Project Based Learning: Day 7

We started class with a Check In meeting. I’ve noticed that many of the students are liking this way of starting class. It gives them an opportunity to connect with each other and to transition more effectively.

I had two tasks for them to complete today. First, I wanted them use the results from the Skill Set Survey that they produced to create groups of mixed abilities, learning styles, and personality types. If they had time left, I wanted them to get in their groups and begin brainstorming ideas in response to the driving question.

To avoid any chance of manipulation, I made copies of the Skill Set Surveys with the student names removed from each class. Then, I had 10th grade analyze the surveys from 9th grade and vice versa. I refrained from telling the students how to organize the groups. Instead, I had students volunteer to lead  the discussion while I coached them throughout the discussion process.

LeadershipIt was interesting to see the approaches that each grade took. The 9th grade class classified each survey according to their primary and secondary skills. From the results, they found Builder, Artist, and Writer to be the three basic skills. They used a grid to rank each survey and created mixed ability groups representing all three basic skills. The 10th grade class did something a little different. They classified each survey using 9 to 10 different categories. Then they realized how complicated that would be to create groups based on so many categories. They also noticed that some of the categories could merge into a broader category. Ultimately, they reduced their categories to four different skills: Leader, Builder, Artist, and Writer.

With the time left in the period, I had the students break up into their groups and begin thinking about the driving question. I wrote the driving question on the dry-erase board and encouraged them to begin wondering. This might be a small point to share, but I’ve been very intentional in the vocabulary that I use with the students. For example, I’ve purposely used words like wander, imagine, create, develop, and explore to inspire divergent thinking.

As their homework assignment, I asked them respond to these two questions:

  • What is your group’s plan? If your group doesn’t have a plan yet, what have they talked about?
  • What did you work on at home tonight?

I’m still considering other options for homework. From my research, some Project Based Learning programs de-emphasize homework. I’ve considered doing the same, but I also think it’s important for students to develop their metacognitive awareness. I plan to continue researching this before making a final decision. Any suggestions would be most appreciated!


Integrating Student Engagement in Instructional Design

instructionalDesignThere are a number of methods that I use to determine if my instructional design is responsive to the needs of each student as they access the Common Core Standards. First, I hold daily class meetings to check in with the students. I have been holding these meetings since the beginning of the year and have found them very successful. Second, I hold one-on-one meetings with each of the students every two weeks. This allows me time to ask questions and learn more about the student’s level of understanding and level of interest. Third, I host online office hours using Google Docs to give students the option of asking questions or expressing their concerns through text rather than through voice. Fourth, I have students maintain progress journals, where they reflect on their learning. Since these are their journals, I allow them to fill them out how they choose. This also allows me to see how they are processing their learning.

While the methods I listed above may not produce some type of a numerical value, they provide insight into the students’ level of engagement with the subject.  For example, one student from my Algebra II class had struggled with the material covered during the first semester. Her scores on tests and quizzes ranged from 60% to 80%. She submitted most of her homework. When I spoke to the student, she said that she wanted to understand the material better, but she did not know how to study. We tried several methods, and nothing worked. I investigated further and discovered that she really loved creating art on the computer. So, I introduced her to several programs online that she could use to explore three-dimensional modeling. She got excited and started working on them instantly. After a week, she found herself struggling to make some of the objects the right size or place them in the right position. That is when I introduced her to the mathematics used in three-dimensional modeling. Instantly, she wanted to learn as much as she could about graphing two- and three-dimensional equations. What I learned from this experience was that engagement is crucial to the learning process. The only way that this could have been this successful was by consistently engaging the student using the methods listed above. By working closely with students and helping them explore the material in their own way (i.e. differentiated instruction), we can facilitate the learning process more effectively. 

Integrating Technology and Differentiated Instruction


Using technology in the classroom not only allows the teacher to differentiate instruction but to foster the development of Digital Age Fluencies. In essence, it serves a dual purpose; however, using technology requires more than enhancing a lesson with a PowerPoint or Prezi presentation or providing the students with time to play an educational game on their iPad. In order to develop Digital Age Fluencies, using technology requires students to understand the complex and dynamic nature of the technological landscape from the perspectives of the user and of the developer. Taking this perspective has made it easier to provide differentiated instruction while still developing Digital Age Fluencies.

Ed tech

This idea actually came to me before I started teaching this school year. I knew I would be teaching Geometry and Algebra II, and considered how I would integrate technology into the curriculum in a way that would really enhance the students’ technological understanding and fluency. One of the ways that I did this was by introducing the students to Microsoft Excel. Many of them had already used Excel to create simple graphs. Instead, I taught the students how to create their own graphing calculators. First, we started small and worked with basic formulas. Then, I introduced them to Macros and created simple calculators for some of the equations we were working with. Finally, I introduced them to Visual Basic and had them work on creating their own graphing calculators.

Another way that I could integrate technology into the curriculum in order to differentiate instruction and develop Digital Age Fluencies is by taking a Project Based Learning approach. I have actually started this with a number of classes and have already seen students more engaged. In Geometry, for example, the students are creating model cities inspired by different two-dimensional shapes. Some of the students had struggled with the relationships between two-dimensional shapes and three-dimensional objects prior in the first semester. For these students, I introduced them to simulators online that helped them visualize the relationships. For the other students, who were able to identify the relationship, but struggled with visually representing it, I introduced them to three-dimensional modeling software. After spending much time with both of these forms of technology, the students understood the project better and the mathematics necessary for creating their model cities.  

