Standard C.2. Pedagogical Knowledge and Practices for Teaching Mathematics
Well-prepared beginning teachers of mathematics have foundations of pedagogical knowledge, effective and equitable mathematics teaching practices, and positive and productive dispositions toward teaching mathematics to support students’ sense making, understanding, and reasoning.
C.2.1. Promote Equitable Teaching
C.2.2. Plan for Effective Instruction
C.2.3. Implement Effective Instruction
C.2.4. Analyze Teaching Practice
C.2.5. Enhance Teaching Through Collaboration With Colleagues, Families, and Community Members
Middle level learners must see the relevance and intrigue of mathematics, including its connections to the other content they are learning. Well-prepared beginning teachers of mathematics at the middle level have beginning repertoires of student-relevant contexts for each topic they teach and understand the importance of using contexts to engage students in the content. As described in the Association of Middle Level Education Standards (AMLE, 2012), beginning middle level teachers “facilitate relationships among content, ideas, interests, and experiences by developing and implementing relevant, challenging, integrative, and exploratory curriculum” (Standard 2, Element c).
ML.3. Strategies to Support Early Adolescents
Well-prepared beginning teachers of mathematics at the middle level use strategies to support a range of early-adolescent learners and engage other educational professionals within their settings to support student learning. [Elaboration of C.2.1 and C.2.5]
Students in the middle school years are reaching a developmental stage that involves new biological and psychological experiences that may lead to sudden changes in interests and behaviors (AMLE, 2015). Well-prepared beginners are knowledgeable about the nature and developmental needs of early adolescents. For example, when children reach early adolescence, their cognitive development in mathematics sometimes far exceeds their biological or psychological development. Well-prepared beginners look for and support mathematical thinking, recognizing that a learner’s behaviors might make determining what they actually know and can do challenging. Students in the middle grades, like other grades, also represent a spectrum of learners that includes students with extraordinary talents and gifts for mathematics, students with cognitive or psychological disabilities, and students for whom structures to help them meet their potentials as learners of mathematics were not in place.
Well-prepared beginners understand the specific needs of their learners, recognizing that they must hold high expectations for each and every student and employ resources such as specialists and readings to ensure that they are providing optimal environments in which each and every student can learn. They recognize mathematics-specific linguistic and cultural considerations in teaching middle level mathematics and seek ESL specialists and other resources to ensure they meet the needs of their emerging multilinguals. Well-prepared beginners seek to and are able to recognize mathematically promising students as well as create learning environments that help all learners excel. They have the disposition to seek out specialists to support and challenge students in their classrooms as well as suggest enrichment options beyond the classroom, such as clubs (e.g., The National Junior Mathematics Club [n.d.], Odyssey of the Mind [n.d.], Creative Adventures in Mathematics) and competitions (e.g., MATHCOUNTS, the American Mathematics Competition [MAA, n.d. a], Mathematical Olympiad [MAA, n.d.b]). Well-prepared beginners seek to support and challenge students with disabilities or learning challenges, accessing specialists to support their efforts. Because approximately 13% of all public school students receive special education services (National Center for Education Statistics, 2015) and because middle level mathematics learning requires significant mathematics expertise, well-prepared beginners value and seek to co-teach with special education teachers, recognizing the benefits of collaboration to support student learning. Vignette 6.2 briefly describes the roles and benefits of co-planning and co-teaching mathematics lessons.
Vignette 6.2. Co-planning and Co-teaching to Support Every Student
Context. Mr. Garza is a sixth-grade mathematics teacher with 28 students including five with special needs. Mary, Angela, Morgan, and Richard have intellectual disabilities, including difficulty with reading comprehension, and Jackson is autistic and exhibits difficulty with social skills but functions well cognitively. Ms. Harris, a special education teacher, co-teaches with Mr. Garza. He has begun a unit that includes expressing one quantity, the dependent variable, in terms of the other quantity, the independent variable. This lesson is designed to help students reason about the relationship between two variables, using concrete situations and graphs. Students will receive cards with situations in words (e.g., “height of a ball thrown straight up into the air from the time it was thrown until it hits the ground”) and cards with graphs to be matched with the stories.
Co-planning. The day before the lesson Mr. Garza and Ms. Harris reviewed the lesson plan and anticipated that Mary, Angela, Morgan, and Richard might struggle to read the scenarios. Jackson will likely need support in discussing why he chose his graph. Mr. W anticipates that all his students might struggle with understanding the variables that might be used for the two axes of the graphs. The teachers decide to scaffold the activity, beginning with a whole-class activity, then having students work in groups of four. Ms. Harris and Mr. Garza will be sure to monitor the learners who might struggle with reading. Mr. Garza will approach Jackson so that he can practice his explanation before the whole-class discussion.
