Beyond Good Teaching reflections

Chapters 1 and 4

The ability to justify their work and mathematical reasoning is vital for students today. In fact, it is so vital that it has been built into the Common Core State Standards, Standards for Mathematical Practice. Specifically, our students must be able to reason abstractly and quantitatively and construct viable arguments and critique the reasoning of others (Common Core State Standards, n.d.). The ability to justify their work also allows students to gain a more in depth understanding of the material, while allowing teachers to more accurately assess the students’ mathematical development.

While this skill is vital for all students, it can be especially difficult for English Language Learners, or ELLs. There are four elements that make up an effective mathematical environment for ELLs: setting high expectations, taking time to listen, observe, and learn, understanding cultural value systems, and affirming ELL’s cultures in the classroom (Celegon-Pattichis and Ramirez, 2012, p. 48). By incorporating these elements into our classrooms, we can more effectively support ELLs in justifying their work. This in turn will support their learning of the English language, their ability to work cooperatively in the classroom, and the ability of the teacher to accurately assess their development.

References

Common Core State Standards Initiative, Preparing America’s Students for College and Career. (n.d.). Mathematics standards. Retrieved August 28, 2018, from http://www.corestandards.org/Math/

Celegon-Pattichis, S., Ramirez, N.G. (2012). Elements of an effective mathematics community for ELLs. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 47-53). Reston, VA: National Council of Teachers of Mathematics

Chapter 5 Case Studies

Case 1 reflection

EDUC 361 / 461

28 September 2018

Chapter five, case one involves an ELL student, referred to in the text as student 3. In the video, Ms. Field is supporting student 3 as she explains how she solved a math problem. Ms. Field supports the student by encouraging the other students to listen and encouraging student 3 to use her white board and hundreds chart for support. I believe that Ms. Field could have supported student 3 further by increasing her wait time between questions. There are several points in the video where Ms. Field posed a question to student 3, but she did not leave any time for student 3 to answer the question before asking another. For instance, at one-point Ms. Field asks “And then what’s your answer? How did you do it? Did you count them? Show me. How did you do it?” (Celedon-Pattichis & Turner, 2012, p. 59). The four questions and request to demonstrate were all posed so quickly that the student did not have a chance to respond to any one question. The student answered this bombardment of questions by pointing to the hundreds chart, which started Ms. Field on a new string of questions. I think the student would be further supported if she is given at least a few moments between questions. In that time the student must translate the question so that she can understand it, think about the response, and then translate the response into English so that the teacher can understand it. Rapid fire questioning does not permit the time that students need to go through the translation process. While this teacher was supportive of student 3’s problem solving, increasing wait time between questions would offer further support.

The second vignette in the chapter focuses on a bilingual teacher who is posing a story questions to her students in Spanish. This teacher is supporting the math learning of her students; first, by using the students’ primary language, and second, by framing the division problem in a context the students understand from what is happening in their home lives. This is evident when the students get distracted talking about the Easter baskets that their parents are purchasing. While students should be supported to learn English, I see no reason why their mathematics learning should not be delivered in their primary language. That is, as long as the teacher delivering the instruction is fluent in the language. If the teacher is not fluent, then she will not understand the thinking of the students. Creating a division question that draws on the recent activities of the students supports them because they do not have to figure out what the story is talking about. The students were engaged in the activity and excited to talk about the Easter baskets. The teacher further supported the students with the use of white boards to work out the problem. This support is important for all students.

Both vignettes illustrated the importance of creating challenging and appropriate tasks for students. A student’s inability to speak English does not mean the student is not capable of complex mathematical thinking. Tasks should be challenging and engaging, which supports all students in the classroom. Students should also be supported by allowing them access to appropriate materials. In the first video, this was accomplished with the use of white boards, counting cubes and a hundred chart. In the second video, this was accomplished with the use of white boards. Hands on materials support all students but are essential to ELL students.

References

Celedon-Pattichis, S. & Turner, E. (2012). Using storytelling to pose word problems in kindergarten ESL and bilingual classrooms. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 56-62). Reston, VA: National Council of Teachers of Mathematics

Case 2 reflection

EDUC 361 / 461

28 September 2018

Chapter five, case two examines the learning of students from the perspective of a teacher, who was herself an English language learner. This teacher describes her experiences as a student learning math and her experiences as a teacher teaching math. The author describes her experiences as a student as being “treated as though we were invisible” (Kinzer & Rincon, 2012, p. 63). This feeling of invisibility has been described by other researchers as well. Mohr and Mohr (2007) described ELLs observed in their study as unengaged and neglected. When the teacher’s explained as giving the students an extended silent period to allow them to adjust (Mohr & Mohr, 2007). Further, Kinzer and Rincon (2012) describe the practice of “dumbing down” the curriculum for ELL students. This is never appropriate. Feeling the need to alter the curriculum for ELL students shows a major bias on behalf of the teacher. The language an individual speaks has zero barring on their ability to reason with mathematics. Language only effects the delivery of the mathematical reasoning, by this I mean what language the information is delivered in.

