Standard C.4. Social Contexts of Mathematics Teaching and Learning 

Wellprepared beginning teachers of mathematics realize that the social, historical, and institutional contexts of mathematics affect teaching and learning and know about and are committed to their critical roles as advocates for each and every student. 
C.4.1. Provide Access and Advancement C.4.2. Cultivate Positive Mathematical Identities C.4.3. Draw on Students’ Mathematical Strengths C.4.4. Understand Power and Privilege in the History of Mathematics Education 
Elaboration HS.3, Supporting the Opportunity to Learn by all High School Mathematics Teachers, states that wellprepared beginning high school teachers need to understand that their beliefs about what learning mathematics entails along with their beliefs related to students’ cultural backgrounds, ability levels, and other defining characteristics affect how they interact with students. Such interactions largely determine how students engage in mathematics and how much students advance. In addition to elaborating Standard C.2 for the high school level, HS.3 also has significant implications for Standard C.4. The following elaboration further addresses the need to cultivate positive mathematical identities in each and every student.
HS.4. Cultivating Positive Mathematical Identities in Each and Every Student
Wellprepared beginning teachers of mathematics at the high school level draw on students’ strengths to cultivate positive mathematical identities. [Elaboration of C.4.2 and C. 4.3]
Within their programs, high school teacher candidates need to see diverse students reasoning and making sense of mathematics—students who differ in socioeconomic status, race/ethnicity, gender performance, immigrant status, religious background, ability, linguistic background, and in other ways. In a methods course, Vignette 7.3 can be used as a catalyst around developing positive student mathematics identities.
Vignette 7.3. Mathematical Identity
I am Michael Davis, an African American junior in a geometry class. I am motivated to do well in this class because I want to go to college, and if I make good grades, I might get a scholarship. I actually like this class because I am given opportunities to solve problems in groups with my peers. I like the discussions that we have, especially debates when we do not agree on a solution. I like activities that allow me to discover important connections between different topics of mathematics, between my life and mathematics, and between mathematics and what is going on in social media and the world. I also like making conjectures and testing them to find out if they are true.
Mathematics teacher candidates need opportunities to reflect on the identities that students develop and the factors that contribute to the contexts in which those identities arise. Questions the vignette could prompt include the following: Why might Michael feel the way he does about mathematics class? Given that Michael is a Black male, what might be noteworthy about his experience? Why might he enjoy working in a group? What does it take to develop meaningful groupworthy problems for students to solve? What would support the development of a classroom culture in which group work is valued? What are the differences between problems and exercises? Why might Michael care about relating mathematics to other topics, his life, social media, and the world? Should these connections be made between things with which he is already familiar, or can they also expand this knowledge? How does a teacher orchestrate this discussion well? In other words, what problems might be considered in this classroom beyond those teachers might consider realworld problems or curricular materials they might think include examples of realworld problems, but that a student sees as contrived or “not my real world?”
Furthermore, teacher candidates need opportunities within their programs to facilitate the mathematics learning of students from the aforementioned groups. Also, when possible, high school mathematics teacher candidates should collaborate with specialneeds teachers and teacher candidates who are focusing on students with special needs. Mathematics teacher candidates need to learn to read Individual Education Plans (IEPs) for students and work with specialneeds teachers to understand how to differentiate instruction, use multiple entrylevel tasks, provide appropriate accommodations, and use other strategies to ensure that mathematics learning occurs in meaningful ways for each and every student.
High school teacher candidates need opportunities to interact with emerging multilingual students and to understand that just translating words for students is not enough; they need to create environments and learning situations in which the students can understand and own the mathematics for themselves. For example, students may be allowed to use algorithms from their home countries or to show their work in ways that are not standard practice in the United States (Civil & Menéndez, 2011; Civil & Planas, 2010; Gutiérrez, 2015).
High school teacher candidates need to understand how mathematics structures and vocabulary can challenge emergent multilingual students in multiple ways: Words have multiple meanings, different meanings in different contexts, and different pronunciations when used as noun versus as adjective. Thus, beginning high school teachers need to be aware of the ways in which language influences learning, including recognizing that all mathematics students are language learners (e.g., the word function does not mean the same thing to a monolingual English speaker when stated in the context of a broken washing machine at home versus in a mathematics classroom). Vignette 7.4 highlights a high school teacher candidate’s revelations in learning to meet the needs of her emerging multilingual students.
This story illustrates the importance of not only having teacher candidates work with a diversity of learners but also having them reflect on and process those experiences to better understand how they can build on the cultural and linguistic resources that students bring to the classroom to support the learning of each and every student. This vignette may be used in a methods course to help teacher candidates consider what they would do if faced with a similar situation.