Co-teaching. Mr. Garza begins by engaging the class in an example of a (real) ball thrown in the air. He then has students read the scenario. He asks what the variables are and how the scenario might look on a graph. He shows two graphs and asks students to tell why one matches the situation and the other one does not. During small-group time, Ms. Harris notices that many students are having difficulty with the reading, so she encourages these groups to summarize the meanings of different scenarios, giving the gist first and then adding details. All students are able to match scenarios to graphs. Mr. Garza and Ms. Harris take turns calling on a representative from each group to explain one scenario and the associated graph to the class, and each provides support feedback and comments. Jackson (prompted to rehearse his response by Ms. G) accurately and willingly shares a rationale for a match.
Well-prepared beginners know that strategies such as using multiple representations of concepts and multiple means of student action and expression are particularly important to middle level learners who are transitioning to more abstract mathematical concepts. Although they may seek guidance from instructional specialists for students identified for such services, they also are disposed to continuously find, try, and evaluate their own strategies to engage, inspire, and support every student. They recognize the critical importance of relationships for middle level learners and seek to establish relationships with each student so that they are better able to build on that student’s strengths and interests to develop that student’s mathematical skills and identity.
ML.4. Meaningful and Interdisciplinary Contexts
Well-prepared beginning teachers of mathematics at the middle level understand how to engage middle level learners in meaningful and interdisciplinary contexts, including the use of mathematical modeling. [Elaboration of C.2.2]
Many contexts, interesting and accessible to middle level learners, can be investigated using mathematics. And many of these contexts can be connected to middle level content in the other disciplines (science, language arts, social studies, as well as other content). Well-prepared beginning teachers of mathematics at the middle level consider ways to design interdisciplinary instruction and are able to engage in interdisciplinary conversations, offering ideas for how important mathematics can be connected to other disciplines (AMLE, 2012). They distinguish between using mathematics as a computational tool and using mathematical reasoning or modeling, and they seek to find meaningful connections for their students.
Well-prepared beginners are knowledgeable about context-based mathematical modeling and design-based activities for middle level learners. Mathematical modeling provides not only opportunities for interdisciplinary instruction but also authentic contexts for engaging in mathematics and building mathematical understanding (Hirsch & Roth McDuffie, 2016; Usiskin, 2015). The modeling task in Table 6.7 involves a problem about the consequences of melting (de Carvalho Borba, Villareal, & da Silva Soares, 2016).
Table 6.7. Interdisciplinary Modeling Task for the Middle Grades
Melting of a Glacier
The problem was to observe the percentage of reduction of a glacier, in this case, the Puncak Jaya glacier located in Indonesia . . . . This theme interested us because mankind is destroying the environment. Our hypothesis is that the glaciers have diminished to the point of almost disappearing. But to prove this hypothesis, we should consult diverse sources.
Note. Adapted from "Modeling Using Data Available on the Internet" by M. de Carvalho Borba, M. E. Villareal, and D. da Silva Soares, 2016, Annual Perspectives in Mathematics Education: Mathematical Modeling and Modeling Mathematics, pp. 145. Reston, VA: NCTM. Copyright 2016 by National Council of Teachers of Mathematics.
Such a task could be implemented in collaboration with teachers of English language arts (ELA), social studies, and science. The composition and production of a final report can include persuasive argument, the process of changing policies and advocating to elected officials or members of the public, and measurement and data gathering, meeting ELA, social studies, and science standards.
Well-prepared beginners recognize that a task's implementation in a classroom influences how meaningful and engaging it is for students. For example, the following list describes effective teaching practices for supporting engineering design (and mathematical modeling), a design that engages students in authentic reasoning and problem solving:
- Pointing out limitations of the class models as a whole (e.g., if none of the models include a mechanism for motion, a teacher might request that students consider motion in a revised model);
- Providing information students would be unable to discover on their own (e.g., explaining or contrasting the mathematical concepts of mean and median as measures of center); and
- Encouraging individual teams of students to pursue specific design challenges to extend their models in general ways (e.g., considering how the function of the object under investigation is similar to and different from a familiar related object).
(Adapted from Engineering in K–12 Education: Understanding the Status and Improving the Prospects (p. 124), by the National Research Council, 2012, edited by L. Katehi, G. Pearson, and M. Feder. Washington, DC: National Academies Press. Copyright 2012 by the National Research Council.)
Middle level learners in classrooms using this engineering design were able to design functional models and data representations. Significant time is required for teachers to develop robust understandings of ways content can be integrated and ways in which engineering design and modeling activities can support mathematical content goals within any given classroom. For example, weighing the cost-benefits of paint, ensuring that lead [e.g., in paint] does not affect safety, provides a meaningful experience that addresses science and social studies. Although teachers throughout their careers will continue to learn about these relevant applications, well-prepared beginners know and have access to resources that provide engaging interdisciplinary mathematics investigations and mathematical modeling activities.