# Creating New Pathways Through Visualization

I recently introduced my Geometry class to the basic concepts of Trigonometry. When I taught my lesson on finding the measurement of the missing angle using ratios, I noticed that many of my students were still struggling with the concepts of adjacent and opposite. I thought maybe they were confusing adjacent with hypotenuse since the hypotenuse is also adjacent to two of the angles, but they all were able to identify the hypotenuse. Even when I shared the adage SohCahToa (or SOHCAHTOA), they still struggled at identifying the adjacent leg and the opposite leg to the angle of reference. So, I created a different way to approach this. Instead of focusing purely on the values assigned to each leg, I had them represent the legs used in determining each ratio.

First, have the students draw a dot (preferably in a bright color) indicating the angle of reference.

Second, using SohCahToa, have the students determine which leg is used in the numerator and which leg is used in the denominator of the trigonometric ratio.

Third, have the students draw whichever leg is used in denominator in blue.

Fourth, have the students draw whichever leg is used in the numerator in black.

I had my students complete this process every time they worked on trigonometric ratios and it greatly helped their ability to visualize and identify the adjacent leg and the opposite leg to the angle of reference. Visualization is such a critical skill to understanding mathematics. We rely so much on visualization when we solve problems. In the early years it is one of the primary ways that we teach students to approach mathematics. What I found, though, is that visualization has a much more profound role in mathematics than in problem solving. Visualization allows us to create new pathways in our understanding of mathematics.

After having my students work through these visualization strategies, I found them identifying patterns (similar to trigonometric identities) without any knowledge of the identities themselves.

# 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.

# 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.

# Integrating Student Engagement in Instructional Design

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 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.

While the methods I listed above may not produce some type of a numerical value, they provide 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 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 instruction), we can facilitate the learning process more effectively.

# 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.

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

# 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