*Written for an Applied Linguistics Master’s Course: Cross-Cultural Perspectives*

**Part 1: Educational Setting**

The classroom observation took place at a primary school located in Nottingham, England. This primary school has recently combined two previously separate schools into one, larger school: a lower grades (Key Stage 0 and 1) and an upper grades primary school (Key Stage 2). The grades taught here are the equivalent to a pre-Kindergarten to Grade 6 school in the United States, with students ranging from three to eleven years in age. The school is currently undergoing major changes to its academic curriculum and school systems in response to the recent merger, with goals for progress focused on teacher development, teaching methods and approaches and student behavior. These goals for improvement are in direct response to a recent report conducted during the previous school year by the Office for Standards in Education (Ofsted), a government division dedicated to working directly with public schools to improve education in the UK.

The students at this school largely come from lower economic status homes. Ofsted (2014) provided data to confirm this in a report which gave information on free school meals, gender and ethnicity within the student body. There are approximately 463 students currently attending this primary school, and approximately 74.4% of that student body receive free school meals. There is a generally equal balance between male and female genders in the student body (51.4% female, 48.6% male). Regarding ethnic backgrounds, the majority of the student body is White British or European (69%) with the remaining ethnicities representing various Asian, Caribbean and African backgrounds, or representing a combination of these.

The classroom I observed was a Key Stage 2, Year 4 class. The students ranged from eight to nine years in age, which is equivalent to a Grade 3 classroom in the U.S. educational system. Within the specific classroom I observed, there were a total of twenty-four students in the classroom. The same ethnicities represented on the whole school level were also represented within this classroom. Most students came from a White British background (89%) with the remaining students from full or mixed backgrounds of Asian, Caribbean and African ethnicities. Fourteen of the students in this classroom were female and ten were male.

The observed lesson took place in a primary school classroom, so it is important to note that students within these types of classroom settings meet with a single classroom teacher on a daily basis from approximately nine in the morning to three in the afternoon, with some time allotted for additional classes with other teachers (Art, Physical Education, etc.). The particular lesson I observed took place during a Monday morning classroom session that occurred after a whole school morning assembly. The observed lesson was the first of two subjects taught during the class’ morning session. The first lesson focused on math and lasted for approximately one hour. The second lesson, which was not observed, took place after the students’ morning recess and was focused on literacy.

The students’ desks were organized into different groups for the observed math lesson. Each group had approximately four to five students seated in various areas around the classroom. For this lesson, the groupings were homogeneous. Meaning, the students were grouped with others who shared similar math skill levels. Generally, the level of instruction did not change in response to the varied skill levels of the students since all students were given the same math tasks to perform in their groups. However, because the math lesson was based on solving a general problem, and because it required students to access and utilize various math skills in solving the problem, the level of instruction effectively adapted and accommodated to the students regardless of the math skills they possessed. The teacher differentiated her instruction in response to varying skill levels by circulating and facilitating groups while they worked, by stopping the general group activity at intervals to review their progress whole class and by using whole class discussions as an opportunity to present ideas visually for lower level math performers to use as a reference. She also provided the groups with number cards to manipulate as they chose while they worked out the presented math problem.

**Part 2: The Lesson**

The lesson plan structure followed a more traditional format, and the main goal for the lesson was for students to collectively use various math skills in order to solve a given problem. The teacher began by introducing the rules for the math activity. The students separated into small groups and were given number cards with the numbers one through ten written on them. Students had to separate these cards into five different sets, and each set had to equal the same sum. The teacher gave the students ten to fifteen minutes to explore and test their ideas to solve the problem. During this time period, the teacher circulated to different groups and asked questions about their progress to help generate more effective discussion within the groups with her help. Next, the teacher had students share their progress as a whole class, focusing on what problems they faced in creating the five sets and how they overcame those problems. The teacher used this whole class discussion time to write down the students’ ideas on an interactive white board and to directly introduce additional math skills the students had not yet connected to the activity. The students were then given an additional ten to fifteen minutes to solve the problem in their groups. Again, the teacher circulated and asked questions amongst the different groups to gauge their progress in solving the math problem and to generate more effective discussions. Finally, students shared how they solved the problem in another whole class discussion, and the teacher recorded their ideas on the interactive white board. The students and teacher worked together in this final whole class discussion to show how to group the numbers and to discuss what math concepts to use to justify the solution. The teacher also contributed ideas and information the students had not yet shared. The students used this shared information to record their work in their math books at the end of the problem solving activity. This math activity and lesson format was repeated for a new math problem with a new set of number cards for the second half of the lesson.

