Literacy Development in the Classroom: Round Robin Reading

Edutopia - 11 Alternatives to Round Robin ReadingIn 11 Alternatives to “Round Robin” (and “Popcorn”) ReadingTodd Finley mentions “Round Robin” Reading, an overly-criticized and antiquated method of oral reading, and provides alternative strategies to literacy development.

When I first started in the classroom, “Round Robin” Reading was highly encouraged. I tried it, but would always see students disengaged. The focus was primarily on one student while the other students needed some type of motivator to keep them engaged. What I also noticed was that many of my students struggled with reading aloud. So, I decided to read aloud to them, emphasizing all the prosodic cues within the text. Suddenly, they were all engaged. Taking this response, I began coaching my students to read with the same level of vigor. Still, I had a few students (mostly my visual learners) who were not as engaged as I had hoped. For them, I developed a strategy that allowed them to draw what they heard the student read aloud.

For more reading on “Round Robin” Reading and alternative strategies to this method:

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.



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.

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.

Writing Addition and Subtraction Expressions – A Linguistic Analysis

Prior to starting a lesson on algebraic expressions, I administered a pre-assessment as a means of assessing background knowledge. Using the document camera and the LCD projector, I wrote five different algebraic expressions on the board. Two of the expressions involved addition, “n + 5” and “4 + p,” two more involved subtraction, “10 – m” and “x – 12,” and the last expression was a multi-step expression involving both addition and subtraction “(x + 10) – 5.” I instructed the students to copy all five expressions down on a piece of paper and write as many verbal phrases as they could to represent the expressions.

On the first two expressions involving addition, all the students used the word “plus” correctly and grammatically. For the expression, “n + 5,” they wrote “n plus 5” as opposed to “5 plus n,” maintaining the syntactic order of the factors. Most of the other phrases given were generally correct, but they were not grammatical. For example, one student wrote, “n added to 5,” for “n + 5.” Because mathematics is read from left to right, the factor “5” would have to be added to the factor “n.” Thus, the student should have written “5 added to n.” Another phrase that was written was, “5 increased by n.” Even though “increased by” is a correct phrase to use to represent addition, it should have been written, “n increased by 5.” Again, the syntactic roles of the factors were switched. Interestingly, the students that used “added to” and “increased by,” wrote “added to” first before writing “increased by.” Based on the syntactic order of both phrases, “increased by” follows the same order as “plus” (i.e. “n plus 5” and “n increased by 5”), while “added to” reverses the order of the factors. This could have caused the students to reconsider the order of the factors when using the phrase, “increased by.” Their confusion could have also originated with the use of prepositions. The phrases “added to” and “increased by” both possess prepositions, and are thereby classified as verbal phrases. If the students made this connection, they would have reversed the syntactic roles of the factors using the phrase, “increased by.” It could also be that the commutative nature of addition implies a sense of bi-directionality in the syntactic order of the phrase given. Thus, if “n + 5” is equal to “5 + n,” then “n increased by 5” should be equal to “5 increased by n.”

The students’ phrases for “10 – m” and “x – 12” were much the same as those involving addition. All the students used the word “minus” correctly and grammatically. About half of the students wrote the phrases “10 subtracted by m” and “10 decreased by m” correctly and grammatically. The rest of the students wrote “10 subtracted from m” and “10 take away m.” All four phrases have maintained the order of the factors, “10” and “m”; however, this order does not maintain the grammaticality of the expression, “10 – m.” This strict adherence to the order of the factors probably stems from the lack of commutativity of subtraction. Still, a phrase like “subtracted from” requires the factors to be used in reverse order. For example, “10 – m” should have been written as “m subtracted from 10.” Even though the phrases “subtracted by” and “subtracted from” include the same verb “subtract,” their prepositions require different syntax. The preposition “by” implies agency. Thus, the factor being subtracted (i.e. “m”) should follow the preposition “by” (i.e. “by m”). On the contrary, the preposition “from” implies movement, with the origin of that movement (i.e. “10”) following the preposition (i.e. “from 10”), and the agent of that movement (i.e. “m”) prefixing the verb (i.e. “m subtracted”), resulting in the phrase, “m subtracted from 10.” Without a strong background in syntax and the use of certain prepositions, it seems that the students are ordering the interaction between conflicting constraints. First, they maintain the lack of commutativity of subtraction by upholding the order of the factors used in the algebraic expression. Second, they correctly write verbal phrases using transitive verbs with an expressed subject and direct object (i.e. “10 minus m”). Finally, if a preposition follows a verb, they uphold the first two constraints at the risk of forming ungrammatical phrases.

The results from this pre-assessment revealed a disjunction between the students’ understanding of algebraic expressions and the linguistic structure of phrases that represent these expressions. If anything, the rubric designed to evaluate the students responses failed to include the grammaticality of the phrases provided by students. Instead, it focused on the students’ ability to represent algebraic expressions using correct verbal phrases without any consideration for the syntactic order of the phrases. Based on this, I would revise the rubric to include grammaticality (see revised rubric below).

