fact, there are a number of websites that demonstrate this the same way that I did (see below).

- Math Open Reference – http://www.mathopenref.com/constequilateral.html
- WikiHow – http://www.wikihow.com/Draw-an-Equilateral-Triangle
- TeacherTube Math – https://www.youtube.com/watch?v=t-ZtoNhEYWQ
- Khan Academy – https://www.khanacademy.org/math/geometry/geometric-constructions/polygons-inscribed-in-circles/v/constructing-equilateral-triangle-inscribed-in-circle

After having the students practice drawing their own equilateral triangles, I introduced them to a **two-part activity**.

**First,** I gave them each a blank piece of paper and asked them to construct an equilateral triangle with each side equal to six inches. With a paper that is 8.5 in x 11 in, this was a fairly simple task for the students. Many of them were able to complete in a short amount of time. Those that struggled with it were allowed to ask their neighbor for support.

**Second,** I gave them a second blank piece of paper and asked them to construct an equilateral triangle with each side equal to twelve inches. At first, they looked at the paper and thought it was impossible. They struggled with it for quite a while on their own, sketching out different designs. Then, they asked if they could talk to one of their neighbors to brainstorm and explore different ideas. Soon, the students ended up in groups of three and four, thinking through all the mathematics they knew, trying to figure out how it could be done. Finally, one group asked if they could construct the triangle in parts, cut it out, and piece it back together. That question changed the whole atmosphere of the class. Quickly, every group saw the solution and were eager to share their approach with the class.

**Steps for constructing an equilateral triangle with sides equal to twelve inches.**

**Step One** – Measure six inches from the long side of the paper on both ends and mark the paper.

**Step Two** – Draw a line through both marks.

**Step Three** – From the bottom right corner of the inner rectangle, measure twelve inches so the twelve-inch mark of the ruler crosses the opposite side (thereby creating a diagonal).

**Step Four** – Measure the distance from the left side that the diagonal crosses the top side and mark that measurement below. Draw a line through both marks.

**Step Five** – Cut the two adjacent triangles and piece together to form an equilateral triangle.

The great part about this activity was that it led perfectly into our discussion of 30-60-90 triangles. Before exploring the properties of the 30-60-90 triangles, the students were already able to see their use in constructing (and forming) other geometric figures.

# Best Practices: Diagonal of a Rectangular Prism

In class, I drew the following figure on the board:

I asked the students if they could determine the length of the diagonal *d* given the information we’ve been recently covering. I’ve been talking to them about special right triangles (i.e. 45-45-90 and 30-60-90 triangles) at length to help them gain a deeper understanding of their properties as I prepare them for trigonometry next year. I even showed them how to use the distance formula to find the length of the hypotenuse. Yet, with all this knowledge and understanding, they weren’t even sure how to begin working on the problem.

I thought about this over the weekend and wondered if there were a different way, a more engaging way, that I could use to help them through this problem. So, I went to the store and bought several spools of string. I divided the class into groups of four and gave them each a spool of string. Then, I marked two opposite corners of the classroom and presented the challenge.

**Group Challenge**

*Cut the string provided using only one cut so it may touch both corners when pulled tight. The groups could use any of the measuring devices provided (i.e. 12-inch ruler, yard stick, or measuring tape).*

This required them to figure out how to use the measurements of the floor and the walls in determining the length of the string. By giving them the condition of only being able to cut the string once, they had to attend to precision (see CCSS.MATH.PRACTICE.MP6). Also, with a variety of measuring devices provided, they needed to determine which device would give them the most accurate measurement (see CCSS.MATH.PRACTICE.MP5).

Needless to say, this offered the students a different way of thinking about finding the diagonal of a cube. Some were able to derive the distance formula on their own as they worked through the calculations. Others admitted that they understand more now about square roots and working with triangles.

# Creating a Culture That Engages Students in Learning

The school culture significantly impacts student learning and achievement in a variety of ways. By providing a safe learning environment, the students will be encouraged to develop personally, socially, and academically, at a pace that is consistent with their needs. By setting high expectations and providing rigorous academic opportunities, the students will be engaged in more meaningful learning. By providing the students with personal and academic supports, they will be able to develop strong connections with the staff and the school.

In the midst of all this, it is important for a teacher to understand his/her role. From the first day of school (or before the first day), the teacher has already begun creating a culture for teaching and learning. Usually, it is expected that teachers design and decorate their classroom. For some, this may mean rearranging their student desks in a way that best fits the teacher’s pedagogical style. For others, it may mean designing their walls and distributing supplies. Creating a syllabus and discussing it the first week of school sets the tone in many ways. From the first week to the first month, every moment spent teaching, is as much a moment of teaching as it is a moment of modeling, coaching, and leading.