Ostensibly, students had to access and use their knowledge of addition, subtraction and division to solve the problem. However, there were also more critical math and general problem-solving skills they had to access and use. The most important skills involved organization, collaboration, reviewing, reflecting, testing, hypothesizing, analyzing and rechecking. These more critical skills reflected an understanding of how to challenge students to utilize their knowledge in authentic, problem-solving formats.

The amount the teacher talked during the course of the lesson was limited to short periods of time. During the introduction of the lesson, the general rules of the game were quickly introduced by the teacher and more time was allotted for the students to talk and confirm their understanding of the rules. Whole class discussions occurred so that the students could brief each other on their progress in solving the problem with the teacher suggesting additional math skills that might help them reach a solution more quickly. This approach to teacher instruction led to the majority of the lesson time being dedicated to students working together and with the teacher in their small groups and as a whole class, negotiating solutions and asking questions to reach a solution. This was a stand-alone lesson where the main learning goal was to get students to work collaboratively, to use their prior knowledge, to work systematically on solving the problem and to purposefully record their work.

The students in this lesson had definitive control over their learning. The teacher was a facilitator and a participator, communicating, guiding and working with students collaboratively. The students were able to freely explore the methods and approaches to solving the given problem before any teacher help was provided. A lot of the thinking came from the students in both their group work and whole class discussions, with the teacher participating by asking questions and introducing other math skills that had not yet been used or considered. However, these additional math skills were ones the students already knew, and the introduction of those skills by the teacher still left room for the students to make the final decision on how they would use those skills to reach a solution.

**Part 3: Critical Pedagogy in the Class**

The format of the lesson was thoughtfully organized and implemented. The main learning goals for the students during this math lesson were to allow students to work systematically, to explain how they solved problems and to record their solutions. Students were permitted to work at their own pace in groups of their choosing in order to solve a problem, using the rules presented to them by the teacher. The construction of knowledge, the opportunities for students to be included and empowered in their learning and the communication techniques utilized by the teacher were all elements that contributed to a positive and engaging learning environment.

Knowledge was constructed through student questioning and discovery learning. Students were free to use their prior knowledge to understand and negotiate how to solve a given problem within their groups. The teacher served as a facilitator, asking students questions to gauge their comprehension, to generate guided discussion, to remind them of additional math skills that could help in solving the problem (mental math, skip counting, etc.) and to bring various math displays in the classroom to their attention that could help their thinking (number boards, math vocabulary boards, etc.). However, the teacher presented these ideas in an open manner. Meaning, the teacher contributed ideas as possible options and let the students decide how they would use the information to solve the problem. The teacher and students in this classroom lesson definitively reflected the type of transformative classroom that Paulo Freire (1970) discussed in his comparison of banking and problem-posing education. The students in this observed lesson were “…critical co-investigators in dialogue with the teacher” (Darder et al, 2009, p. 57) because of the freedom in which they could explore solutions to the problem with the teacher serving as a guide and participating as an additional problem solver.

Student interaction, involvement and empowerment were additional elements of learning observed in this lesson. There was a lot of student to student interaction supplemented by teacher to student interaction during small group and whole class discussion. Most of the interactions were positive, where students worked cohesively with each other as they contributed their personal knowledge and experience to the activity. This kind of authentic, active learning reflected the critical pedagogy classroom illustrated by Lisa Delpit (1990): students in this lesson were able to use more than one interpretation or understanding of various math skills to solve the given problem (Darder et al., 2009, p. 333), and though the math operations were not explicitly connected to problems in daily life, as with the wheel that students had to fix using critical math skills in Delpit’s (1990) example lesson, the students in the observed lesson were still using critical skills – collaborating, analyzing, hypothesizing, etc. – that were all connectable to the skills that students will use in their daily lives both inside and outside of the classroom. (Darder et al., 2009, p. 334).

Students, in general, were included in the lesson. Most participated in whole class discussion and group work. However, situations did arise where student inclusion was an issue: sometimes, more vocal students dominated discussions. In these situations, the teacher explicitly intervened to give less vocal students more opportunities to speak out. Since classroom seating was already organized into homogenous groups based on math skill level, students ultimately grouped themselves in a similar fashion when choosing their partners for the math activity. These groupings were not always helpful because they were often too large for all students to participate consistently and equally, and the lower level groups did not always have the confidence to work through problem solving as quickly as the more adept groups. Despite these setbacks, the students were empowered by the teacher because she encouraged them to freely explore and collaboratively work together on solving the problem as they wanted. There was little to no teacher interference or correction when students were exploring solutions. Rather, she participated in the discussions to generate more effective thinking in all groups.