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

Teaching Mathematics Content to English Learners

In teaching, it is essential that the planning of a lesson and the pedagogical methods used during that lesson address the diversity of learning needs represented by the students in the classroom. In the mathematics classroom, it is easy to assume that by teaching the processes involved in solving a particular problem, students will fully understand the concept being taught; however, students, who are considered English learners (EL), continue to struggle with the way the concepts are explained and applied in various settings.  It is a misconception to think that mathematics is simply a list of calculations or a study of numbers. Mathematics is the study of the logical order that explains the relations that exist numerically, represented by a language that integrates both mathematical and colloquial vernacular interdependently. The IRIS Center for Training Enhancements expands on the use of mathematical and colloquial vernacular in understanding mathematics, “Math involves more than numbers and includes vocabulary terms such as numerator, quotient, and simplify. Furthermore, words like table and round have meanings in math different from their more common definitions.”  In other words, the study of mathematics requires not only a fluency of numbers, but also a fluency of the language used to describe the relationships of those numbers.

OLYMPUS DIGITAL CAMERATo facilitate the learning of mathematics and address the learning needs of ELs, I would integrate a variety of Specifically Designed Academic Instruction in English (SDAIE) strategies throughout each lesson. For example, if I was talking about bisecting an angle, I would start by talking to students about the word “bisect.” I would write the word on the board and ask them if they know of any words that start with “bi-.” This also helps access the students’ background knowledge, allowing them to make connections with their current learning. If students mentioned a word with “bi-” meaning “two,” I would emphasize this on the board. It would even be beneficial to bring in pictures of a bicycle, and other objects representing the concept of “two” with the prefix “bi-.” If students mentioned words like “biology” that start with “bi-,” but not necessarily the “bi-” prefix, I would clarify this and create a graphic organizer to emphasize the difference. I would do the same for the part of the word “sect.” After the students understand that “bisect” means to cut in half, I would begin teaching them the concept of bisecting an angle. In my experience, it has been helpful to explain that bisecting in mathematics means to divide into two equal parts. By teaching students that bisect simply means “cutting into two parts,” they do not understand that the two parts must be identical.

Accessing the students’ background knowledge is another important strategy for addressing the learning needs of ELs, “Background knowledge helps students make connections with new information and helps them understand concepts” (IRIS Center for Training Enhancements). While strategies for accessing background knowledge may be simpler to convey visually through the use of graphic organizers, it is best to take a multimodal approach when possible. For example, when discussing the concept of combining like terms, I would start with a simple activity that every student could relate to. I would place a number of containers at each table, with various items placed in each container. Then, I would divide the class into small groups and instruct them to organize the items. After they completed that, I would begin a discussion where each group could share how they organized their items and their reason for organizing them that way. Using one of their solutions, I would write it on the board and review their steps for organizing the items while introducing some of the new mathematical vernacular. Instead of “organizing,” I would begin using “combining.” After a few demonstrations, I would then lead into the concept of combining like terms.

Multiculturalism 2Another important strategy to include is providing students with a variety of
opportunities to engage other students in discussion of the concept being taught. This will increase the amount of exposure students have using the pertinent mathematical and colloquial vernacular while developing the students mathematical fluency. In particular, “Teachers can support student learning by allowing them multiple opportunities to participate in classroom discussion and by encouraging them to explore and share their own perspectives” (IRIS Center for Training Enhancements). One way to do this is by dividing the students into groups and engaging them in formative assessment tasks. Unlike summative assessments, formative assessments tasks would allow students to share their perspectives and collaborate with each other as they work toward solving a variety of tasks. Throughout the activity, I would circulate from group to group, facilitating the collaboration process.


Divergent Thinking – Changing Paradigms

After watching the video Changing Paradigms, Divergent ThinkingI was especially interested in the study on divergent thinking that Sir Ken Robinson described. He mentioned that after being tested in Kindergarten, the same students were tested later and declined in their ability to engage in divergent thinking. He attributed this decline to these students having been educated. When the education system is built on a foundation that was established in the tradition of industrialization, it is easy to see why a system this antiquated is not meeting the current learning needs of students. When I was a university student, I reflected on my educational upbringing. I noticed that the educational system was very similar to a factory. After graduating from the university, I had my theories on learning, but wanted to experience it from the perspective of a teacher. I taught for a number years, first in charter schools, then in private schools. After observing what I did, I made it a goal of mine to make a change. As a mathematics teacher, it was a challenge to encourage students to think divergently when most of them assumed there is only one solution to a problem. That is when I thought to even reconsider the manner in which I taught the content. Why resort to teaching the examples in the textbook exactly as they are given. If I am not willing to think divergently about teaching the concept, why should I expect my students to think divergently?

In the after school setting, I have been given an amazing opportunity to explore different ways of re-establishing the students’ ability to divergently think. In mathematics, I have students analyzing the linguistic aspects of word problems. They start with words that they commonly use and construct their own word problems to describe situations to which they can relate. I have also thought about the students’ struggle with seeing numbers in different ways. Similar to the study that Sir Ken Robinson mentioned, I asked students to write down as many variations of a number as they possible could. For example, I would use the number “10” and would get a few additions examples, subtraction examples, multiplication examples, and fraction examples. In all the situations that I asked this question, I never once saw “10 + 0.” I actually posted that on the board and students were quick to tell me that I never said they could add “0.” I realized that somewhere along their years of education, a sense of restriction was built in them. This video reaffirms my belief that as educators, it is our role to inspire divergent thinking.


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.


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.