One area that I think is especially important for teachers to exercise their role in creating a culture of teaching and learning is in their level of energy. For example, I love mathematics. At first, the students would chuckle at my excitement over the problems that I would challenge them with, but soon, they felt the same excitement. Interestingly, many of them doubted themselves in the beginning and refused to work on the challenging problems. Now, they wouldn’t have it any other way. In fact, in a recent class meeting, the students reflected on their level of confidence and efficacy and noted how much it has improved over the past few months.

# Lesson Planning: Integrating English Language Development with Mathematics

The lesson planning process is just that, a process. At one point, I thought that I could follow a checklist, fill in the blanks, and the result would be a solid lesson plan. I thought that maybe if I included a few strategies that addressed the learning needs of EL students and students with special needs, that I would have developed a higher quality lesson plan, but even that was not so. Developing a high quality lesson plan requires more than filling out a checklist. In mathematics, for example, the lessons follow a sequence that ultimately ties into a main concept. Knowing this, it would be beneficial to integrate the collaborative model into the lesson through problem-based learning as students connect what they have learned and apply this to a number of challenges. Through a model that thrives on communication and social interaction, it would not only help students develop their critical thinking skills, but provide them with a language-rich environment for improving their English language proficiency. According to Shahzia Pirani-Mellstrom (n.d.), “Due to this interaction students not only advance their language skills, but also learn how to be better critical thinkers by examining material together and sharing various perspectives.” This is particularly important in mathematics, where students easily grasp the calculations, but struggle with the language used to explain the concepts and read the problems given to them. By increasing the amount of time students spend communicating and collaborating with their peers, they are simultaneously developing their critical thinking skills and their English language proficiency.

Considering the diversity of students that are represented in many of our classrooms, it is important to include strategies of English language development that focus on reading, writing, speaking, and listening. Even though mathematics may not seem to be the most likely subject for including English language development, it does provide an excellent means for developing English language proficiency and mathematical fluency. Using collaborative models that thrive on problem-based and project-based learning are ideal ways of engaging students through social interaction, while providing a way to formatively assess the students’ understanding of the material. Time management is another crucial aspect of designing an effective lesson plan, “An accurate allocation of time for activities during lesson planning is critical for the lesson plan’s successful implementation” (Serdyukov & Ryan, 2008, p. 122).

In my lesson plan, I was planning to address the concept of percentages by teaching students some of the strategies for decoding the language of percentages and then working in collaborative groups to create their own posters for explaining how to use these strategies depending on the problem given. At the end of the lesson, they would have the opportunity to share their posters with the rest of the class and discuss why they chose to provide the explanations they did. Considering the time management piece of the lesson, I am still debating whether the students would be able to complete the posters in time to share their results with the class. Again, this is why managing time is so essential to the quality of a lesson plan.

**References**

Pirani-Mellstrom, S. (Interviewee). (n.d.). Successful teaching practices in action: Project-based learning for English language learners. [Interview Transcript]. Retrieved from http://ediv.alexanderstreet.com.ezproxy.nu.edu/View/1641205

Serdyukov, P. & Ryan, M. (2008). *Writing effective lesson plans*. Pearson: United States

# Project Based Learning: Day 6

Instead of starting class with a Check-In meeting, I decided to start class with a collaborative activity. I had all the students silently redesign the room to accommodate a class meeting. They were not allowed to talk to each other, but they could write notes to each other and use hand signals. The caveat was that they could not make a sound or else they lost the challenge. I did this to drive home the importance of communication and collaboration. They completed the challenge and did so successfully. After starting the meeting we reflected on the activity and they all really enjoyed it, especially what they learned from it.

The night before, I asked students to respond to the following two questions:

- Imagine a unique and creative way that you could document all that you’ve experienced and learned throughout the Project Based Learning process. Describe, draw, and/or design your idea in order to present it to the class on Tuesday. Use any medium that you wish, as long as your idea is unique and creative.
- Research different PBL activities online and find (5) unique project ideas that you would have a lot of fun doing. Your (5) project must be described in your own words (i.e. copying from the internet will not be acceptable). If you found the project idea online, provide the url somewhere in your description. To receive full credit, your (5) project ideas must be different from everyone else’s. How you all decide to compare project ideas, I’ll leave that up to. If you decide to create a document of some sort on Google, I’d ask that you share that with me.

I had all the students take a seat and share one of their project ideas. I premised this by telling them not to present their idea, but to sell us on their idea. In other words, I wanted them to consider us to be their potential investors and they had to convince us to invest in their idea. Even though they listed five project ideas for their homework assignment, I had them share one of those ideas so that I could model how the students to question or comment on each other’s ideas. As we went around the room, I engaged each student with questions to further explore their ideas.