Teacher intervention was another element observed in the lesson. Opportunities to praise students were done enthusiastically, particularly when students expressed a clear understanding of how to solve the problem. However, all thoughts were praised and appreciated by the teacher in some way, even if they deviated from the lesson at hand. In this way, students felt encouraged to share their thoughts in a safe, non-judgmental learning environment. Scolding did take place during the course of the lesson, particularly when students were talking over the teacher during whole class discussions. However, at times this teacher correction was not a product of student misbehavior, but rather, it was a result of miscommunication: at one point during the lesson, the teacher clearly stated that groups had the choice to engage in whole class discussion or to continue work in small groups. However, the students who chose to continue working in their small groups were scolded by the teacher for being “too loud,” despite the fact that they were given the choice to continue working while others engaged in whole class discussions. In this way, the teacher fell back into habits of “banking education” as demonstrated by Paulo Freire (1970), where she reflected the kind of teacher who talks while students meekly listen or who disciplines while students meekly accept discipline (Darder et al., 2009, p. 53). In the context of this miscommunication, it would have been helpful for the teacher to allow students to explain why they were talking “loudly” and over her during the whole class discussion, since they were given the option to do so by the teacher herself.

Overall, the lesson was very effective in engaging students in discovery thinking and learning. Students were able to work on various math skills and critical skills that will benefit them in and out of the classroom. Collaborative, problem-solving activities that connect to real life contexts are imperative for both the student and the teacher to learn and develop in a classroom. In this way, teachers learn how to work collaboratively with students and how to create a more democratically-based classroom while students are better equipped and empowered to use the knowledge they possess and the knowledge they gain from others to solve issues in their classrooms, their schools, their local communities and, perhaps eventually, their national and global communities.

**Part 4: Redesign the Lesson**

The lesson on the whole was well designed, organized and implemented. It was clear from the construction of knowledge, the opportunities for student empowerment and the teacher and student roles that the teacher intended for the lesson to be exploratory in nature. In this respect, students were allowed to learn through discovery, accessing their prior knowledge and experience in an effort to collaboratively work together to solve a given problem. As with any educator, there are difficulties in consistently maintaining a transformative learning environment that allows for students and teachers to work together equitably in the classroom. Overall, though this lesson was empowering and effectively implemented, there were a few elements that I thought could be improved upon in an effort to better create effective and authentic learning. The use of manipulatives and visuals, the way in which students were grouped, the questioning techniques used and the way in which students reflected on their work could all be modified to move towards a more effective and engaging learning environment.

The manipulatives and visuals used enhanced the lesson during small group work and whole class discussion. The students were provided with number cards in order to engage in each problem solving activity. In a math lesson of this nature, I think that the use of number cards is essential for students to be able to work with manipulatives that can aid them in their thinking. However, I would take the use of visuals and manipulatives further by mirroring them on the interactive whiteboard. For the course of the lesson, the teacher used the interactive whiteboard as a space to record student and teacher thinking shared in whole class discussions. A pre-made interactive whiteboard where various pages are made to scaffold student thinking would be helpful in creating a more cohesive transition from what students worked on in their groups with the number cards to what they saw and shared through visuals on the board.

Organizing visuals and manipulatives in a more premeditative way could also empower students to take on the role of a teacher amongst their peers as they could come up to the front of the class and use the whiteboard manipulatives to explicitly show their thinking to the class. Additionally, having the manipulatives mirrored on the board would make thinking clearer for students with lower level math skills: if they are able to see what is being done on the board in a way that mirrors the materials in front of them, they may be able to see more clearly where their thinking could be modified as they move closer to solving the given problem. Moreover, I think it would have been helpful to provide a choice of manipulatives – number cards, colored math cubes, etc. – so that students who visualize numbers in various formats could have different ways to organize their thinking. In particular, students who struggle with numerical representations of numbers may have benefitted from a more abstract and visual representation of numbers (i.e. instead of a number card with “1” on it, they may have been better at organizing various sets of colored cubes – 1 green cube, 2 red cubes, etc. – to organize the number sets for the math activity). I think that providing options here would allow a teacher to further cater to different modes of student thinking – some students prefer numerical representations while others may prefer more abstract, visual representations when counting.

