Over the past two weeks, the students have been learning about three-dimensional modeling. After reviewing the different characteristics and different types of two-dimensional shapes, they began exploring three-dimensional objects. Many of the students struggle with visualization, especially involving three-dimensional objects, so they spent some extra time on this concept to help them visualize the relationship between these two classes of objects (i.e. two-dimensional shapes and three-dimensional objects). First, the students had to name all the two-dimensional shapes they could see in different three-dimensional objects. Then, they observed cross-sections of all the three-dimensional objects and identified each cross-section as a two-dimensional shape. Finally, they considered ways of creating three-dimensional objects other than using a geometric net. For example, when asked how to construct a cone, most of them suggested a circular base and a sector of a circle. Using circles of different sizes, each with a slightly smaller radius than the next, the students observed the teacher place the circle with the largest base on the table. Then, the teacher stacked the circles with successively smaller radii. By time the students saw the fifth circle placed on top, they already suspected that the stack of circles was forming a cone.

Before moving on to the next concept, it was imperative that the students were able to visualize three-dimensional objects and identify two-dimensional shapes in the faces and the cross-sections of three-dimensional objects. This particular skill addresses Common Core State Standard Geometric Measurement and Dimension (G-GMD) 4, “Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects” (California Department of Education, 2013, p. 74). A test was created (as a summative assessment) that would assess the students’ ability to identify two-dimensional shapes that were represented by three-dimensional objects. The test included fifteen different three-dimensional objects that the students have seen before and have worked with on numerous occasions during the first semester. The test also included three sets of adaptations.

The first set of adaptations was designed to address the learning needs of for English Language Learners, like Student A. Student A is a 9th grade student at Orion International Academy. Her family arrived in the United States from Mexico when she was 8 years old. She is bilingual in Spanish and English. She is the oldest of three siblings. Her parents own their own business and work evenings. She is responsible for helping her two younger siblings with their homework. Student A speaks only Spanish at home and her parents depend on her to translate everything. Her CELDT results placed her at intermediate in speaking and listening and early intermediate in reading and writing. She struggles when she reads the mathematics textbook and tends to look confused when I am explaining a concept with too much mathematical jargon. Her STAR test results placed her Below Basic in mathematics and ELA. Aside from school, Student A enjoys playing sports, especially soccer and is planning to try out for the girls’ soccer team. She has very few friends at the school, mostly because the other students complain that she can be a little dominant.

Based on Student A’s needs, two different adaptations were implemented. First, the directions were read aloud, with clarification provided for the meaning of “indicate” and “represented.” Examples were given for “two-dimensional shapes” and “three-dimensional objects” to help differentiate between the two concepts. The difference between “two-dimensional” and “three-dimensional” is that “three-dimensional” implies an additional dimension of physical space, providing depth. This is also implied in the use of “shape” and “object.” For an English Language Learner, this subtle difference may cause confusion and frustration in responding to the questions. Therefore, it felt necessary to reiterate that semantic difference and clarify any misunderstanding. Second, English Language Learners were provided with a supplementary piece of paper that included different two-dimensional shapes and their respective names. One of the areas with which Student A seemed to struggle was differentiating between the use of “triangle” and “rectangle.” Both words are similar in their phonology and their morphology. For example, both words end with the morpheme “angle” and possess the letters “t” and “r” in the first syllable.

On her assessment, Student A scored 20 points out of a possible 25 points, earning her an 80%. Not only did she raise her overall grade in the class from a 70% to a 74%, but she also performed above the class average of 72%. Even though two-dimensional shapes were reviewed with the class prior to the test and Student A was provided with a supplementary sheet that included illustrations of two-dimensional shapes and their respective names, she used “triangle” and “rectangle” incorrectly for three responses. Interestingly, the first three questions involve pyramids, all three containing triangular sides, but each with a different base. Student A correctly indicated that triangles are represented in all three pyramids. She was also able to correctly indicate the two-dimensional shapes used as bases in the second and third question, but in the first question, which had a triangular base, she referred to the base as a “rectangle.” After reflecting on Student A’s responses, it seems that three of her responses were directly related to her English language development. Instead of requiring Student A, an English Language Learner, to have to write her response, it could have been more beneficial to allow her to draw the two-dimensional shapes that were represented by the given three-dimensional objects. Thus, if she wrote “rectangle,” but drew a triangle and understood it to be a triangle, then her error would be linguistic and not conceptual.