After going around the room once, the students gained a better idea not only of how to sell their ideas, but also how to respond to the other student’s presentations. I gave them a few minutes to review their four other project ideas before having them share these ideas with the class. The students who seemed to struggle most with this part of the activity were encouraged by their peers to share whatever came to mind. Taking whatever the student said, the class brainstormed the idea to give the student something to explore.

At this point, the students were curious why we spent so much time recording and sharing five unique project ideas. This is when I introduced them to the **Genius Hour** (related to the **80/20 principle**). Essentially, **Genius Hour** is a time set aside in the schedule for the students to actively pursue their own interests and explore their passions. While most of my students were absolutely excited about the idea, I noticed that a few students seemed a little stressed out about it. They felt that it lacked the structure of a regular classroom. They were also concerned that whatever they chose to pursue would not meet my expectations. I realize that much of this stems from their response to years of learning within structured environments. So, I structured it a little more for these students to help scaffold their transition to this other type of learning.

We had about 10 minutes left. For last night’s homework, I asked the students to brainstorm different ways they could document their Project Based Learning experience. I felt that they could probably share out some of their ideas and they could vote on the best way.

Some of the ideas they suggested were:

- Picture poster/wall
- Scrapbook (tangible or virtual)
- PowerPoint presentation
- Class PBL website
- Class PBL blog
- Notebook (similar to a Lab Notebook)

After discussing the merits of each idea, the class made two decisions. First, a small group of interested students would develop a class PBL website, documenting each group’s progress. This small group would attend weekly website develop workshops that I would host during their lunch period. Second, each student would be given the freedom to choose how they document their Project Based Learning experience. The only restriction is that they must consistently update it and somehow show me their updates.

So far, everything has been working out great with these past few days dedicated to introducing the students to Project Based Learning. I’ll be researching **Genius Hour** and developing a guide for that as I go. Stay tuned for updates on that! Also, if you have any ideas or suggestions for rolling out a successful Genius Hour session, please share!

Here are some of the websites that I’ve been using to research about Genius Hour.

- Genius Hour – http://www.geniushour.com/
- LiveBinders – http://www.livebinders.com/play/play?id=829279
- WikiSpaces – https://geniushour.wikispaces.com/
- Engage Their Minds (WordPress) – https://engagetheirminds.wordpress.com/genius-hour-resources/
- Cybraryman – http://cybraryman.com/geniushour.html
- “Genius Hour and the 6 Essentials of Personalized Education” (Edutopia) – http://www.edutopia.org/blog/genius-hour-essentials-personalized-education-nichole-carter

# Student Intervention: Case Study #1

A couple years ago, I had been teaching an 8^{th} grade science class. In the back of the class sat a student, who never seemed to take class seriously. She was always talking to her friends, drawing in her notes, or sneaking text messages on her cell phone. I spoke to her parents about her behavior and they expressed the same frustration with her at home. Her grades were quickly declining, so I scheduled a parent meeting and discussed the option of afterschool tutoring. Shortly after, I began meeting with the student afterschool and tutoring her in science. A couple weeks had passed and I noticed something. She had a difficult time reading the material. I realized this after having her read one of the paragraphs aloud. I asked her about her ability to read, and she said that she needed help, but was too embarrassed to ask for help. She didn’t want her friends to make fun of her. Then, I noticed something else. Whenever I wrote anything on the marker board, I saw her straining her eyes to see what I wrote. I had her sit closer. She had to move to the front row to clearly see what I had written on the board. I asked her about her vision and she admitted that her vision was poor. She said that her parents had been struggling financially and she didn’t want to ask them to buy her glasses. I continued to tutor her afterschool, but I also made a few changes to my classroom. Since she didn’t want to lose her status among her friends, I moved all the students around and arranged it so that her group was closer to the front. I also planned my teaching to incorporate more group collaboration and rely more on reciprocated teaching. This not only helped her improve her grades in class, but it made the topic more engaging for everyone. By the end of the school year, she became one of my top performing students.

# Literacy Development in the Classroom: Round Robin Reading

In *11 Alternatives to “Round Robin” (and “Popcorn”) Reading*, Todd Finley mentions “Round Robin” Reading, an overly-criticized and antiquated method of oral reading, and provides alternative strategies to literacy development.

