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.
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.
It 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!
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
California Department for Education (2013). California Common Core State Standards, Mathematics. Retrieved from: http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf
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 http://ediv.alexanderstreet.com.ezproxy.nu.edu/View/1641205
Serdyukov, P. & Ryan, M. (2008). Writing effective lesson plans. Pearson: United States
Day 3 – Introductory Week
I started the class off with a Check-In meeting. While I modeled the basic order of the meeting, I also wanted the students to experience a sense of urgency. Unlike the Check-In meetings I modeled the past couple days, this meeting moved at a much faster pace.
During the Check-In meeting, I went around the room and asked how the students were feeling. It sounds simple, but it’s always a good idea as a group to know how everyone is feeling at the start of any endeavor. I followed by acknowledging some of the great things that students were doing yesterday and addressed any concerns that I had. Finally, I reviewed the day’s tasks and ensured that all the students understood what needed to be done and when each task needed to be completed.
Ultimately, I want the students to take ownership of this class and be agents of their own learning. In order to encourage a Project Based Learning experience in its truest essence, I thought it crucial that the students be a part of the creative process. So, I had them consider what we needed as individuals, as groups, as a class, to function effectively and successfully. They conducted research and discussed their ideas with their peers before sharing the following three items:
- Group Contract – A contract for each student within a group to sign in agreement with the expectations set forth at the beginning of each project. (Initial Accountability)
- Peer Reflection – A form that allows the students to assess their group members’ performance as well as their own. (Ongoing Accountability)
- Skill Set Assessment – A form that assesses each student’s learning style, skill set, and personal preferences. The results from each student’s assessment will be used to create heterogeneous groups of mixed abilities.
When asked to provide reasoning for deciding on these three items, the students said that they were most concerned about group dynamics and individual accountability. They felt that both areas were essential to successful collaboration.
With those suggestions, I assigned them the tasks of completing rough drafts of each by the end of class so we may present them for peer review the next day. I divided the class into three sections (designated by each of the three forms) and directed the students to choose a section that they were most interested in creating, with the condition that all three sections must have an equal (or near equal) amount of students.
After the students moved into three equal groups, they began the process of creating each of the three forms. I circulated throughout this time, keeping a fairly comfortable distance so the students wouldn’t feel like I was hovering over them. When necessary, I engaged a group with questions to help drive their thinking.
Within the last five minutes of class, I called the students together and led a Debrief Meeting1. I asked the groups to provide updates on their progress and had the students reflect on the process of collaborating. Some of the questions that I asked were:
- What did you notice that worked well?
- What do you think could have been improved?
- How could we improve that next time?
- Did we have the right tools?
They struggled a little bit with these questions, so I helped provide a few examples to get them thinking and reflecting.
- 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.
Differentiation is a way of teaching that addresses the diverse academic needs and learning styles of students. It requires teachers to continually assess their students and respond to their learning needs in order to plan lessons that will maximize student learning. Essentially, it provides students with equity of opportunity.
Differentiation involves planning and designing a set of interrelated activities for students to work on individually, in small groups, or as a whole class. It does not involve creating unrelated activities for students to work on individually. For example, a lesson that I taught recently in Geometry required students to classify and organize a number of different polygons. A number of my high achieving students could already complete this assignment without any support, so I gave them a challenge. Instead of creating a completely unrelated assignment for them to work on, I had them investigate the more obscure polygons and determine if certain properties applied to them. Then, I had them research the platonic solids and their unique relationship to regular polygons. This addressed the learning needs of the high achieving students without deviating too far from the theme of the lesson. In fact, the assignment enhanced their understanding of polygons.
Another area in mathematics that differentiation addresses the diverse academic needs and learning styles of students is in their approach to solving problems. Too often teachers demand that students solve a problem a particular way. This not only squelches creativity, but it establishes mathematical procedures as rote operations. Differentiating mathematics to allow students the freedom to solve problems the way with which they feel most comfortable personalizes the experience of mathematical thinking. The benefit of this type of differentiation is witnessed in the level of critical inquiry that follows. Instead of every student following the same approach, they bring a different perspective to the problem – their perspective. While at first, some students may value the perspectives of others, they soon begin to appreciate the merit and strength of each perspective, thereby adding to a more comprehensive understanding of the mathematical content.
Throughout the year, I administer a number of performance-based tasks. At first, students work through the task independently. This helps them develop their own thoughts and arrive at their own solution. Then, they partner with another student, share their ideas, and create a combined solution. If time permits, the pairs of students partner with another pair of students, share their ideas, and create a combined solution. While the students present only one combined solution per group in front of the class, they have had the opportunity to share their own ideas to a number of students and listen to other students’ ideas for solving the task. In one example, the students were given a simple linear programming task to solve. Most of the students were attempting to create and graph a system of inequalities to achieve the desired results. One student, however, chose to visually represent the problem by drawing all the components. His solution was simple, much simpler than the algebraic approach. At first, the other students tried dismissing the approach, but then they considered a new approach that combined the simplicity of the one student’s approach with the procedural fluency of the other’s algebraic approach. If I were to have had the students approach the problem the same way, the one student may have struggled with the approach, while the other students may have never taken the time to reconsider their own approaches.
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.
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.
I 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?
In years past, I tried using a Multiple Intelligences Inventory like the surveys that many of us have seen, but found it to be counterintuitive. Instead, I use centers as a way of assessing and identifying my students’ needs. At the beginning of the school year, I arranged the classroom into centers with different types of math activities for them to explore. For example, one center focused on visual learners by engaging the students in an art activity involving mathematics. Another center focused on linguistics learners and involved word problems and logic puzzles. A third center focused on kinesthetic learners and had manipulatives for the students to explore mathematical concepts. At each of these centers, I had activities for the students to engage in independently (i.e. intrapersonal learners) or with a friend (i.e. interpersonal learners). By allowing the students to choose the center and activity they would like to participate, I was able to assess their style of learning as well as their learning needs.
In one class that I did this, I learned that many of my students appealed to the center that involved art, while a small number of them appealed to the center that involved word problems and logic puzzles. For this class, I approached mathematics from a more visual perspective, while providing the students with resources online for them to engage in further reading. I had students complete daily math journals where they had to summarize their learning by writing or illustrating their thoughts. I also provided the students with the means to solve problems visually (e.g. bar diagrams) or linguistically (e.g. written explanations).
In another class, I found that many of my students were absolutely uninterested in the logic games. Interestingly, this was my Geometry class, which addresses deductive and inductive logic throughout most of the course. After reflecting on the students’ needs, I decided to steer away from the traditional axiomatic approach to Geometry, and embraced a more project-based approach. Many of these students were interpersonal, linguistics, and bodily-kinesthetic learners. Therefore, I designed a variety of activities for them to explore the concepts of Geometry while collaborating in groups on projects that they designed and built.
For more information on the application of Multiple Intelligences Theory in Mathematics Education: