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!

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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. 

Student Engagement Strategies: Chunking Information and Cooperative Learning,

EngagementIt is highly critical for students, as agents of their own learning, to be actively engaged in the processing of information. When presenting students with new information, the different strategies used by the teacher to engage students in the learning process facilitate the students’ learning while providing them with greater access to developing their fluency in the content area. One way to encourage active engagement is by teaching content in small chunks. Similar to deconstructing a standard into different learning targets, dividing content into smaller chunks of information reduces the cognitively load necessary for processing the new information and allows the teacher to explore the content more rigorously. According to Marzano (2007, p. 44), “Learning proceeds more efficiently if students receive information in small chunks that are processed immediately.” Another way to encourage active engagement is to integrate cooperative learning groups, “Groups should be established to facilitate active processing of information during a critical-input experience.” (Marzano, 2007, p. 43). One way to do this in a mathematics class is to have students attempt a solution on their own and then share their work with a partner. With that partner, they would have to develop a better solution together. Afterward, they would find another set of partners and altogether they would develop an even better solution. Having students continually share their solutions with their peers and collaborate on developing a better solution allows them to verbalize their reasoning and critique the reasoning of others.

Marzano, R. (2007). The art and science of teaching: A comprehensive framework for effective instruction. Alexandria, VA: Association for Supervision and Curriculum Development.

Instructional Strategies: Project-Based Learning

PBL 2Project-based learning (PBL) is another student-centered model of learning that emphasizes 21st century learning skills (i.e. communication, collaboration, creativity, and critical thinking). According to the West Virginia Department of Education, “When engaged in standards-focused Project Based Learning (PBL), students are working in teams to experience and explore relevant, real-world problems, questions, issues and challenges; then creating presentations and products to share what they have learned.” There are many ways to implement PBL, but they all begin with problem situation. From that situation, the students analyze the situation provided and investigate possible solutions. They gather more information through research and assess their findings. Together, they develop a solution and prepare a presentation to share with the class.

PBL is not unique to mathematics education. In fact, PBL can be used in many content areas. Using PBL in mathematics education allows students to explore mathematical concepts in a more engaging way than working through a series of problems. Project-based learning provides a means of engaging students through social interaction, while providing a way to formatively assess the students’ understanding of the material. 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 mathematical literacy.

Approaching Mathematics in a Less Than Mathematical Way

I recently had the opportunity to teach a lesson to a group of 6th grade students on writing algebraic expression involving addition and subtraction. The group consisted of 18 students, of which 10 were classified as English language learners two were classified as students with special needs. I started the lesson by introducing myself and the objective of the lesson that we would be addressing throughout the lesson. Then, I moved on to the pre-assessment by posting the five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. The students wrote all five of the expressions on their papers and began translating. I circulated around the room and noted a few observations. Some students worked through each expression sequentially, while others provided at least one phrase for each expression before adding more phrases. Interestingly, the first phrase that every student wrote for addition expressions included the word “plus” and for subtraction expressions included the word “minus.”

Algebra WritingAfter administering the pre-assessment, I modeled how to represent algebraic expressions using visual models. One of the models that I wanted to focus on was the bar diagram. I used the bar diagram to differentiate between the operations of addition and subtraction and to illustrate the commutative property of addition. Since there were 18 students, I divided them into six groups of three students each and gave each group a set of algebraic expressions to represent using a bar diagram. I circulated around the groups and engaged them with questions to further their thinking. After five minutes passed, I brought the class together and had the groups share their solutions with the rest of the class.  This provided an excellent opportunity to record their responses using the document camera and LCD projector. I had the students help me organize the responses to see the order of complexity of each response. For example, for the expression “n + 5,” we categorized “n plus 5” as a simple statement and “n increased by 5” as a complex statement. Then, we looked at the phrase “5 more than n” and contrasted it with the first two based on the order of the factors. I had each group brainstorm other ways of translating the expressions into verbal phrases. We did the same for “10 – m” and highlighted the impact of commutativity on writing verbal phrases. This led into our discussion on multi-step algebraic expressions and the syntax necessary to translate these expressions into verbal phrases.

I administered the post-assessment by posting a set of five different algebraic expressions using the document camera and LCD projector and directing the students to translate each expression into as many different phrases as they could. I changed the factors from the expressions used during the pre-assessment and gave the students several more minutes to complete this task before concluding the lesson. Compared to the pre-assessment, the students took almost double the amount of time to consider verbal phrases to represent each algebraic expression with during the post-assessment, signaling the students increased level of understanding. I also observed an increase in the number of grammatically written phrases used to represent the expressions given. Unlike the syntactic errors that the students made on the pre-assessment, they rearranged the factors in the correct syntactic order.

After reflecting on the pre- and post-assessments that I administered, I would consider changing the post-assessment. For the post-assessment, I would have the students organize a list of verbal phrases and explain their reasoning for organizing the phrases the way they did. Instead of repeating the same style of assessment as the pre-assessment, this type of assessment would require the students to apply their critical thinking skills as they consider the application, the syntax, and the logical implication of each verbal phrase. This could lead into a nice introduction for the following lesson by having the students work with a partner, reflecting on each other’s responses and critiquing their partner’s reasoning.

