# Differentiated Instruction: Engaging Students at a Whole New Level

Differentiated instruction provides insight into the students’ level of engagement with the subject.  For example, one student from my Algebra II class had struggled with the material covered during the first semester. Her scores on tests and quizzes ranged from 60% to 80%. She submitted most of her homework. When I spoke to the student, she said that she wanted to understand the material better, but she did not know how to study. We tried several methods, and nothing worked. I investigated further and discovered that she really loved creating art on the computer. So, I introduced her to several programs online (e.g. Blender , Desmos, and Scratch by MIT) that she could use to explore three-dimensional modeling. She got excited and started working on them instantly. After a week, she found herself struggling to make some of the objects the right size or place them in the right position. That is when I introduced her to the mathematics used in three-dimensional modeling. Instantly, she wanted to learn as much as she could about graphing two- and three-dimensional equations. What I learned from this experience was that engagement is crucial to the learning process. The only way that this could have been this successful was by consistently engaging the student using the methods listed above. By working closely with students and helping them explore the material in their own way (i.e. differentiated learning), we can facilitate the learning process more effectively.

Scratch is a free programming language where you can create your own interactive stories, games, and animations.

Graph functions, plot tables of data, evaluate equations, explore transformations, and much more – for free!

Blender is a professional free and open-source 3D computer graphics software product used for creating animations.

# Planning and Implementing Differentiated Instruction

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.

# 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.

# Alternative Assessments: Multiple-Choice Tests

Have you ever given a multiple-choice test and wondered whether restricting your students’ creativity to a small number of available options truly assessed their comprehension of a concept? I have, but I didn’t toss the multiple-choices out. Call me an optimist, but I always try to find the benefit of something.

I asked myself, “What could possibly be the benefit of giving a multiple-choice test?” The last time I created a multiple-choice test, I found it far more involved than creating other types of assessments. Once I typed out the question, I knew the correct answer and struggled to write three more false answers for a total of four possible answers.  So, I wondered if I could harness some of that critical thinking that went into creating the multiple-choice test. What if I could take a different approach to this type of test that required a higher level of critical thinking? I thought for a while about it and decided to create a complete multiple-choice test, but without the questions.

That’s right! I gave my students a multiple-choice test with four answer choices (with the correct answer marked) for each question, but without the questions. Using the answer choices provided, they had to figure out what the question could have been. The first time that I tried it, the students seemed extremely engaged in the different way of thinking. What I liked about it was that it really got them to critically think about the material before writing down a question.

What alternative assessments have you created to inspire critical thinking?

# Multiple Intelligences: A Different Type of Survey

In years past, I tried using a Multiple Intelligences Inventory like the surveys that many of us have seen, but found it to be counterintuitive. Instead, I use centers as a way of assessing and identifying my students’ needs. At the beginning of the school year, I arranged the classroom into centers with different types of math activities for them to explore. For example, one center focused on visual learners by engaging the students in an art activity involving mathematics. Another center focused on linguistics learners and involved word problems and logic puzzles. A third center focused on kinesthetic learners and had manipulatives for the students to explore mathematical concepts. At each of these centers, I had activities for the students to engage in independently (i.e. intrapersonal learners) or with a friend (i.e. interpersonal learners). By allowing the students to choose the center and activity they would like to participate, I was able to assess their style of learning as well as their learning needs.

In one class that I did this, I learned that many of my students appealed to the center that involved art, while a small number of them appealed to the center that involved word problems and logic puzzles. For this class, I approached mathematics from a more visual perspective, while providing the students with resources online for them to engage in further reading. I had students complete daily math journals where they had to summarize their learning by writing or illustrating their thoughts. I also provided the students with the means to solve problems visually (e.g. bar diagrams) or linguistically (e.g. written explanations).

In another class, I found that many of my students were absolutely uninterested in the logic games. Interestingly, this was my Geometry class, which addresses deductive and inductive logic throughout most of the course. After reflecting on the students’ needs, I decided to steer away from the traditional axiomatic approach to Geometry, and embraced a more project-based approach. Many of these students were interpersonal, linguistics, and bodily-kinesthetic learners. Therefore, I designed a variety of activities for them to explore the concepts of Geometry while collaborating in groups on projects that they designed and built.

For more information on the application of Multiple Intelligences Theory in Mathematics Education:

# 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.

• Parentheses only refers to embedded operations, so we left that aside.
• Exponents were essentially a form of multiplication, since  $x^2$ is equivalent to  $x \cdot x$.
• Multiplication and Division 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  $\frac x 2$  could be written as  $x \cdot \frac 1 2$  (i.e. multiplying x by the multiplicative inverse of 2).
• Addition and Subtraction 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,  $5-3$  could be written as  $5 + (-3)$.

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  $4(3^2) + 3^2 + 4$. 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  $3^2 = 9$). 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.

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