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 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


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.

Differentiated Instruction: Assessment Adaptations

Geometric Solids 2

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.

Geometry - Student AOn 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.

Geometry - Student BOn 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.

Geometry - Student CIn 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


California Department for Education (2013). California Common Core State Standards, Mathematics. Retrieved from:

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

Project Based Learning: Day 4

Day 4 – Introductory Week

feedback_for_teachersI 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
  • Task(s) Overview

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.

Student DiscussionAfter 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.


  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.

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.