Throughout this whole experience, I really appreciated the students’ level of engagement. The students were eager to participate in class discussions and provide responses that facilitated the critical thinking process. They were also open to the concept of analyzing mathematics through language. As I circulated around the classroom, I heard groups engaged in collaborative discussions, using their communication skills to create solutions that made sense to everyone in the group. They shared ideas and built consensus as they brainstormed different verbal phrases to represent the algebraic expressions given. It was great to see so many students receptive to approaching mathematics in a less than mathematical way.

Solving Multistep Percent Problems (Sample Lesson Plan)

Lesson Plan - Multistep Percent Problems (Thumbnail)In designing this lesson plan, I wanted to approach the concept of multi-step percent problems in the context of problem situations. According to the Common Core State Standards Initiative (2012), “Mathematically proficient students make sense of quantities and their relationships in problem situations.” This means that for students to be mathematically proficient, they need to be able to decontextualize and contextualize quantitative problems. By teaching students to decontextualize problem situations and represent them symbolically, students will develop their quantitative reasoning skills. The lesson explores ways of decontextualizing problem situations that relate to percentages (i.e. sales on merchandise). Students are then shown how to use different operations to compare sale prices during the Teacher Presentation and practice decontextualizing problem situations in collaborate multi-ability groups during Class Activities.

For English Learner students, the process of decontextualizing problem situations requires a certain level of English language proficiency that may be beyond their proficiency level. In order to facilitate their English language development while lowering their anxiety, I integrated the collaborative model into the Class Activities portion of the lesson. The collaborative model is beneficial not only for English Learner students, but for all students. As students interact and engage their peers in this model, they become active agents of their own learning and experience learning at a whole new level, “Educational experiences that are active, social, contextual, engaging, and student-owned lead to deeper learning” (Cornell University, 2013). For English Learner students and especially for Jonathan, the profiled English Learner student, collaborating offers an opportunity to interact with other students, allowing them to develop their English language proficiency through listening and speaking with other students. While developing their English language proficiency, English Learner students will also be able to engage in a model of learning that develops their critical thinking skills, “The fact that students are actively exchanging, debating and negotiating ideas within their groups increases students’ interest in learning. Importantly, by engaging in discussion and taking responsibility for their learning, students are encouraged to become critical thinkers” (Dooly, 2008, p. 2). In order to facilitate the active exchange of ideas within these collaborative groups, I integrated the use of posters, strategically placed around the room, with a list of discussion and sentence starters. John Larmer (2013) suggests the use of discussion and sentence starters as a way of helping English Learner students acquire 21st century learning skills and develop their English language proficiency, “Help English learners access the project by thinking carefully about the language functions called for in specific 21st century learning activities; place them in supportive teams; provide models such as sentence starters and graphic organizers to help with collaborative discussions.”

At the beginning of the lesson, I engaged the students in brainstorming as a way of activating background knowledge of the word “sales.” Exposing English Learner students to new concepts through vocabulary correlations facilitates their learning and their English language development. By activating background knowledge, the students will be able to make connections to the learning that are meaningful and personally engaging. It also helps to engage the students in activating background knowledge to ensure that students are approaching the new concept with a basic foundational understanding of the concepts necessary for learning the new concept. Using a Circle Map to record the students’ responses help graphically organize the responses for the students, especially the English Learner students, to visualize their relationships to the word. For Jonathan, the profiled English Leaner, the Circle Map will facilitate his acquisition of the meaning of the word “sales” and the relationships of the students’ responses, especially considering his interest in creating art. Individuals with spatial intelligence enjoy creating art and easily learn through visual presentations. Giles, Pitre, and Womack suggest that teachers utilize visual presentations to accommodate the learning needs of students with spatial intelligence, “Teachers can foster this intelligence by utilizing charts, graphs, diagrams, graphic organizers, videotapes, color, art activities, doodling, microscopes and computer graphics software.” I have also integrated PowerPoint in the Introduction and the Teacher Presentation as another way of visually presenting the concepts, example percent problems, and the steps for solving each problem.

During the Introduction, I emphasized the time (or extended time) necessary to wait on students to provide their responses to questions. It is crucial to allow students enough time to think about a question and develop a response that they are confident in sharing with the class. For English Learner students, sharing with the class may already be a high-anxiety situation. There is no need to exacerbate the situation by requiring speedy responses. In order to lower the students’ anxiety, I emphasized implementing a longer wait time. This not only provides the English Learner students and Jonathan, the profiled English Learner student, with extra time to process content, but also to linguistically plan their response.

