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

Well-prepared 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

C.4.5. Enact Ethical Practice for Advocacy

As suggested by Dewey (1910), improving the public’s knowledge of mathematics and science and cultivating critical reasoning skills are important goals for education and essential for a democratic citizenry. Middle grades become important in a child’s development for bringing numerical and other mathematical reasoning tools to bear on the decisions citizens must make in a democracy, even when research shows that strongly held beliefs and cultural affinities may disable individuals’ use of reason and sense making to inform decisions (Kahan, Peters, Dawson, & Slovic, 2013). Mathematics, in particular, provides young, future citizens with useful tools and practical lenses to examine social and personal issues that arise throughout their lives. Middle school mathematics teachers must communicate and model the power and utility of mathematics for decision making and personal growth.

ML.6. Respond to the Needs of Early Adolescents

Well-prepared beginning teachers of mathematics at the middle level understand the developmental needs of early adolescents and use this knowledge to create and implement culturally relevant mathematical experiences for their students. [Elaboration of C.4.3]

Well-prepared beginning teachers of mathematics at the middle level are able to cultivate the development of students’ positive mathematical identities and draw on students’ cultural and linguistic strengths as well as their individual interests and passions as part of the middle school mathematics instructional program. As an example, using personal-information surveys can help early-adolescent learners to examine data and measurement units that are plausible (Lovett & Lee, 2016). Similarly, proportional reasoning skills can be developed by examining socially relevant questions about equity and fairness in the contexts of consumer issues, population growth, and crime rates (Simic-Muller, 2015). Drawing on how different cultures employ mathematical ideas provides students opportunities to learn about and honor their own and other cultures as exemplified in how students from immigrant families might participate in a culture-laden lesson that develops “a compassionate understanding of their classmates from different backgrounds and [fosters] an atmosphere of respect, solidarity, and collaboration” (Taylor, Rehm, & Catepillán, 2015, p. 106). Additionally, students’ language can be used as a resource, as illustrated in Vignette 6.3.

Vignette 6.3. Attending to Algebra Content and Language in a Classroom With Emergent Multilingual Students

Rosa, a student teacher, was thrilled to be teaching with Marie, a 6^th-grade teacher in a school designed specifically for recent immigrants. Students in her class represented 13 countries and 11 languages. Although numerous students were Spanish speakers, some students were the only ones speaking their native languages. Together, Rosa and Marie designed a 2-week unit on algebraic thinking, with a focus on generalizing patterns, writing algebraic expressions and equations, and connecting representations (situations, generalized rules, equations, and tables).

With the students’ diverse backgrounds and need for visual supports, Rosa and Marie decided to infuse children’s literature as a tool for building common background. The first book, Two of Everything (Hong, 1993), was a Chinese folktale that delighted the sole Chinese student and was well received by everyone. Knowing which students had stronger understandings of English and felt comfortable in front of the class, Marie asked four students to come to the front and act out the story while she read the book (she had brought some basic props from home). In the story, a magic pot is found that doubles everything that goes in it.

After reading the story, Rosa asked students to tell their shoulder partners what happened in the story. Along with other aspects of the story, they talked about the magic pot doubling whatever was dropped into it. Because the students spoke different languages, they often spoke to each other in everyday English. During planning, Marie had encouraged Rosa to build meaning for the mathematical language, using their everyday languages as a resource. During the lesson, Rosa recorded a vocabulary table on the board:

	Everyday Meaning	Mathematics Meaning
Rule
Expression
Sentence/Equation

Rosa asked students to talk to their shoulder partners about the meanings of the words when they use them outside of math class. Rosa recorded their examples of the everyday meanings. For each one, Rosa referred to the pattern in the magic pot and added in the language or symbols connected to the story.

	Everyday Meaning	Mathematics meaning
Rule (words)	“Quiet” “No cell phones”	“Multiply by 2”
Expression (symbols/numbers)	“How beautiful!”	2 × a 2 ● a a × 2 2a
Sentence/Equation (equation is a number sentence)	“How beautiful are your eyes!”	n = 2 ● a 2 × a = n

After completing the table, Rosa had students say each word together. Then she explained the magic pot had changed its rules, and it would be their job to determine the new rules and to write expressions and equations for each new rule. She distributed an activity page that had several input-output tables (labeled as In-the-Pot and Out-of-the-Pot), with places to record rules, expressions, and equations. These tables were then solved and discussed.

In reflecting, both Rosa and Marie noted every student used the three identified words when they discussed the new magic-pot rules. Marie noticed the Chinese student had spoken in English without first using his translator for the first time. While the unit progressed, students continued to use appropriate language and to explore increasingly complex linear relationships across a range of situations, including generating their own stories.

This vignette provides an example of maintaining high expectations and providing strong support for students. It shows the value of co-planning and co-reflecting, and the effect this can have on the teacher candidate as well as on the students.

In the middle school mathematics classroom, early adolescent learners can examine complicated challenges they will confront at school and in their lives, using the analytical and logical tools provided by mathematics. Well-prepared beginners must be careful to avoid advocacy of a particular point of view but can examine how mathematics informs opinions and decisions about topics that are current, relevant, or of particular interest to learners (such as climate and the environment, health and human sexuality, bullying, or lotteries) and thereby empower students to use mathematics for critical thinking and authentic purposes. Well-prepared beginners realize that such careful selection and enactment of tasks shapes students' emerging mathematical identities and influences the decisions they will make in terms of continuing in mathematics, pursuing careers, and selecting college majors.

ML.7. Equitable Structures and Systems in Middle Schools

Well-prepared beginning teachers of mathematics at the middle level are aware of structures that support and inhibit opportunities for learning in schools and systems. [Elaboration of C.4.1, C.4.4, and C.4.5]

Well-prepared beginners recognize that current systems and structures do not provide equitable opportunities for early adolescents to learn mathematics. Tracking is one such structure that is typically initiated at the middle level (Loveless, 2016). Tracking, the practice of grouping students in classes on the basis of perceived ability levels, leads to enrollment of some students in algebra courses in the middle grades and other students being kept from enrolling in such courses. Tracking has been demonstrated to create and reinforce social inequities because African American, Latinx, and children living in poverty are underrepresented in the accelerated tracks (Boaler, 2011; Larnell, 2016).

Different from tracking, but related, is differentiation through enrichment and acceleration. Enrichment provides opportunities within courses to deepen student understanding; acceleration is a faster pace through a curriculum. Rushing elementary learners to algebra and high school learners to calculus is generally identified as acceleration and is counter to ample research (Bressoud, Mesa, & Rasmussen, 2015). This practice has contributed to practices at the middle level that are inconsistent with the principles of middle level education articulated by the Association for Middle Level Education (AMLE, 2010) and runs counter to principles of equity. Both enrichment and acceleration are used in middle level to differentiate instruction. Well-prepared beginners recognize the distinctions between enrichment and advancement and recognize also that although the intent might be to provide support and challenge to all students, many efforts to provide enrichment or acceleration have resulted in inequities and denied students access to important mathematics. Well-prepared beginners realize that they can advocate for more equitable practices related to advancement. For example, they might advocate that eighth-grade Algebra I placement assessments to be based on not only high-stakes-test scores but on multiple measures based on student potential, creativity, and interests. Well-prepared beginners are committed to implementing enriched curricula, recognizing the importance of developing the mathematical practices and positive mathematical dispositions for each and every student.