The second set of adaptations was designed to address the learning needs of students identified with special needs, like Student B. Student B is a 9th grade student at Orion International Academy. She has been diagnosed with the dyslexia and requires extra time on assignments and assessments. In fact, her California English Language Development Test (CELDT) results placed her at early intermediate in reading and writing. Her STAR test scores have always placed her at the Basic level in mathematics. She performs well on her homework and tests when it only involves calculations. When there are word problems or multi-step directions, she struggles and gets frustrated. She has very few friends and complains that the students, who are her friends, are not always nice to her. In class, she usually works alone. When she does work in groups, she applies herself only when it involves mathematical calculations or drawing. She enjoys art and tends to draw during class and needs to be constantly re-engaged by the teacher. Student B is good at dance and socializes with her friends, but has never tried out for any of the school’s sports teams.

Student B was provided the same adaptations as Student A, plus an additional adaptation. It was requested by the school that Student B receive extra time to complete all assignments and assessments. The other students were allowed 20 minutes to complete the assessment. Student B was allowed an extra 20 minutes (for a total of 40 minutes) to complete the assessment. These extra 20 minutes helped her significantly. By the end of the first 20 minutes, she had only completed the first eight questions. She still had almost half of the test to complete.

On this assessment, Student B scored a 14 out of 25, earning her a 56%, 16% lower than the class average of 72%. After this assessment, her grade for the class lowered from a 75% to a 72%. After analyzing Student B’s incorrect responses, many of them were found to be random and without any reference. It was also noticed that many of her incorrect responses were written outside of the provided response box. It seems as if Student B had started writing random names of two-dimensional shapes, hoping that she would not miss identifying any of them. In retrospect, it could have been more beneficial to Student B if she was asked to only indicate one of the two-dimensional shapes represented by the given three-dimensional objects. In fact, every two-dimensional shape that she listed first in the response boxes was a correct response. By limiting Student B to only indicating one of the two-dimensional shapes represented by each of the three-dimensional objects, it would have reduced the level of mental processing necessary for visualizing three-dimensional objects.

The third set of adaptations was designed to address the learning needs of students identified as gifted, like Student C. Student C is a 9th grade student at Orion International Academy. She is heavily involved in afterschool sports and clubs. She played on this year’s volleyball team and recently made it on the school’s basketball team. When she is not playing sports, she writes articles for the school newspaper and is treasurer for the student council. She also volunteers her time tutoring other students after school. Her parents are actively involved in the PTA and regularly volunteer their time at school events. Her STAR test results place her in Advanced in both math and ELA. She completes all of her homework on time and usually scores in the top percent on all tests and quizzes. She is always engaged and actively participates in class. During group work, Student C is usually taking the lead and assigning tasks to everyone in the group.

In addition to the directions on the assessment, Student C was also asked to describe each two-dimensional shape as specifically as she can, using mathematical vocabulary to classify the shapes. For example, if one of the faces of a three-dimensional object had a triangle with all three sides of equal length, Student C would need to specify the triangle as an “equilateral triangle.” Usually, Student C is finished with an assessment before the other students in class. Adding this extra requirement to the directions for Student C extended the time she used to complete the assessment to the full 20 minutes. On this particular assessment, Student C scored a 25 out of 25, earning her a 100%, 28% higher than the class average of 72%. After this assessment, her grade for the class increased from a 98% to a 99%.

Overall, the class average for this assessment was a 72% or 18 correct solutions out of a possible 25. Most of the students (approximately 90% of the students) experienced difficulty with the three-dimensional objects that include pentagons and hexagons (see questions #3, 8, 9, 10, and 13). Other areas that students (approximately 50% of the students) experienced difficulty were with the tetrahedron (see question #1) and the octahedron (see question #7). Even in class, many students found these two objects confusing

References

California Department for Education (2013). California Common Core State Standards, Mathematics. Retrieved from: http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf

# 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.

# Project Based Learning: Day 5

Day 5 – Introductory Week

Again, I started out class with a Check-In meeting. When we got to the tasks for the day, I wrote them out on the board. Essentially, there were two tasks that I wanted the students to complete: (1) Take the Skill Set Survey and (2) participate in a collaborative activity. The Skill Set Survey took most of the students about 10 to 15 minutes to complete. After the survey, I introduced the students to the collaborative activity. The collaborative activity is actually more of a challenge. I divided the class into three groups and distributed their materials. Then, I gave them the directions that they must construct a table from the materials provided that stands at least eight inches tall and could withstand the weight of a textbook. This was actually the first challenge. With the possibility that one or more of the groups could surpass this challenge, I created two more challenges. The second challenge requires their table to withstand the weight of three textbooks, while the third require their table to withstand a three feet book drop.