Student groupings were another element that I would slightly change within a lesson of this nature. Homogenous groupings were the primary way in which students were organized. Some element of this organization was based on student choice, but it was a natural choice for students to make since they were already seated next to students with similar math skill levels prior to the lesson and were not encouraged to get up and move around the classroom when choosing groups. If I were conducting the lesson, I would have also allowed students to choose their groups during both phases of the problem-solving activity, but I would have encouraged them to move around the classroom to choose their group members. I would also have had students choose and work with a different group of students for each phase of the math lesson. Additionally, I would be sure to limit the groupings to three students maximum. In this way, I think it would allow the students to work with a selection of peers who may have different ways of thinking to their own which would create more interesting changes to problem solving methods. It would also allow students to have more experience in working with different people which better mirrors how one needs to adapt to working with different people in real life. Additionally, I think limiting the group numbers to a three person maximum could create opportunities for less dominant students to feel more secure in speaking out during problem solving with their peers.

Questioning techniques on the part of the teacher was another element that I would modify in some ways in this lesson. The teacher did circulate amongst the students during their group work and asked questions about what they were doing. However, she limited some of her questioning to basic, closed questions that did not provide opportunities for students to build on their thinking at the group level. She did ask more open-ended questions during whole class discussions. However, I felt that this didn’t provide students with lower level math skills ample time to build on their thinking during small group work. I think I would have created some specific questions that worked on a closed- to open-ended scale so that students could explore their thinking in a more scaffolded way. In doing this, the teacher could better gauge how students are progressing towards solving the given problem and could better gauge how to guide them in their thinking in a more targeted manner. It would also allow for students with lower math skill levels to explore their thinking with teacher guidance in a smaller, more structured context before sharing their thinking with an entire class. Additionally, I would have allowed for phases within the lesson for students to ask questions during whole class or group discussions. In this way, students would be empowered to help each other: questions that several students have in common might help them feel like they are not alone in their thinking; further, a student might share a question that one of his/her peers could answer. This kind of dialogue through questioning mirrors the roles and interactions that Paulo Freire (1970) exemplifies in the “teacher – student” and “student – teacher” dynamic; a teacher can learn from his/her students in the same way that students can learn from their teacher or from each other in a collaborative, problem-posing educational environment (Darder et al., 2009, pp. 56-57). I think targeted questioning in this manner, in addition to scaffolding student thinking in a more effective way, would allow the teacher to create equal opportunities for all students to vocalize their thoughts in small and whole group discussions. Of course, this kind of questioning also relies on the equal distribution of teacher facilitation time as s/he moves from group to group to ask questions and participate in discussions.

Reflection was another part of the lesson that I would modify in some ways. Though the teacher did pointedly take time to discuss and reflect on how students solved problems during whole class discussions, I think the reflections were a little too vague at times. I would have had students explicitly tell me the general steps they followed in order to most efficiently solve the given problem. This student thinking could be recorded visually on the interactive whiteboard at the end of each problem-solving activity. Students could then see the steps that needed to be followed in a more visual format and could add in any additional ideas to the steps for the solution they outlined as a class. Reflection on thinking is important for any learning situation as it provides students to ask additional questions, share different ways of thinking and decide as a group which approaches they think are best and why in a more open-ended, dynamic format. What a teacher can observe from this type of student to student interaction could also aid him/her in formatively assessing how well students connected to the activity, and whether or not they understood the math skills and processes used which could be particularly helpful when assessing how students with lower level math skills progressed in the context of the problem-solving activity.

No matter how well a lesson is constructed or how experienced a teacher is, the transformative and critical elements of teaching as they relate to the recommended modifications are pedagogical goals teachers must continually work on achieving. Though there were some setbacks to how the lesson was organized and implemented, it was clear that the teacher wanted students to be as engaged and empowered as possible. She allowed students to explore their thinking in a more or less safe learning environment with a posed problem that students could potentially encounter in a real life context.

**References:**

Delpit, L. (1990)**.** Language diversity and learning. In Darder, A., Baltonado, M. P., & Torres, R. D. (Eds.), *The critical pedagogy reader* (pp. 324-337). Routledge: New York.

Freire, P. (1970). From pedagogy of the oppressed. In Darder, A. Baltonado, M. P., & Torres, R. D. (Eds.), The critical pedagogy reader (pp. 52-60). Routledge: New York.

Office for Standards in Education. (2014, June 9). *Section 8 inspection report*. Retrieved from http://reports.ofsted.gov.uk/inspection-reports/find-inspection-report/provider/ELS/137863

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