The videos for this case study highlight the strategies that the teacher discusses in the case study, such as beginning lessons by drawing on students’ prior knowledge. In the first video the teacher draws on the students’ prior knowledge when she asks them what they know about fractions. This strategy benefits all students, not just ELL’s. The teacher provides further support during group work by asking students to explain their thinking and the strategies that they used. Again, this practice supports the learning of all students. The final two videos on decomposing numbers to make multiplication easier demonstrate how all students can be supported in the classroom. In the first of the two videos, a student explains the process that she used to solve a problem. She uses a variety of strategies and supports to help her solve the problem, which she explains to the group as she solves it. In the second video, a student explains to the class her method of decomposition. The teacher supports her by encouraging her as she works and asking her to work additional examples. She supports the rest of the class when she has the student scaffold the final example with input from the rest of the class.

In sum, the teacher provides supports that support the mathematical development of all students in the classroom. She makes sure to check for understanding by checking in with students as they work. Finally, she encourages all students to participate in mathematical discourse in the classroom.

References

Kinzer, C. & Rincon, M. (2012). Fostering an equitable classroom for English language learners. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 63-69). Reston, VA: National Council of Teachers of Mathematics

Mohr, K. A.J., & Mohr, E. S. (2007). Extending English language learners’ classroom interactions using the response protocol. Retrieved from http://www.colorincolorado.org/article/extending-english-language-learners-classroom-interactions-using-response-protocol

Case 3 reflection

EDUC 361 / 461

28 September 2018

Chapter five, case three is from the perspective of an English language learner turned teacher. Ricardo Rincon arrived in the United States as a young adult, so his perspective is a little different from an individual whom arrived in the country as a child. However, he also found that his teachers based their conceptions of his content level learning on his English language proficiency. Upon beginning his education, it was believed that his English proficiency was not high enough to allow him to take core classes at the community college level. Even after reaching a level of proficiency that enabled him to take classes, he found that his teachers assessed his performance on his language use instead of his content area knowledge (Kinzer & Rincon, 2012). This was at the college level! This shows that the misconceptions common to the teaching of ELL’s extend beyond the elementary school years and impact a vast number of students.

In the video, Mr. Rincon uses a graphic organizer to record what his students know about number lines. He extends the conversation and allows students to draw on prior knowledge by asking about number lines that exist outside of school. Using the graphic organizer and connecting new knowledge to previous learned material are two ways of supporting students in the classroom. These practices support all students but are especially important for ELL’s. The list of timelines the students generated was impressive. They were able to extend what they knew about timelines to come up with examples that the teacher did not even consider. The teacher also supports his students by having them clarify and expand upon each other’s ideas. This supports the language development of the students. Finally, Mr. Rincon supports his students by utilizing long wait times, which allow students to fully consider their responses before speaking.

References

Kinzer, C. & Rincon, R. (2012). Fostering an equitable classroom for English language learners. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 63-69). Reston, VA: National Council of Teachers of Mathematics

Case 4 reflection

EDUC 361 / 461

28 September 2018

Chapter five, case four illustrates the effectiveness of what we have been learning about in all of our classes. I have wondered if I am going to be an effective teacher for English language learners (ELLs), as I do not speak any of the other languages found in my area. With the help of Jennie, the math coach, David transforms his classroom into a discourse community. This starts with added visual supports, which help students understand the concept even if they do not understand the words (Ramirez & Confer, 2012). This made the math content more accessible to the students. The visual models not only support ELLs but the rest of the students as well. David and Jennie went through the lesson and carefully considered the strategies the students would use and the visual models that would support those strategies (Ramirez & Confer, 2012). This is the type of intentional planning that should occur for every lesson plan.

David and Jennie considered the language development and needed supports when crafting their lesson plan. Providing the students with the vocabulary and supports during the work stage, gave them the confidence they needed to share during discussion (Ramirez & Confer, 2012). I have worked with several quiet students, some shy and some who were apprehensive of their abilities to the extent they felt uncomfortable sharing. Usually I would speak with these students before discussion to make sure that they understand the concepts and their thinking was on the right track. However, that is where my support ended. I had not considered helping the students feel confident enough to share with the whole group. I think that some of the students that I work with now would share more if I provided additional support on how to explain their thought process.

When I read about David and Jennie supporting the students and the number of students who wanted to share, I felt like I was sharing their excitement. To have so many students feel confident enough to share in one lesson was amazing. I was especially moved when Anisa said, “I like how I sounded…I felt smart” (Ramirez & Confer, 2012, p. 77). Providing the supports that she needed enabled Anisa to participate in the discussion, which not only increased her confidence in math, but her self-esteem as well. This is what teaching is all about, lifting up our students and supporting them to do their best; building their esteem and helping them to develop a growth mindset.