The lesson incorporates a number of strategies, supporting the 1/3 Plus Model. The students are intentionally placed in collaborative multi-ability groups to allow the top third the opportunity to lead the collaborative discussion and provide support to his/her peers. During the Introduction, the top third may offer more advanced vocabulary when they think of the word “sales.” The middle third of the students will learn because they have a basic understanding of the concepts underlying percent problems and will be able to participate in the collaborative discussions. During discussions, they may respond on a voluntary basis or they may wait to hear another student respond first. By giving students extra time to respond after asking them a question, the middle third (and bottom third) will be more encouraged to participate in class discussions. The bottom third will be first engaged by the activation of background knowledge. This will facilitate their learning and help them make connections to any new vocabulary or concepts. Using graphic organizers, like the Circle Map, will help the bottom third visualize vocabulary and their relationship to the concept being taught. Including PowerPoint as a means of visually highlighting key aspects of translating percent problems and providing images for students to relate to facilitates the learning process. By collaborating in multi-ability group, the bottom third are able to work with other students, developing their mathematical and English language proficiency through observational learning and social contexts. The questions asked at the end of the lesson are progressively leveled according to Bloom’s taxonomy to encourage higher order thinking of all students. Having students participate in a think-pair-share to engage their learning and have them share their responses helps the bottom third and middle third take the time to reflect on their learning and hear other students’ responses. It helps the top third reflect and consider further investigations while sharing their ideas with other students.


PowerPoint Presentation Used with Lesson

Providing for Diverse Student Populations

Re: “Multiculturalism’s Five Dimensions” by Banks, J. A., & Tucker, M. (1998)

One of Dr. Banks’ five dimensions of multicultural education is knowledge construction. In order to provide every student with “the implicit cultural assumptions and frames of reference and perspective” of mathematics, it is important to analyze the language used (Banks & Tucker, 1998). For example, I am currently working on a different approach to solving word problems. Instead of focusing on the operations required to solve word problems, we will focus more on the language used in the word problem.

Another one of Dr. Banks’ five dimensions of multicultural education is equity pedagogy. In this dimension, Dr. Banks suggest that teachers change their methods to accommodate diverse learning needs (Banks & Tucker, 1998). Ever since my first year of teaching, I have attempted different methods of teaching. Specifically, I have incorporated cooperative learning, direct interactive instruction, differentiated instruction, and problem-based learning into every lesson. I plan to continue using these methods as a means of promoting student engagement and increasing the opportunity for every student to excel.

Lastly, Dr. Banks addressed the question of maintaining our unity as a community, a community that is inclusive of all its many perspectives (Banks & Tucker, 1998). This made me think about the open-mindedness essential to successfully teaching any subject, especially as we transition to the Common Core State Standards. In mathematics, for example, students always suggest other ways of solving certain problems. Instead of dismissing their suggestions, I have found it beneficial embracing their suggestions and encouraging further thought. This not only motivates them to think more creatively about mathematics and share their differences, but it sets up the classroom as a community of learning and exploration.

Multiculturalism

Active Student Engagement

It is highly critical for students, as agents of their own learning, to be actively engaged in the processing of information. When presenting students with new information, the different strategies used by the teacher to engage students in the learning process facilitate the students’ learning while providing them with greater access to developing their fluency in the content area. One way to encourage active engagement is by teaching content in small chunks. Similar to deconstructing a standard into different learning targets, dividing content into smaller chunks of information reduces the cognitively load necessary for processing the new information and allows the teacher to explore the content more rigorously. According to Marzano (2007, p. 44), “Learning proceeds more efficiently if students receive information in small chunks that are processed immediately.” Another way to encourage active engagement is to integrate cooperative learning groups, “Groups should be established to facilitate active processing of information during a critical-input experience.” (Marzano, 2007, p. 43). One way to do this in a mathematics class is to have students attempt a solution on their own and then share their work with a partner. With that partner, they would have to develop a better solution together. Afterward, they would find another set of partners and altogether they would develop an even better solution. Having students continually share their solutions with their peers and collaborate on developing a better solution allows them to verbalize their reasoning and critique the reasoning of others.

Marzano, R. (2007). The Art and Science of Teaching: A Comprehensive Framework for Effective Instruction. Alexandria, VA: Association for Supervision and Curriculum Development.

Integrating Filmmaking and Mathematics: The Life of Zero

The Life of Zero is a collaborative film project that addresses an abstract mathematical concept through a theatrical display of performing arts. The movie associates the positive and negative undertones of the number line with real life conflicts that students face every day. Almost entirely conducted through interpretive dance, the students were able to combine movement and dramatic display as a way of translating the discordant nature of the positive and negative integers. The students that participated in the film project were involved at all levels of production from conceptualizing the story and designing the storyboard to creating props, designing the sets, helping with choreography, providing lighting and special effects, and acting.

Rigor, Relevance, and Relationship

The emphasis in mathematics education is on shifting the classroom dynamic to promoting mathematical reasoning. Too much time has been placed on rote learning, particularly in mathematics. With the transition to the Common Core State Standards, we are beginning to integrate rigor, relevance, and relationship, in a way that enhances the students’ procedural knowledge while fostering greater conceptual understanding. For example, embracing the Standards for Mathematical Practice as a set of guiding principles that explore the many facets of mathematical reasoning allows students to become active and engaged learners.  Through rigor, they will exercise higher order thinking and develop their metacognitive skills. Through relevance, they will contextualize the abstract nature of mathematics by forming connections with real life experiences, personalizing their learning. Through relationship, we as teachers must know our students in order to create a highly engaging learning environment.

3Rs Venn Diagram