As the students worked on the activity, I circulated around the classroom, keeping a reasonable distance away from the groups. I wanted to avoid the path of becoming a “helicopter” teacher. I was excited to see how engaged each student was, but I also wanted to let them learn. I could see how this can be challenging for teachers. We’re so used to wanting to jump in, but in this moment, I learned that it’s far more beneficial to let them learn from each other. Hence, my role has begun to change from that of a “content master” to a “learning facilitator.” To be honest, I am absolutely okay with this change. If anything, it highlights my importance in helping my students along the process of learning and not necessarily on the product. Intermittently, I would step in and engage one of the groups with a series of critical thinking questions. I generally used these to help the group consider the scientific and mathematical reasoning of their design.

At the end of the activity, one group passed the first two challenges, another group passed the first challenge, and the third group was still building their table. I didn’t want to have the students leave class without considering what we just experienced, so I led them in a quick reflection. I asked them the following reflection questions:

• How well did your table do?
• How well did your group talk to each other?
• How well did your group work together?
• How could you improve this next time?

One of the things that was mentioned by a student was the need for student jobs. With the few minutes we had left, we explored all the areas of responsibility that would need to be addressed and created a list of student jobs.

Student Jobs used with PBL:

• Crew Leader (1 student) – Ensures that all students workers are fulfilling their responsibility.
• Assistant Crew Leader (1 student) – Assists the Crew Leader in all duties.
• Supply Supervisor (1 student) – Organizes supply storage facility.
• Supply Recovery Team (3 students) – Retrieves all supplies distributed and delivers to Supply Supervisor.
• Project Storage Team (3 students) – Transports projects between the work tables and the project storage facility.
• Facilities Management Team (2 students) – Ensures that the PBL environment is neat and clean upon entering and leaving the room.

Since we have a marker board in the classroom, I’ve been using that to write down a list of their daily tasks. I’m thinking of creating a PBL Resource Board with calendars, task lists, and other useful resources. I’ll take a picture and post it here as soon as I’m finished with it.

# Project Based Learning: Day 4

Day 4 – Introductory Week

I started the day by leading the students with a Check-In meeting. During this meeting, I modeled how to start with a quick question (like “How’s everybody doing today?”) to get a feel how people are feeling and what energy level they’re bring to the start of class. After the opening question, I moved on to the acknowledgements section. I feel it’s important that the students learn how to identify and acknowledge strengths in each other’s work. Plus, who doesn’t want to hear that they’re doing a good job? Following acknowledgements, I open the floor for anyone to share suggestions or vent frustrations regarding any group’s or any student’s work performance. Before starting this at a meeting, I introduced this strategy to the students clearly explaining that for this to work, we would all need to have an open mind. It actually helped me talk to the students about the concept of constructive feedback. For feedback to really be constructive, the receiving party must be open to seeing the feedback as an opportunity for growth. This is when I introduced the students to the benefits of a Growth Mindset (an idea discovered and developed by Dr. Carol Dweck of Stanford University).

Finally, we got to the main part of the meeting, the peer review process. This process is critical to developing a strong Project Based Learning program. I distributed copies of each group’s rough drafts and had the students read the first form. As they read through the form, I advised them to make notes of parts that they liked and parts that they thought could be improved. For the parts that could be improved, I asked them to provide suggestions for possible changes to the text. Once everyone got to read the form and make any necessary notes, we went around the room and shared with the group what we liked most and what we thought could be improved. During this time, I instructed the group, who authored this form, to follow along and take note of each student’s feedback. We did this for all three forms (the Group Contract, the Peer Reflection Form, and the Skill Set Assessment). By then, the period had already ended.

Even though we’ve been focusing on developing the foundations for Project Based Learning, all of the students are engaged. What’s especially important is that they’re involved in every aspect of the learning experience. They know that their opinion counts. In four days, I’ve already seen students developing a stronger presence in the classroom, taking more and more ownership of their learning.

By the way, here’s a sample agenda of a basic Check-In Meeting:

• Opening Question/Activity
• Acknowledgements
• Constructive Feedback/Opportunities for Growth

It’s a simple agenda and will probably be altered later on, but for now it serves its purpose.

