Classroom Management: Student Involvement

Classroom management is essential in providing students with an environment that is conducive to learning. According to Marzano and Marzano (2003), “One of the classroom teacher’s most important jobs is managing the classroom effectively.” There are several factors that highly impact the effectiveness of how a classroom is managed. First, the teacher must utilize instructional strategies and curriculum design that foster student engagement and accommodate the students’ learning needs, “Effective instructional strategies and good classroom curriculum design are built on the foundation of effective classroom management” (Marzano, Marzano, & Pickering, 2009, p. 4). While the teacher should be knowledgeable of current pedagogical trends, it is important that the students understand the instructional strategies being used and participate in planning the curriculum to be covered. By involving students in the selection of the topics to be covered, the students are more intrinsically motivated to participate in the lessons. Likewise, the number of topics decreases, allowing for deeper exploration of each topic. According to Glasser’s concept of a quality curriculum, “Quality learning requires depth of understanding together with a good grasp of its usefulness. Learning a smaller number of topics very well is always preferable to covering many topics superficially,” (Charles, 2005, p. 78). In fact, fewer topics allow for greater clarity and specificity.




Digital Literacy: Developing Greater Comprehension of Online Texts

Reading OnlineI’m wondering if any teachers have already seen this as students begin using technology to access primary sources online. I think this trend prefaces a growing need to further develop strategies to foster biliteracy. While digital reading sources may not necessarily be presented in a different language, it does require a different set of comprehension building skills.

(See article – Digital Reading Poses Learning Challenges for Students by Benjamin Herold)

I realize that this sounds like it has nothing to do with mathematics, but I think it does. When we consider the comprehension building skills that students need to understand the mathematics that they read, we are talking about a different set of skills than what they would use to read literature. Instead of teaching students to perform calculations and expecting them to understand a problem by simply decoding the reading, we need to consider develop a combined approach to helping our students develop mathematical literacy through the integration of mathematics and English Language Arts.



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.

Integrating Technology: Visualization using Aurasma


It seems like there is always some new app or some new device that has come out on the market. Staying current with and integrating all this new technology is a challenge, but it builds credibility among your students and it makes learning more engaging. For example, all my students knew how to use their devices to read QR Codes, but none of them had ever used Aurasma.

AurasmaAurasma is similar to a QR Code Reader, but it allows the viewer to see a different image or a video when scanning a particular picture. So, for my class, we created interactive walls where students prepared pictures and video clips to play when someone scanned particular pictures or words on the walls. I never heard about this app until I attended a recent conference that I attended on transitioning to the Common Core. I participated in a workshop specifically on integrating technology with mathematics and learned all about this app.


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.