Ch. 19 Classroom Centered Practices in Mathematics

Culturally Responsive Teaching

To achieve a high quality public education for all students, all educators must be able to work effectively in diverse settings. To become effective in diverse contexts, educators must be willing to learn about systemic racism and inequities in the public education system and to develop culturally competent skills and mindsets (EOGOAC, 2017). Professional learning opportunities aimed at increasing cultural competencies are focused on increasing educators’ knowledge of student cultural histories and contexts (as well as family norms and values in different cultures), the ability to access community resources for community and family outreach, and developing the skills for adapting instruction to align to students’ experiences and identifying cultural contexts for individual students (RCW 28A.410.260). In accordance with best practices regarding family engagement, districts should make every effort to ensure cultural competence training programs are developed and implemented in partnership with families and communities (EOGOAC, 2017).

When considering mathematics teaching practices to reach students who have not yet met grade-level standards in mathematics, it is important to consider the positive impact of culturally responsive teaching in order to better support all students in mathematics. Studies have shown that culturally responsive teaching, defined as teaching that leverages students’ cultural knowledge to facilitate learning, has positive effects on students’ learning. Furthermore, teachers having respect for cultural diversity positively influences the students’ motivation to learn.

Margery Ginsberg suggests a motivational framework for culturally responsive teaching which can support learning. The framework is made up of four essential motivational conditions, which Ginsberg has found to act “individually and in concert to enhance students’ intrinsic motivation to learn.” The conditions are:

1. Establishing Inclusion—the teacher creates a learning environment in which students and teachers feel respected by and connected to one another.
2. Developing a Positive Attitude—the teacher creates a favorable disposition among students toward learning through personal cultural relevance and student choice.
3. Enhancing Meaning—the teacher creates engaging and challenging learning experiences.
4. Engendering Competence—the teacher creates a shared understanding that students have effectively and authentically learned something they value.

Teaching mathematics with a culturally responsive lens means that the teacher creates an inclusive environment, makes the learning relevant with some aspects of student choice, plans and enacts learning activities that are engaging and challenging, and supports his or her students in knowing what they have learned and why it is of value.

When classrooms and schools are staffed with culturally competent educators, schools are more likely to effectively work towards closing the opportunity gap and increasing student achievement. OSPI has created a toolkit to support educators as they integrate students’ funds of knowledge in the classroom. Additional resources that support culturally responsive practices include: Culturally Responsive Teaching Matters!, Culturally Responsive Classroom Management, and Culturally Responsive Teaching.

References

• Ginsberg, M. B. (2015). Excited to learn: motivation and culturally responsive teaching. Thousand Oaks, CA: Corwin Press.
• Ladson-Billings, G. (1994). The Dreamkeepers: Successful teaching for African-American students. San Francisco: Jossey-Bass.
• Pewewardy, C. (1994). Culturally responsive pedagogy in action: An American Indian magnet school. In E. R. Hollings, J.E. King, & W. C. Hayman (Eds.), Teaching diverse populations: Formulating a knowledge base (pp. 77–92). Albany, NY: State University of New York.

Teacher and Student Relationships

Good relationships between teachers and students help improve academic success. Students try harder, knowing someone cares about the outcomes. Students feel more comfortable seeking help when the relationship is positive and supportive. Teachers who have high expectations for their students and positive attitudes about mathematics positively influence student outcomes.

Students who report having a more supportive relationship with their mathematics teachers were willing to exert more energy learning the lesson and helping their peers. The relationships, either positive or negative, had long-lasting effects on students.

Developing a Growth Mindset

The beliefs people have about intelligence play a big role in mathematics. Some believe intelligence remains the same, this is a fixed mindset. Others believe in a growth mindset, where intelligence changes throughout your lifetime. People with a fixed mindset believe they are good at certain things and bad at others. With a growth mindset, a person could work hard enough and become good at whatever they want.

A person with a fixed mindset and who is good at mathematics will be able to be successful most of the time, but when they come to an obstacle, they tend to give up quicker than those with a growth mindset.

Students with a growth mindset see math as something to work at. When it gets difficult, which it will, they persevere. They believe that the brain is like a muscle, the harder one works, the stronger it gets.

In this model, students are first taught about the brain and how growth mindset works. Then they apply this mindset to learning mathematics or other topics.

Explicitly teaching students how growth mindset works is a foundational skill for success. Growth mindset instruction should be an integral part of both core programs and intervention programs.

References

• Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons.
• Dweck, C. (2008). Mindsets and Math/Science Achievement. Prepared for the Carnegie Corporation of New York-Institute for Advanced Study Commission on Mathematics and Science Education.
• Motivating Students to Grow their Minds. Copyright © 2008–2012. Retrieved from Mindset Works.