# Project Based Learning: Day 3

Day 3 – Introductory Week

I started the class off with a Check-In meeting. While I modeled the basic order of the meeting, I also wanted the students to experience a sense of urgency. Unlike the Check-In meetings I modeled the past couple days, this meeting moved at a much faster pace.

During the Check-In meeting, I went around the room and asked how the students were feeling. It sounds simple, but it’s always a good idea as a group to know how everyone is feeling at the start of any endeavor. I followed by acknowledging some of the great things that students were doing yesterday and addressed any concerns that I had. Finally, I reviewed the day’s tasks and ensured that all the students understood what needed to be done and when each task needed to be completed.

Ultimately, I want the students to take ownership of this class and be agents of their own learning. In order to encourage a Project Based Learning experience in its truest essence, I thought it crucial that the students be a part of the creative process. So, I had them consider what we needed as individuals, as groups, as a class, to function effectively and successfully. They conducted research and discussed their ideas with their peers before sharing the following three items:

• Group Contract – A contract for each student within a group to sign in agreement with the expectations set forth at the beginning of each project. (Initial Accountability)
• Peer Reflection – A form that allows the students to assess their group members’ performance as well as their own. (Ongoing Accountability)
• Skill Set Assessment – A form that assesses each student’s learning style, skill set, and personal preferences. The results from each student’s assessment will be used to create heterogeneous groups of mixed abilities.

When asked to provide reasoning for deciding on these three items, the students said that they were most concerned about group dynamics and individual accountability. They felt that both areas were essential to successful collaboration.

With those suggestions, I assigned them the tasks of completing rough drafts of each by the end of class so we may present them for peer review the next day. I divided the class into three sections (designated by each of the three forms) and directed the students to choose a section that they were most interested in creating, with the condition that all three sections must have an equal (or near equal) amount of students.

After the students moved into three equal groups, they began the process of creating each of the three forms. I circulated throughout this time, keeping a fairly comfortable distance so the students wouldn’t feel like I was hovering over them. When necessary, I engaged a group with questions to help drive their thinking.

Within the last five minutes of class, I called the students together and led a Debrief Meeting1. I asked the groups to provide updates on their progress and had the students reflect on the process of collaborating. Some of the questions that I asked were:

• What did you notice that worked well?
• What do you think could have been improved?
• How could we improve that next time?
• Did we have the right tools?

They struggled a little bit with these questions, so I helped provide a few examples to get them thinking and reflecting.

Notes:

1. Debrief meetings are extremely beneficial in providing students with an opportunity to engage in reflection. They also provide a great way for the teacher to check for understanding. Throughout the semester I plan to introduce my students to different methods of debriefing. As I do, I’ll share each method on my blog for future reference.

# Project Based Learning: Day 2

Day 2 – Introductory Week

Today, I had the students gather in a circle in the middle of the classroom and I engaged them in a discussion of the following questions:

• What is Project Based Learning?
• What does a Project Based Learning classroom look like?
• How is Project Based Learning different from group projects?
• How have other schools integrating Project Based Learning?
• How are tests used with Project Based Learning?
• How is learning assessed with Project Based Learning?
• What are some examples of Project Based Learning activities?
• What is the student’s role in Project Based Learning?
• What is the teacher’s role in Project Based Learning?

Prior to this discussion, I distributed this list of questions to all the students and encouraged them to learn what they could about Project Based Learning using these questions to guide their investigation.

Since I want my students to feel more empowered, I started by coaching one of my students to lead the discussion, while another student kept record of all that was said during the discussion. (We later defined this role as the Discussion Leader.) As the students explored each question, I already saw a difference in engagement. Instead of two or three hands, I saw every hand raised up in the air when a question was asked. I saw an eagerness to share ideas and express creativity. The discussion even led to a few more questions we wrote on the board to explore further. For example, the students were interested in clearly differentiating between projects and project based learning as well as between project based learning and problem based learning.

Throughout the discussion, the students also shared a few best practices, i.e. using group contracts, peer reflections, and project calendars. With three designated work areas, I assigned each area with one of the best practices and asked the students to sit down in the area they would like to develop. I gave them the timeframe for working on each of these items, and gave them some space to brainstorm, research, share, and create. After 25 minutes, I checked in with all three groups and already saw rough drafts of each item. I alerted them that they have 5 more minutes to work on this and that rough drafts should be ready to share during tomorrow’s class for the other groups to review.

From what I’ve already learned about Project Based Learning, I feel that these two days have been very productive. The students have been showing so much excitement and have already been thinking about ways of making this experience an absolute success.

