Creating New Pathways Through Visualization

Trig - Sin Cos Tan (1)

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.

 Trig - Sin Cos Tan (3)

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.

 Trig - Sin Cos Tan (4)

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

 Trig - Sin Cos Tan (5)

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

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

Blender (logo 2)Blender is a professional free and open-source 3D computer graphics software product used for creating animations.

Planning and Implementing Differentiated Instruction

set_veen_diagram

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.

Hands-on Critical Thinking: Equilateral Triangles

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

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

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

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

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

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

EquiTri2

Step Two – Draw a line through both marks.

EquiTri4

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

EquiTri5

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

EquiTri7

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

EquiTri8

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

Best Practices: Diagonal of a Rectangular Prism

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

Diagonal of a Cube

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.

Project Based Learning: Day 7

We started class with a Check In meeting. I’ve noticed that many of the students are liking this way of starting class. It gives them an opportunity to connect with each other and to transition more effectively.

I had two tasks for them to complete today. First, I wanted them use the results from the Skill Set Survey that they produced to create groups of mixed abilities, learning styles, and personality types. If they had time left, I wanted them to get in their groups and begin brainstorming ideas in response to the driving question.

To avoid any chance of manipulation, I made copies of the Skill Set Surveys with the student names removed from each class. Then, I had 10th grade analyze the surveys from 9th grade and vice versa. I refrained from telling the students how to organize the groups. Instead, I had students volunteer to lead  the discussion while I coached them throughout the discussion process.

LeadershipIt was interesting to see the approaches that each grade took. The 9th grade class classified each survey according to their primary and secondary skills. From the results, they found Builder, Artist, and Writer to be the three basic skills. They used a grid to rank each survey and created mixed ability groups representing all three basic skills. The 10th grade class did something a little different. They classified each survey using 9 to 10 different categories. Then they realized how complicated that would be to create groups based on so many categories. They also noticed that some of the categories could merge into a broader category. Ultimately, they reduced their categories to four different skills: Leader, Builder, Artist, and Writer.

With the time left in the period, I had the students break up into their groups and begin thinking about the driving question. I wrote the driving question on the dry-erase board and encouraged them to begin wondering. This might be a small point to share, but I’ve been very intentional in the vocabulary that I use with the students. For example, I’ve purposely used words like wander, imagine, create, develop, and explore to inspire divergent thinking.

As their homework assignment, I asked them respond to these two questions:

  • What is your group’s plan? If your group doesn’t have a plan yet, what have they talked about?
  • What did you work on at home tonight?

I’m still considering other options for homework. From my research, some Project Based Learning programs de-emphasize homework. I’ve considered doing the same, but I also think it’s important for students to develop their metacognitive awareness. I plan to continue researching this before making a final decision. Any suggestions would be most appreciated!

Integrating Student Engagement in Instructional Design

instructionalDesignThere 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. 

Integrating Technology and Differentiated Instruction

mooc-learn

Using technology in the classroom not only allows the teacher to differentiate instruction but to foster the development of Digital Age Fluencies. In essence, it serves a dual purpose; however, using technology requires more than enhancing a lesson with a PowerPoint or Prezi presentation or providing the students with time to play an educational game on their iPad. In order to develop Digital Age Fluencies, using technology requires students to understand the complex and dynamic nature of the technological landscape from the perspectives of the user and of the developer. Taking this perspective has made it easier to provide differentiated instruction while still developing Digital Age Fluencies.

Ed tech

This idea actually came to me before I started teaching this school year. I knew I would be teaching Geometry and Algebra II, and considered how I would integrate technology into the curriculum in a way that would really enhance the students’ technological understanding and fluency. One of the ways that I did this was by introducing the students to Microsoft Excel. Many of them had already used Excel to create simple graphs. Instead, I taught the students how to create their own graphing calculators. First, we started small and worked with basic formulas. Then, I introduced them to Macros and created simple calculators for some of the equations we were working with. Finally, I introduced them to Visual Basic and had them work on creating their own graphing calculators.

Another way that I could integrate technology into the curriculum in order to differentiate instruction and develop Digital Age Fluencies is by taking a Project Based Learning approach. I have actually started this with a number of classes and have already seen students more engaged. In Geometry, for example, the students are creating model cities inspired by different two-dimensional shapes. Some of the students had struggled with the relationships between two-dimensional shapes and three-dimensional objects prior in the first semester. For these students, I introduced them to simulators online that helped them visualize the relationships. For the other students, who were able to identify the relationship, but struggled with visually representing it, I introduced them to three-dimensional modeling software. After spending much time with both of these forms of technology, the students understood the project better and the mathematics necessary for creating their model cities.  

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.

Engagement (2)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

Lesson Plan

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