Academic Language

Academic language (also referred to as academic English, disciplinary language, scientific language, critical language, and language of school) helps define school success for all students. It is the language of textbooks and homework, the language found in assessments, and the language students hear and see in all classrooms. This language is different in register (the words, phrases, and expressions used to talk about content-specific concepts), structure, and vocabulary from everyday language. Academic language is at the heart of grade-level curriculum across content areas (Gottlieb & Ernst-Slavit, 2014). Academic language includes: vocabulary, representing information, and student discourse.

It is important for educators to be aware of the challenges students face in mathematics with regard to academic language. Language development is not limited to vocabulary instruction, but also includes “instruction around the demands of argumentation, explanation, analyzing purpose and structure of text, and other disciplinary discourse” (Zwiers et. al., 2017). To support the development of academic language in mathematics, learning environments should include speaking, writing, diagramming, and gesturing. Access to learning, that promotes conceptual development, is necessary for all students (Walqui, 2009).

Mathematical vocabulary is more likely than ever to have an impact on students’ math success because students need to understand math-specific words, words with multiple-meaning, and mathematical symbols to develop proficiency in math vocabulary (Pierce & Fontaine, 2009). Explicit teaching (direct instruction) should address words that have multiple meanings, concepts that can be represented with multiple terms, awareness of symbols and diagrams as they relate to mathematics, and the connection between mathematics vocabulary and everyday vocabulary (Roberts & Truxaw, 2013). To learn the math vocabulary needed for success, educators should engage students in rich and lively activities. These activities should encourage deep processing of word meanings and provide a range of opportunities to encounter math vocabulary (Pierce & Fontaine, 2009).

Teaching students to interpret and represent information in mathematics is complicated, as it requires more than reading and writing text. Students must learn to interpret and demonstrate their mathematical thinking through written explanations, symbols, and graphic representations. Educators must teach students the skills needed for success. Teaching sentence structures in mathematics is important to comprehension since often every word within mathematical texts or word problems is essential (Adoniou, 2014). Students might know the meaning of certain academic math words. However, if they cannot put them in a comprehensible sentence, knowledge of academic words alone will not help them be successful
(OSPI academic language toolkit).

When students engage in mathematics and are taught to provide meaningful explanations, higher level thinking and reasoning is promoted. Meaningful mathematical discussions help build knowledge and support the mathematical learning of all students in a math-talk community (Wagganer, 2015). The National Council of Teachers of Mathematics (NCTM, 2014) Principles to Actions includes communication as a process strand that highlights the importance of language skills in mathematics classrooms. Students need multiple opportunities to use academic language by engaging in meaningful discourse. The Common Core’s Standards for Mathematical Practice (SMP) state that students should engage in discussion that constructs viable arguments–SMP 3, critiques each other’s reasoning–SMP 3, and communicates with precision–SMP 6 (CCSSI 2010, p. 6-7). Academic discourse helps to develop conceptual understanding and improve language use (Hill & Miller, 2013). Conversations for students developing mathematical language may serve as scaffolding because opportunities to make and communicate meaning are provided (Zwiers et. al., 2017). Students benefit from collaborative discussions because mathematics conversation provides:

• Meaningful discussion.
• Oral language practice.
• A way for students to clarify what is being asked and what is happening in a problem.
• Time to process information and hear the thinking of others.
• Opportunities for teachers to model academic language, appropriate vocabulary use, thinking processes.
• Build common understandings and shared experiences (Echevarria, Vogt, & Short, 2009; Zwiers et. al., 2017).

Support, Structure and Scaffolding

Educators should provide structure and support for students by intentionally teaching how to participate in these types of math conversations. Students benefit from learning how to question, reason, make connections, solve problems, and communicate solutions effectively (Echevarria, Vogt, & Short, 2009).

Providing a variety of scaffolds that foster students’ participation supports students both in organizing their thinking and making sense of the mathematics. Examples include:

• Sentence frames, which provide tools to support mathematical conversations.
• Teacher modeling and think-alouds.
• Word walls and posters displaying commonly used terms, operations, and math processes.
• Graphic organizers, which provide visual representations of mathematical information.
• Artifacts and Manipulatives upon which to build shared meaning and support sensemaking.
• Structured peer interactions, to communicate ideas and clarify understanding (Echevarria, Vogt, & Short, 2009; Zwiers et. al., 2017).

Academic language is critical to student outcomes in both mathematics and English language arts.