References

Ramirez, M. A. & Confer, C. (2012). From “plussed” to “added”: Supporting English language learners. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 63-69). Reston, VA: National Council of Teachers of Mathematics

Case 5 reflection

EDUC 361 / 461

29 September 2018

Chapter five, case five focuses on a teacher named Sara. Sara has shown an extraordinary ability to work with English language learners, or ELLs. She does so by establishing high expectations for all students, providing visual supports and creating an environment of collaboration and learning.

From the beginning of the year, the students are exposed to complex mathematical vocabulary. Sara introduces the vocabulary using oral and written repetition (Chval, 2012). This allows the students to connect the new vocabulary terms to their prior knowledge. The use of repetition provides the students with a variety of opportunities to hear the vocabulary terms used in context. The use of complex mathematical terms is one of the ways that Sara expresses her high expectations for the students. She also expresses her high expectations when she requires the students to justify their reasoning during discussion (Chval, 2012). The students know where the bar is set, and they work to reach it consistently in Sara’s classroom.

Sara provides additional supports for her students by making use of the calculator’s buttons as visual and communication aids (Chval, 2012). The use of the calculator extends beyond the traditional use, which general requires students to record their process after they have completed it. Sara requires her students to plan their process before they begin. This allows the students to express their ideas before they attempt to solve the problem. Having the students share their planed keystrokes helps them to share their mathematical thinking when they do not know the words to express what they are thinking. It also helps the students to communicate with one another about the different methods that they used to solve the problem.

Finally, Sara exemplifies what we have read about creating a community of learners in the classroom. She explicitly teaches the students how to appropriately communicate with each other, including how to handle disagreements or incorrect solutions (Chval, 2012). The transcripts provided in the case study really show the work that Sara puts into teaching the students how to appropriately discuss math in the classroom. Sara scaffolds this support, which is shown in the way she moves her position in relation to the board as the students become more efficient communicators and problem solvers.

This case study was another great example of how to accomplish the practices that we learn about in our courses. Sara establishes expectations early in the year which enable students to learn and communicate efficiently. This is evident in her text scores. She is a shining example of all we should aim for as teachers.

References

Chval, K. (2012). Facilitating the participation of Latino English language learners: Learning from an effective teacher. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 77-88). Reston, VA: National Council of Teachers of Mathematics

Chapter 7

Chapter 7, written by Marta Civil and Jose Maria Menendez, discusses the importance of parent partnerships in the education of ELL students. What I found most surprising in the chapter, is the way that some of the teachers responded to the parents about the alternate methods that the parents use. The teachers were so closed to methods other than their own that they made the parents feel as if their way of doing this was not valued in the classroom (Civil & Menendez, 2012). There are so many different, viable ways of doing math that it seems ridiculous to me that any teacher would discount an alternate method. The chapter specifically mentions the way that some of the parents learned how to divide while attending school in Mexico. This method is very similar to the algorithm that we use here, only the subtraction is done mentally instead of as a written step. I thought of this as a more advanced form of the algorithm, with the way we do it serving as a scaffold until the student can perform the subtraction mentally. The reading cites the problem with this method is that teachers cannot tell if the student cheated by using a computer of calculator, but I feel that this would not be much of a problem. If I had a student turn in something to me that uses this method, I would have them show me how it works with a new problem. If the student could show me all the steps without using a computer or a calculator, then I would know that the student is capable of using the method without cheating. I would encourage the student to double check their subtraction if they are having trouble, but I would not discourage the student from using the method that they learned from their parents.

Another thing that I found interesting about this chapter is the approach to interacting with parents. Before changing my major to elementary education, I was an early childhood education major. The approach to working with parents is totally different in the early childhood setting, with parents being viewed as partners in the education of their children. Parents are also thought of as experts when it comes to their children. This is totally different from the way that the parents were interacted with in the textbook.

I hope to create partnerships with the parents of the students in my classroom. I will do this by establishing contact with parents at the beginning of the year and making sure that every parent has a way to contact me that works for them. I also plan to keep the parents apprised of what we are working on in the classroom throughout the year. When I taught preschool, I had an open-door policy with parents. This policy stated that any parent could come into my classroom at any time to help or observe. This is more challenging in the elementary and middle school system, due to the required background checks. However, I will do my best to make sure that every parent feels welcome. If a parent has a different method for doing something, I plan to invite them into the classroom to teach their method. If they are not comfortable with that because of language barriers, then I will do my best to learn the method myself so that I can share it with the class. I will also encourage the students to teach the class the methods that they have learned from their parents.

The bottom line is that parents know their children better than we will ever know them as students. We get them for a year, they get them for life. We need to treat all parents, regardless of language differences, as partners in the education of their children. If we view the alternate methods of doing math as tools, then parents are great resources to supply those tools.

References

Civil, M. & Menendez, J.M. (2012). Parents and children come together: Latino and Latina parents speak up about mathematics teaching and learning. In Celegon-Pattichis, S., Ramirez, N.G. (eds.), Beyond good teaching, Advancing mathematics education for ELLs (pp. 127-138). Reston, VA: National Council of Teachers of Mathematics