When I first started in the classroom, “Round Robin” Reading was highly encouraged. I tried it, but would always see students disengaged. The focus was primarily on one student while the other students needed some type of motivator to keep them engaged. What I also noticed was that many of my students struggled with reading aloud. So, I decided to read aloud to them, emphasizing all the prosodic cues within the text. Suddenly, they were all engaged. Taking this response, I began coaching my students to read with the same level of vigor. Still, I had a few students (mostly my visual learners) who were not as engaged as I had hoped. For them, I developed a strategy that allowed them to draw what they heard the student read aloud.

For more reading on “Round Robin” Reading and alternative strategies to this method:

# Rethinking the Order of Operations (i.e. PEMDAS)

I was talking to my students the other day about the Order of Operations (i.e. PEMDAS). I quickly reviewed the Order of Operations as we worked through several algebraic expressions. Then, I asked my students why we followed the Order of Operations. They said, “It’s just what we’re supposed to follow.” I decided to investigate this further and I asked them, “But why does the Order of Operations follow the order that it does?” They didn’t know.

Why do we follow the order that makes up the Order of Operations? I asked several other people and got the same response — their teacher told them it’s what they were supposed to follow. Welcome to the era of education that focused primarily on the product. Why study the reason behind the Order of Operations, when teaching students that they’re supposed to follow it works? That is like asking, “Why teach an archer about trajectory physics, when you could just teach the archer to hit a specific target?”

So, I entertained the challenge of understanding the Order of Operations further with my students. I wrote PEMDAS vertically on my marker board and analyzed the directionality of each operation.

**P**arentheses only refers to embedded operations, so we left that aside.**E**xponents were essentially a form of multiplication, since is equivalent to .**M**ultiplication and**D**ivision are usually computed together from left to right. I stopped my students at this point to consider the two different operations. Could multiplication be computed bidirectionally? Yes. What about division? If we think of division as multiplication, then could be written as (i.e. multiplying x by the multiplicative inverse of 2).**A**ddition and**S**ubtraction are much like Multiplication and Division. While addition could be computed bidirectionally, subtraction won’t allow for it. Fortunately, we are able to write subtraction as the addition of an additive inverse. For example, could be written as .

Therefore, instead of PEMDAS, we have PMA. This simplifies things a little, but it doesn’t answer the question why we follow the order of PEMDAS (even if it’s written as PMA).

Let’s visualize a situation where PEMDAS would be needed. In the above picture, we have four stacks of squares with nine circles in each square, a single square with nine circles in it, and four circles by themselves. If we had to put this in an expression, we could write it as . If we had to add up all of the blue circles, how would we do it? Would you want to count all the circles one-by-one? This may be a little time-consuming. What if we computed each of the squares as nine circles (from ). Then, if one square equals nine circles, four squares must equal 36 circles. Thus, 36 circles + 9 circles + 4 circles. Interestingly, we just followed PEMDAS. Wait! I don’t remember bringing PEMDAS into this. It seems that the reason for PEMDAS is right in front of us.

Think about it.

Four squares of nine circles + one square of nine circles + four circles

4 ( 9 circles) + 9 circles + 4 circles

36 circles + 9 circles + 4 circles

Basically, we had to convert the original expression into an expression that allowed for addition to happen with quantities of similar terms. There was no way to add a square with four circles. Instead, we had to convert the square to nine circles to allow for addition to be possible. Ultimately, PEMDAS follows PMA, ranking the operations in priority from the embedded operation (P), to multiplication (M), and then to addition (A). If addition is the final operation performed, then all quantities must be similar in order to be added. Likewise, if multiplication is the final operation performed, then all quantities must be similar in order to be multiplied.

In summary, we follow the Order of Operations to allow for multiplication and addition with quantities of similar terms.

# Classroom Management: Social Contracts

Along with involving students in the selection of topics to be covered, I would also involve them in developing behavior guidelines for the class. According to Marzano, Marzano, and Pickering (2009, p. 13), “Probably the most obvious aspect of effective classroom management involves the design and implementation of classroom rules and procedures.” In their book, Discipline with Dignity, Curwin, Mendler, and Mendler (2008, p. 68) agree with the importance of involving student in the development of the classroom rules and procedures, introducing their view of the classroom rules and procedures as a social contract, “The social contract is an agreement between teacher and students about the values, rules, and consequences for classroom behavior.” Using a social contract would not only increase student responsibility, but it would also affect their academic achievement.

**Resources**

- Curwin, R., Mendler, A., & Mendler, B. (2008).
*Discipline with dignity: New challenges, new solutions (3rd ed.)*. Alexandria, VA: Association for Supervision and Curriculum Development.

- Marzano, R., Marzano, J., & Pickering, D. (2009).
*Classroom management that works*. Upper Saddle River, NJ: Pearson Education Inc.