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.


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 Technology and Differentiated Instruction


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.  

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.


Pirani-Mellstrom, S. (Interviewee). (n.d.). Successful teaching practices in action: Project-based learning for English language learners. [Interview Transcript]. Retrieved from

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

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.

Differentiation 3Another 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.

Assessment Accommodations

Pre-assessments, formative assessments, and summative assessments are all methods of assessing, monitoring, and evaluating student learning. Pre-assessments give the teacher insight on each student’s ability or level of understanding prior to starting a lesson. This allows the teacher to make any necessary changes to his/her lesson in to adapt to the learning needs of the students. In mathematics at the high school level, pre-assessments are usually implemented by assigning the students a problem on the board to solve. The students work through the problem on their individual white boards and show the teacher their answer. Another way that this may be implemented is by asking students to analyze any errors in a worked out example to assess their ability to critique the reasoning of others.  Depending on the lesson following the pre-assessment, it may be administered the day before to give the teacher the opportunity to address any misconceptions in the upcoming lesson the next day. One way to accommodate the pre-assessment is to have students explain to their partners what the activity is and what they need to do before attempting it. This will help address any misunderstanding they may have about the activity.

Formative assessments monitor student learning by providing the teacher with ongoing feedback. Instead of waiting until the end of a lesson sequence, the teacher monitors his/her students’ learning throughout the entire process of the learning sequence. They are not only a way of monitoring student progress from the teacher’s perspective, but a way for teachers to guide their students’ learning through reflection. In mathematics at the high school level, formative assessments take place in a variety of ways. For example, the teacher may use the time that students are working with their partners to circulate around the classroom and engage each of the groups. The teacher may periodically have the students use their individual white boards to work on a particular problem. Other forms of formative assessments may be a little more extensive. For example, the teacher could engage the students in a performance-based assessment task or a project-based learning activity. Both of these could last a little longer than a class period, but the level of student engagement and participation provides the teacher with a wealth of information regarding the students’ current level of understanding. One way to accommodate this type of assessment is by grouping particular students with other students who are willing to provide extra help during the assessment. This will allow those students to get the support they need from their peers as they work through the assessment.

Summative assessments are usually given to evaluate student learning at the end of a learning sequence, limiting the teacher’s ability to monitor his/her students’ progress throughout the lesson sequence. Summative assessments that are used more periodically allow a teacher to reflect on the results of those assessments and gauge their teaching appropriately. In mathematics at the high school level, these usually come in the form of a chapter test, but not always. They can also come in the form of weekly quizzes and mid-chapter quizzes. The format of the assessment varies according to the material being covered. One way to accommodate summative assessments is by providing students with a menu of questions, allowing them to choose the questions that they feel more confident answering.

Alternative Assessments: Multiple-Choice Tests

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

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

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

What alternative assessments have you created to inspire critical thinking?

Classroom Management: Social Contracts

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


Classroom Management: Student Involvement

Classroom management is essential in providing students with an environment that is conducive to learning. According to Marzano and Marzano (2003), “One of the classroom teacher’s most important jobs is managing the classroom effectively.” There are several factors that highly impact the effectiveness of how a classroom is managed. First, the teacher must utilize instructional strategies and curriculum design that foster student engagement and accommodate the students’ learning needs, “Effective instructional strategies and good classroom curriculum design are built on the foundation of effective classroom management” (Marzano, Marzano, & Pickering, 2009, p. 4). While the teacher should be knowledgeable of current pedagogical trends, it is important that the students understand the instructional strategies being used and participate in planning the curriculum to be covered. By involving students in the selection of the topics to be covered, the students are more intrinsically motivated to participate in the lessons. Likewise, the number of topics decreases, allowing for deeper exploration of each topic. According to Glasser’s concept of a quality curriculum, “Quality learning requires depth of understanding together with a good grasp of its usefulness. Learning a smaller number of topics very well is always preferable to covering many topics superficially,” (Charles, 2005, p. 78). In fact, fewer topics allow for greater clarity and specificity.



Workshop: Introduction to the Standards for Mathematical Practice

121diagramEarlier today, I had the opportunity to deliver a workshop on the Standards for Mathematical Practice (SMP) to a room of OST staff. The workshop was designed to:

  • Help them develop a basic understanding of the SMP,
  • Familiarize them with the language used in the SMP,
  • Discuss strategies for coaching their staff on the SMP.

After reviewing the workshop’s Agenda and introducing the Learning Goals, I presented them with a picture of a house and asked them, “What do we need to build a house?” Before moving to the next slide, I had the participants share their ideas of what was needed to build a house. I wrote their responses on the board and together we drew arrows between each idea, creating a sequence of events. Then, we moved through the next eight slides, comparing the information on the slides with the ideas that they shared. The order of the slides paralleled the SMP to provide the context for comparing the different steps of building a house with the SMP. I did this so that the participants focused on the process of building a house as a prelude to focusing on the process of solving problems. The emphasis here is on the process of solving than on the product of solving (i.e. the answer). Ultimately, I wanted the staff to witness the learning opportunities that could take place when the teacher focused more on the process of solving problems than on the product (i.e. the answer). I concluded this series of slides with a question that required the participants to consider what building a house has to do with mathematics. I used a general question to allow for greater discussion.

We transitioned to TED Talks video presented by Dan Meyer, in which he discussed the need to rethink how we engage students in mathematics. This led us into an introduction to the SMP. Two slides are provided for each SMP. On the first slide, the SMP is listed with a few bulleted focus items to consider when addressing the SMP. On the second slide, I provided two to three strategies to utilize when addressing the SMP. Instead of reading aloud the information on each slide for the participants, I engaged them in a discussion of what the SMP means to them and how they could address it in the classroom. We used the information provided on the slides to validate their ideas and consider alternatives.

Then, we explored three different problems that are samples provided by the SBAC. Two of the problems provided are third grade level problems and one of the problems is a sixth grade level problem. As we approached each problem, I modeled for the participants how their staff should be guiding their students’ thinking. Instead of disseminating information, I engaged the participants in:

  • Thinking about the problem on their own and writing down their ideas
  • Discussing the problem and their ideas with their partner
  • Conceptualizing the problem using different visual models
  • Sharing their solution and their reasoning with the rest of the class
  • Ask questions and critiquing the reasoning of others
  • Defending their solutions based on evidence provided in the problem

Finally, we concluded the workshop by brainstorming strategies for introducing the SMP to their staff and coaching them on addressing the SMP in their classrooms.

Slideshare link: