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.
There are a number of methods that I use to determine if my instructional design is responsive to the needs of each student as they access the Common Core State Standards. First, I hold daily class meetings to check in with the students. I have been holding these meetings since the beginning of the year and have found them very successful. Second, I hold one-on-one meetings with each of the students every two weeks. This allows me time to ask questions and learn more about the student’s level of understanding and level of interest. Third, I host online office hours using Google Docs to give students the option of asking questions or expressing their concerns through text rather than through voice. Fourth, I have students maintain progress journals, where they reflect on their learning. Since these are their journals, I allow them to fill them out how they choose. This also allows me to see how they are processing their learning.
In class, I drew the following figure on the board:
I asked the students if they could determine the length of the diagonal d given the information we’ve been recently covering. I’ve been talking to them about special right triangles (i.e. 45-45-90 and 30-60-90 triangles) at length to help them gain a deeper understanding of their properties as I prepare them for trigonometry next year. I even showed them how to use the distance formula to find the length of the hypotenuse. Yet, with all this knowledge and understanding, they weren’t even sure how to begin working on the problem.
I thought about this over the weekend and wondered if there were a different way, a more engaging way, that I could use to help them through this problem. So, I went to the store and bought several spools of string. I divided the class into groups of four and gave them each a spool of string. Then, I marked two opposite corners of the classroom and presented the challenge.
Cut the string provided using only one cut so it may touch both corners when pulled tight. The groups could use any of the measuring devices provided (i.e. 12-inch ruler, yard stick, or measuring tape).
This required them to figure out how to use the measurements of the floor and the walls in determining the length of the string. By giving them the condition of only being able to cut the string once, they had to attend to precision (see CCSS.MATH.PRACTICE.MP6). Also, with a variety of measuring devices provided, they needed to determine which device would give them the most accurate measurement (see CCSS.MATH.PRACTICE.MP5).
Needless to say, this offered the students a different way of thinking about finding the diagonal of a cube. Some were able to derive the distance formula on their own as they worked through the calculations. Others admitted that they understand more now about square roots and working with triangles.
There are a number of methods that I use to determine if my instructional design is responsive to the needs of each student as they access the Common Core Standards. First, I hold daily class meetings to check in with the students. I have been holding these meetings since the beginning of the year and have found them very successful. Second, I hold one-on-one meetings with each of the students every two weeks. This allows me time to ask questions and learn more about the student’s level of understanding and level of interest. Third, I host online office hours using Google Docs to give students the option of asking questions or expressing their concerns through text rather than through voice. Fourth, I have students maintain progress journals, where they reflect on their learning. Since these are their journals, I allow them to fill them out how they choose. This also allows me to see how they are processing their learning.
While the methods I listed above may not produce some type of a numerical value, they provide insight into the students’ level of engagement with the subject. For example, one student from my Algebra II class had struggled with the material covered during the first semester. Her scores on tests and quizzes ranged from 60% to 80%. She submitted most of her homework. When I spoke to the student, she said that she wanted to understand the material better, but she did not know how to study. We tried several methods, and nothing worked. I investigated further and discovered that she really loved creating art on the computer. So, I introduced her to several programs online that she could use to explore three-dimensional modeling. She got excited and started working on them instantly. After a week, she found herself struggling to make some of the objects the right size or place them in the right position. That is when I introduced her to the mathematics used in three-dimensional modeling. Instantly, she wanted to learn as much as she could about graphing two- and three-dimensional equations. What I learned from this experience was that engagement is crucial to the learning process. The only way that this could have been this successful was by consistently engaging the student using the methods listed above. By working closely with students and helping them explore the material in their own way (i.e. differentiated instruction), we can facilitate the learning process more effectively.
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 2 – Introductory Week
Today, I had the students gather in a circle in the middle of the classroom and I engaged them in a discussion of the following questions:
- What is Project Based Learning?
- What does a Project Based Learning classroom look like?
- How is Project Based Learning different from group projects?
- How have other schools integrating Project Based Learning?
- How are tests used with Project Based Learning?
- How is learning assessed with Project Based Learning?
- What are some examples of Project Based Learning activities?
- What is the student’s role in Project Based Learning?
- What is the teacher’s role in Project Based Learning?
Prior to this discussion, I distributed this list of questions to all the students and encouraged them to learn what they could about Project Based Learning using these questions to guide their investigation.
Since I want my students to feel more empowered, I started by coaching one of my students to lead the discussion, while another student kept record of all that was said during the discussion. (We later defined this role as the Discussion Leader.) As the students explored each question, I already saw a difference in engagement. Instead of two or three hands, I saw every hand raised up in the air when a question was asked. I saw an eagerness to share ideas and express creativity. The discussion even led to a few more questions we wrote on the board to explore further. For example, the students were interested in clearly differentiating between projects and project based learning as well as between project based learning and problem based learning.
Throughout the discussion, the students also shared a few best practices, i.e. using group contracts, peer reflections, and project calendars. With three designated work areas, I assigned each area with one of the best practices and asked the students to sit down in the area they would like to develop. I gave them the timeframe for working on each of these items, and gave them some space to brainstorm, research, share, and create. After 25 minutes, I checked in with all three groups and already saw rough drafts of each item. I alerted them that they have 5 more minutes to work on this and that rough drafts should be ready to share during tomorrow’s class for the other groups to review.
From what I’ve already learned about Project Based Learning, I feel that these two days have been very productive. The students have been showing so much excitement and have already been thinking about ways of making this experience an absolute success.
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:
I was talking to my students the other day about the Order of Operations (i.e. PEMDAS). I quickly reviewed the Order of Operations as we worked through several algebraic expressions. Then, I asked my students why we followed the Order of Operations. They said, “It’s just what we’re supposed to follow.” I decided to investigate this further and I asked them, “But why does the Order of Operations follow the order that it does?” They didn’t know.
Why do we follow the order that makes up the Order of Operations? I asked several other people and got the same response — their teacher told them it’s what they were supposed to follow. Welcome to the era of education that focused primarily on the product. Why study the reason behind the Order of Operations, when teaching students that they’re supposed to follow it works? That is like asking, “Why teach an archer about trajectory physics, when you could just teach the archer to hit a specific target?”
So, I entertained the challenge of understanding the Order of Operations further with my students. I wrote PEMDAS vertically on my marker board and analyzed the directionality of each operation.
- Parentheses only refers to embedded operations, so we left that aside.
- Exponents were essentially a form of multiplication, since is equivalent to .
- Multiplication and Division are usually computed together from left to right. I stopped my students at this point to consider the two different operations. Could multiplication be computed bidirectionally? Yes. What about division? If we think of division as multiplication, then could be written as (i.e. multiplying x by the multiplicative inverse of 2).
- Addition and Subtraction are much like Multiplication and Division. While addition could be computed bidirectionally, subtraction won’t allow for it. Fortunately, we are able to write subtraction as the addition of an additive inverse. For example, could be written as .
Therefore, instead of PEMDAS, we have PMA. This simplifies things a little, but it doesn’t answer the question why we follow the order of PEMDAS (even if it’s written as PMA).
Let’s visualize a situation where PEMDAS would be needed. In the above picture, we have four stacks of squares with nine circles in each square, a single square with nine circles in it, and four circles by themselves. If we had to put this in an expression, we could write it as . If we had to add up all of the blue circles, how would we do it? Would you want to count all the circles one-by-one? This may be a little time-consuming. What if we computed each of the squares as nine circles (from ). Then, if one square equals nine circles, four squares must equal 36 circles. Thus, 36 circles + 9 circles + 4 circles. Interestingly, we just followed PEMDAS. Wait! I don’t remember bringing PEMDAS into this. It seems that the reason for PEMDAS is right in front of us.
Think about it.
Four squares of nine circles + one square of nine circles + four circles
4 ( 9 circles) + 9 circles + 4 circles
36 circles + 9 circles + 4 circles
Basically, we had to convert the original expression into an expression that allowed for addition to happen with quantities of similar terms. There was no way to add a square with four circles. Instead, we had to convert the square to nine circles to allow for addition to be possible. Ultimately, PEMDAS follows PMA, ranking the operations in priority from the embedded operation (P), to multiplication (M), and then to addition (A). If addition is the final operation performed, then all quantities must be similar in order to be added. Likewise, if multiplication is the final operation performed, then all quantities must be similar in order to be multiplied.
In summary, we follow the Order of Operations to allow for multiplication and addition with quantities of similar terms.