References

• Adoniou, M., & Qing, Y. (2014). Language, Mathematics and English language learners. Australian Mathematics Teacher, 70(3), pp. 3-13.
• Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Council of Chief State School Officers. (2012).
• Framework for English Language Proficiency Development Standards corresponding to the Common Core State Standards and the Next Generation Science Standards. Washington, DC: CCSSO.
• Echevarria, J., Vogt, M. E., & Short, D. (2009). The SIOP Model for Teaching Mathematics to English Learners. Boston: Pearson Allyn & Bacon.
• Ernst-Slavit, G., & Slavit, D. (2015). Mathematically Speaking. Language Magazine.
• Hill, J., & Miller, K. (2013). Classroom Instruction That Works with English Language Learners, 2nd Edition. Denver, Colorado: Association for Supervision and Curriculum Development.
• Pierce, M. E., & Fontaine, M. (2009). Designing Vocabulary Instruction in Mathematics. The Reading Teacher, 63(3), pp. 239-243.
• Roberts, N. S., & Truxaw, M. P. (2013). For ELLs: Vocabulary beyond the definitions. Mathematics Teacher, 107(1), pp. 28-34.
• Slavit, D., & Ernst-Slavit, G. (2007). Teaching Mathematics and English to English Language Learners Simultaneously. Middle School Journal, 39(2), pp. 4-11.
  Background and Philosophy
• Wagganer, E. L. (2015). Creating Math Talk Communities. Teaching Children Mathematics, 22(4), pp. 248-254.
• Walqui, A. (2009). Improving Student Achievement in Mathematics by Addressing the Needs of English Language Learners. NCSM: Leadership in Mathematics Education, No. 6.
• Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the Design of Mathematics Curricula: Promoting Language and Content Development.

Cross-Curricular Teaching Practices

Students use many of the same skills and strategies in mathematics as they do in English language arts. Making explicit connections between strategies across content areas strengthens students’ cognitive processes. To make these connections, educators should point out when a vocabulary word, skill, or strategy has a dual purpose across content areas and model these connections during instruction. One way to model cross-curricular connections is to be intentional when selecting read-alouds. For example, strategies used to make sense of complex language in a mathematical word problem are similar to the strategies used when reading informational text. Activating background knowledge supports student reading comprehension and mathematical reasoning. Students may activate background knowledge about a topic within a mathematical task the same way they would activate background knowledge while reading text. Learning explicit and systematic strategies for receiving and providing feedback benefits students across content areas. For example, providing feedback to justify a strategy used for solving a mathematical problem is similar to providing peer feedback for revisions during writing.

Mathematical Representations and Manipulatives  (CRA)

Mathematics instruction, at all grade levels, should begin with developing a conceptual understanding of mathematical ideas. This can be accomplished through the use of concrete and representational models before moving to abstract representations. When planning instruction, teachers should consider how to sequence the learning to support moving from concrete representations to the symbolic and abstract. Visual representations of the mathematics are critical in laying a strong foundation of mathematical ideas. Students need experience using concrete manipulatives and then moving to representational models to solidify the use of imagery in problem solving before moving to abstract symbols. The connections students make throughout these stages are essential and should be an intentional
design of any lesson.

The first stage is the concrete stage in which students experience math by physically manipulating various objects. The second stage engages students in using representational models to solve math. During this stage, students represent concrete objects as pictures or drawings. Using abstract symbols is the third stage.
Instructional practices should support students moving from the concrete and representational stages to using numbers and symbols to model and solve math problems. Students need opportunities to develop mathematical thinking at each stage and to make connections between the stages to develop the ability to move flexibly among the different representations.

Traditional mathematics instruction has historically focused on computation and students’ ability to apply procedures quickly and accurately. According to National Council of Teachers of Mathematics, procedural fluency includes the ability to apply, build, modify, and select procedures based upon the problem being solved. This definition of procedural fluency pushes the bounds of traditional mathematics instruction, as it requires foundational knowledge of concepts, reasoning strategies, properties of numbers and operations, and problem-solving methods.

*CRA will be addressed further in chapter 20

Mathematically Productive Instructional Routines

Mathematically Productive Instructional Routines (MPIRs) are high leverage instructional routines that focus on student ideas as central to the learning, make student thinking visible, and provide opportunities for mathematical discourse thereby allowing opportunities for students to make sense of mathematics in their own way. Consistently engaging students in these routines can change student’s dispositions about mathematics, support shifts in instructional practice, and deepen mathematical content knowledge and a growth mindset for both students and teachers.

MPIRs can be implemented with students from pre-school to college and are not tied to any curriculum. MPIRs take place at the beginning or end of a math lesson and are designed to take 10–15 minutes. It is beneficial to use routines daily, or multiple times a week. Mathematical routines can cover many different mathematical ideas, and can be used across a variety of concepts and topics.