# Differentiated Instruction: Mathematics

Differentiation is a way of teaching that addresses the diverse academic needs and learning styles of students. It requires teachers to continually assess their students and respond to their learning needs in order to plan lessons that will maximize student learning. Essentially, it provides students with equity of opportunity.

Differentiation involves planning and designing a set of interrelated activities for students to work on individually, in small groups, or as a whole class. It does not involve creating unrelated activities for students to work on individually. For example, a lesson that I taught recently in Geometry required students to classify and organize a number of different polygons. A number of my high achieving students could already complete this assignment without any support, so I gave them a challenge. Instead of creating a completely unrelated assignment for them to work on, I had them investigate the more obscure polygons and determine if certain properties applied to them. Then, I had them research the platonic solids and their unique relationship to regular polygons. This addressed the learning needs of the high achieving students without deviating too far from the theme of the lesson. In fact, the assignment enhanced their understanding of polygons.

Another area in mathematics that differentiation addresses the diverse academic needs and learning styles of students is in their approach to solving problems. Too often teachers demand that students solve a problem a particular way. This not only squelches creativity, but it establishes mathematical procedures as rote operations. Differentiating mathematics to allow students the freedom to solve problems the way with which they feel most comfortable personalizes the experience of mathematical thinking. The benefit of this type of differentiation is witnessed in the level of critical inquiry that follows. Instead of every student following the same approach, they bring a different perspective to the problem – their perspective. While at first, some students may value the perspectives of others, they soon begin to appreciate the merit and strength of each perspective, thereby adding to a more comprehensive understanding of the mathematical content.

Throughout the year, I administer a number of performance-based tasks. At first, students work through the task independently. This helps them develop their own thoughts and arrive at their own solution. Then, they partner with another student, share their ideas, and create a combined solution. If time permits, the pairs of students partner with another pair of students, share their ideas, and create a combined solution. While the students present only one combined solution per group in front of the class, they have had the opportunity to share their own ideas to a number of students and listen to other students’ ideas for solving the task. In one example, the students were given a simple linear programming task to solve. Most of the students were attempting to create and graph a system of inequalities to achieve the desired results. One student, however, chose to visually represent the problem by drawing all the components. His solution was simple, much simpler than the algebraic approach. At first, the other students tried dismissing the approach, but then they considered a new approach that combined the simplicity of the one student’s approach with the procedural fluency of the other’s algebraic approach. If I were to have had the students approach the problem the same way, the one student may have struggled with the approach, while the other students may have never taken the time to reconsider their own approaches.

# Project Based Learning: Day 1

After reflecting on all that happened during the first semester, I decided to take on the challenge of adopting a different approach to teaching, i.e. Project Based Learning. Before even introducing the students to the different approach, I spent some time having them reflect on the past semester. We listed about 9 or 10 different items on the board that contributed to their struggles, narrowing it down to a single culprit that caused most of them to fell the pressure: Procrastination. Instead of stopping there, I had them delve deeper into the reasons underlying their need to procrastinate. They cited reasons such as having too many distractions (i.e. wanting to do something else or socialize with friends), feeling lazy (i.e. feeling the need to wander), or not understanding the material (i.e. not knowing where to look for the right information).

Then, I asked them, “What if I told that you we could address each of those needs if only we took a different approach to learning?” They were absolutely intrigued. I shared with them an example I created in Biology for creating a 3-dimensional version of the animal cell. The students would then be split into two groups, the offensive and the defensive groups. The offensive group would have to design a virus that were capable of successfully infiltrating the cell and defeating its defense mechanisms. The Defense group would have to redesign the animal cell, improving upon its defense mechanisms, to protect it from any possible viral attacks. They became more and more excited as I describe the challenge and were already asking me questions about the challenge. Some even asked if they could compare this situation with what happened on Return of the Jedi and take notes on what happened in the story. In a matter of minutes, I saw students that weren’t jaded by the hint at more work, but were excited and engaged!

Before jumping into the project, I wanted the students to learn a little bit about Project Based Learning. As an educator, I feel it’s important that my students understand the strategies used and reasons behind using these strategies. So, I shared some of the research on Project Based Learning with them. We starting reading a few articles and discussing the benefits and the challenges of Project Based Learning.

Tomorrow, we’ll finish our discussion on Project Based Learning and I’ll start the students on a simple project of redesigning the classroom to be more conducive to Project Based Learning.