Classroom Management: Social Contracts

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


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.

Workshop: Introduction to the Standards for Mathematical Practice

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

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

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

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

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

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

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

Slideshare link:

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.

Instructional Strategies: Collaborative Group Work

It has been a tradition in many mathematics classrooms to follow the direct instruction model, but with the emphasis of the Common Core State Standards on developing students’ mathematical fluency, using a method that requires students to develop their thinking individually and then collaboratively (in that order) is consequential. This is particularly the reason why integrating formative assessments into the sequence of lesson planning is essential to teaching mathematical fluency. This strategy reverses the gradual release of responsibility by having the teacher give the students a task to complete in class or as homework prior to the lesson, in the form of a pre-assessment. The reasoning for this is to provide the teacher with an understanding of the students’ use of mathematics and any areas of opportunity. On the day of the lesson, the teacher passes back the students’ work with feedback or questions provided on each student’s work, encouraging the students to engage in self-reflection, “When we use assessment for formative purposes, students should receive growth-producing feedback and have the opportunity to make adjustments to their work based on that feedback,” (Rutherford, 2009, p. 139). This allows the student to consider their reasoning for using the mathematics that they did. Then, the teacher divides the class into small groups and has them work collaboratively on developing a joint solution. After students have developed a joint solution, the teacher holds a whole-class discussion. Prior to holding the discussion, the teacher should circulate around the classroom and note the types of approaches that each group is using to solve the task. The teacher can use these notes to help guide the discussion and explore the methods used.  If time is available, the teacher could pass out several sample responses to each group, allowing them to analyze the different approaches and compare them to their own approach. Then, the teacher could hold a whole-class discussion on the different approaches used.

TeamworkTeaching mathematical fluency is not only about teaching content, but about teaching students how to use that content, how to think about that content, and how to apply that content to the real world. Collaborating with others offers an opportunity for students not only to interact with their peers, allowing them to develop their mathematical fluency through listening and speaking with other students, but also to engage the content at a deeper level, “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). 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). Because this strategy offers a student-centered approach that builds on collaboration, it is generally applicable to all content areas. However, in terms of mathematics, it offers students a unique approach to mathematical fluency by improving how they think about mathematics. Based on the nature of this strategy, it provides for an excellent means of formative assessment, “Formative assessments…first allows students to demonstrate their prior understandings and abilities in employing the mathematical practices, and then involves students in resolving their own difficulties and misconceptions through structured discussion” (Mathematics Assessment Resource Service, 2013).

Instructional Strategies: Direct (or Explicit) Instruction

The direct instruction (or explicit instruction) model is based on scaffolded learning, gradually releasing the responsibility of learning from the teacher to the student, “Explicit instruction models support practice to mastery, the modeling of skills, and the development of skill and procedural knowledge” (California Department of Education, 2013, pp. 17-18).This model is particularly beneficial in teaching procedural knowledge to students regarding mathematical concepts. Typically, direct instruction begins with the teacher modeling how to perform a particular task. The teacher may utilize the think aloud strategy at this point to model how the students should reason through the task. According to Walqui (2006, p. 170), “When introducing a new task or working format, it is indispensable that the learners be able to see or hear what a developing product looks like.” Then, the teacher engages the students in guided practice, eliciting student participation in performing the next task. At first, the teacher may ask students if the steps taken to perform the task are correct. The teacher may continue with similar tasks or calculation that are progressively more difficult, eliciting greater student participating in performing the task. Following the guided practice, the teacher assigns the students an assignment or activity to complete individually. During this time, the teacher should circulate around the classroom observing student work, asking guiding questions, and offering help when needed. Throughout the lesson, the teacher checks for understanding through a variety of formative assessment techniques. These include asking the students questions, having them respond with whiteboards, or circulating and listening to their conversations as they engage in a think-pair-share. At the end of the lesson and throughout the unit, the teacher tests the students using summative assessments (i.e. quizzes, mid-chapter tests, and chapter tests).

The direct instruction model is not explicitly unique to teaching mathematics. In fact, direct instruction is easily implemented as a model for teaching reading comprehension (Gersten & Carnine, 1986) and science (Adelson, 2004). The focus of the direct instruction model is on the method or procedure used. It is a teacher-centered method that specifically addresses procedural knowledge. The benefit of direct instruction is that students acquire the skills and procedural knowledge for effectively performing certain tasks, “Teacher-centered methods of instruction are often necessary to educate students on difficult material that requires multiple steps, and for procedures which are unlikely for students to discover on their own” (Cohen, 2008, p. 4). Because the direct instruction model is teacher-centered, it does not foster the development of reasoning and metacognitive awareness that is necessary for students to think mathematically.

Rethinking the Use of Real World Examples in Mathematics

While connecting mathematics to the real world may seem like it would be a simple task, it ends up being a real challenge. I always tried to find some application to the real world that the students could use to understand the mathematics better, but it was never that easy. I realized that my struggle was not in finding the real world examples, but in what I was looking for. I usually imagined career-based real world examples. Then, if students asked why they had to learn something in mathematics, I could refer to the type of career that would need that type of mathematical background. Later, I considered re-thinking the types of examples I wanted to use. Instead of career-based real world examples, I explored real world examples that students experience day-to-day. This subtle change made it easier to think of examples. After a while, my students were even thinking of their own examples. I realized that the examples I originally presented them were far removed from anything that they could really relate to. By selecting examples that they could relate to, they were more open to working through the problem.