While there are several different formats for these routines, all Mathematically Productive Instructional Routines share these common attributes:

They are routine. MPIRs are brief and used frequently. Students and teachers engage in these activities often enough that the routine itself is learned and can be engaged in quickly and meaningfully. The predictable structure creates a safe time and space for students to take risks and explore and share their ideas.

They are instructional. While classrooms also rely on routines designed to manage student behavior, transitions, and supplies, MPIRs are routines that focus on student learning. MPIRs provide an opportunity for students to share their mathematical ideas and make connections and deepen their understanding of math concepts as they listen and respond to other students. Routines also provide an opportunity for the teacher to formatively assess students.

They are mathematically productive. Prompts for each MPIR are carefully chosen to opportunities for students to enact the Standards for Mathematical Practice. Student discussions highlight central mathematical ideas. Students gain important insights and develop positive dispositions about engaging in mathematics through their participation in MPIRs.

Mathematically Productive Instructional Routines create a structure where teachers listen to, build on, and respond to student thinking. Using such routines frequently can support the development of a classroom culture in which sense-making is at the heart of all learning, and mistakes are expected, respected, and inspected.

Number Talks

Number Talks are an example of a mathematically productive instructional routine that can support the development of a classroom culture in which students feel encouraged to share their thinking, and teachers become skilled at listening to their students’ thinking. This short mental mathematics routine can be used daily with any curricular materials to promote number fluency as well as develop conceptual understanding of numbers and operations.

In a number talk, students have the opportunity to share their thinking and learn from fellow students about multiple ways of using number relationships and structures, and visual models to perform mental computations. With number talks, teachers must listen to and represent student thinking, which not only provides them with information for determining next steps, but also deepens the teacher’s own understanding of mathematics. Number talks are the best pedagogical method for developing number sense and helping students see the flexible and conceptual nature of mathematics (Boaler, 2015).

In their recent book, Making Number Talks Matter, Number Talks pioneers and researchers Cathy Humphreys and Ruth Parker claim:

Number Talks help students become confident mathematical thinkers more effectively than any single instructional practice we have ever used.… With Number Talks, students start to believe in themselves mathematically. They become more willing to persevere when solving complex problems. They become more confident when they realize that they have ideas worth listening to. And when students feel this way, the culture of a class can be transformed.

Jo Boaler, Stanford University mathematics education professor, provides educators and parents with a 15-minute video about Number Talks that gives a full description of the practice and shares examples to help schools get started with Number Talks in every classroom.

References

• Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons.
• Hiebert, J., & Morris, A. K. (2012). Teaching, rather than teachers, as a path toward improving classroom instruction. Journal of Teacher Education, 63(2), 92–102.
• Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In Instructional explanations in the disciplines (pp. 129–141). Springer US.
• Humphreys, C., & Parker, R. (2015). Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4–10. Stenhouse Publishers

Games

Mathematics games may be used for extended learning time to support instruction and to help students meet the state standards. Some research has found that game-based learning is an effective way to enhance motivation and performance.

Choosing which game to play depends on the instructional goal and learning target. Games can be used both for instruction and practice. Games may also give students the opportunity to apply new learning. Games may not be appropriate in all situations, and are more effective if they are embedded in instruction and include debriefing and feedback. Also, games should be used as adjuncts and aids, not as stand-alone instruction.

Technology

When used strategically, technology can provide students with greater access to conceptual understanding and procedural fluency. Technology can provide students with additional  representations of mathematical ideas, allow inquiry-based exploration, reinforce procedural learning and fluency, and provide efficient screening and diagnostic assessment data. Teachers must monitor student progress and adjust instruction based on formative assessment in all formats. Technology is a tool, not an intervention in and of itself. Technology alone cannot replace effective teaching or intervention activities. It must be a balanced supplement, especially with students who struggle with self-regulation and efficacy. Online mathematics programs, whether purchased or Open Educational Resources, should be aligned with K–12 Learning Standards for Mathematics, adequately scaffold learning, and provide a variety of rich and rigorous tasks.

Chapter Reference

Reykdal, C. (2018). Strengthening Student Educational Outcomes Mathematics Menu of Best Practices and Strategies, State Superintendent, Office of  Superintendent of Public Instruction, Olympia Washington, pages 28-38. Except where otherwise noted, this work by the Office of Superintendent of Public Instruction is licensed under a Creative Commons Attribution License

Graphic, Pixabay License  https://pixabay.com/photos/learn-mathematics-child-girl-2405206/