Vignette 7.4. A Student Teacher’s Revelation Related to Emergent Multilingual Students
On the first day of school, my cooperating teacher learned that three Guatemalan students who did not speak English would join her 7^{th}period algebra course. I was immediately excited and eager to meet them. I have studied Spanish for many years and now have a bachelor’s degree in the language, but I have had few opportunities to use my knowledge in the workplace. I knew that speaking Spanish would be useful for a teacher in the public school system, but I was surprised to get to use it so soon. Because of their language needs, this group of students has worked with me every day since Day 1. This experience has given me insight into the challenges that nonEnglishspeaking students face daily at schools in which teachers have not been prepared to facilitate their learning and to help them achieve success.
During the initial observation weeks of student teaching, I spent my time in 7^{th} period sitting with the Guatemalan students, providing explanations in Spanish of what transpired in class. In the first days of class, during prealgebra review, simple translation of the materials and interpretation of the teacher’s lessons kept these students with the rest of the class. However, when the ideas discussed in class quickly became more complex, I realized that translations did not suffice. They understood the words I was saying, but not the mathematics. I realized then that these students, like any other students, needed instruction that they could understand. I began to shift my focus from translation and interpretation to teaching mathematics—a more difficult job that required much more thought and preparation. The same materials were not appropriate for both languages and both cultures, and I needed to differentiate my instruction. Because I was still observing, I was able to provide these materials and provide oneonone assistance to these students.
While I continued to spend entire periods working with just the Guatemalan students, I wondered how they functioned in their other classes that had no Spanishspeakers to help them. I followed them through their schedule for a day to see what they were experiencing and found that they did nothing all day. They were ignored and not even given assignments in all but one class (Spanish class). I do not know how the teachers should have taught students with whom they could not communicate, but I think that doing nothing is the wrong solution.
When I began teaching 7^{th} period myself, however, I realized that providing all the extra help needed by the Guatemalan students while teaching 25 other students in the classroom was very difficult (even though I spoke Spanish). My ability to help them diminished, and I started to see the real challenge in helping the struggling students without ignoring the other students.
I decided that to be successful, one must not separate the struggling students from the rest of the group. I began designing lessons to engage both groups of students and relied on multiple levels of knowledge to allow students at multiple levels to interact with one another. Additional assistance may be provided, but, in general, we must ensure that our lessons meet the needs of each and every student.
One fact stood out to me the more I worked with them: The Guatemalan students had a strong desire to participate and be a part of the class. I had seen just how little they were able to engage with anyone in their other classes, and I think they saw this mathematics class as one of their few opportunities to interact with other students and teachers. I saw that I could use this desire not only to motivate them to understand and ask questions but also to motivate other students to participate.
The Guatemalan students learned the numbers in English very quickly, and within a few weeks, they were calling out answers. Their English teacher facilitated this process by reinforcing use of words from other courses in her class. The students stopped saying numbers in Spanish and stated them only in English, even when we were talking privately. I helped them to create a mathematical dictionary in their notebooks, entering words like equalities and inequalities; in a short time, they began explaining their processes using these words. I was impressed by their fearlessness in class. Although they did not speak English well, they never hesitated to give solutions in front of the class. I believe that their willingness to share encouraged the other students in the class to speak out and share their own understandings.
One student seemed to have more difficulty than the other two. I learned from him and his Spanish teacher that he had not been to school in several years before coming to the United States. He was weak in mathematics but also not strong in reading, writing, or the Spanish language; his first language was Xinca, a language spoken by indigenous people in Guatemala. I saw that he needed more than the class could provide. After he did poorly on a test, I explained to him that to succeed in the class, he needed extra work. He agreed to work with me before school, and the next morning he was there.
We worked together one on one for 20 or 30 minutes each morning, and he eventually caught on to what we were doing in class. I found knowing when to keep him working on a topic and when to move on with the class challenging, but we tried to use the extra time as wisely as possible. After a few weeks, he came less often before school and became less dependent on me during class. He became more comfortable asking his peers questions and more confident on quizzes and tests. Eventually, he stopped coming in the mornings, and I felt confident that he would be able to work along with the other students in the classroom.
My experience working with this class and the Guatemalan students has taught me a lot about how to effectively support students who have unique language needs. I realized that the students had come into our mathematics classroom eager to participate and wanting to be part of the classroom. I simply needed to find the proper ways to support them in making sense of the U.S. conventions we use and to clarify the language when there were not easily identifiable cognates. I learned that instead of treating all the Guatemalan students as the same, I needed to probe the previous experiences they had in mathematics, how each one learned, what resources and expertise they had before entering my classroom, and what they needed from me to succeed. I have learned the importance of not lowering expectations for students based on their language needs, that they can achieve the same learning objectives in different ways. I have also learned to recognize when a student truly needs extra work, outside class time, to be able to function with peers. It is important to keep in mind that providing the same learning experiences for all students does not necessarily result in an environment that is equitable. All students should have the same opportunities to learn and succeed. Providing these opportunities may require adjustments or supplemental instruction, and incorporating these methods and the resources the students bring into a curriculum to take advantage of and celebrate the diversity of learners benefits not only the targeted students but also students from different backgrounds.