The actions on the 5x8 card may appear somewhat peculiar. Why, for example, would school leaders be looking for students to say a second sentence? The answer is that second sentences are leading indicators of the practices promoted by the CCSS-M. They will happen frequently when students are asked to explain their thinking. They will be the norm when students are expected to give answers that make sense to all of their classmates, even those who don’t initially understand. But they will be absent in classrooms where students are asked to fill in the blank in a teacher’s question; to give an answer rather than an explanation. Without expansive professional development, a principal tuned into the differences between classrooms where there are many and few second sentences will learn a great deal about good instructional practice simply through focused observation.

Explaining one's ideas and hearing the reactions of others promotes learning. Thus in classrooms in which a few students do all the talking, learning opportunities are distributed inequitably. Over time silent students may come to believe they are not expected to talk, and may disengage entirely. When all students are given the time to explain their thinking, a greater investment of every student in the instructional activity is demanded and rewarded, and the opportunity for students to serve as learning resources for each other is maximized.

Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40(3), 259-281.

Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In. J. Boaler & J. G. Greeno (Eds.), Multiple Perspectives on Mathematics Teaching and Learning (pp. 171-200). Westport, CT: Ablex Publishing.

Boaler, J., Wiliam, D., & Brown, M. (2000). Students' experiences of ability grouping-disaffection, polarisation and the construction of failure. British Educational Research Journal, 26(5), 631-648.

Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42-47.

Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed‐ability approach. British Educational Research Journal, 34(2), 167-194.

Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. The Teachers College Record, 110(3), 608-645.

Cohen, E. G. (1994). Designing groupwork: Strategies for the heterogeneous classroom. New York: Teachers College Press.

Moschkovich, J. N. (2012). How equity concerns lead to attention to mathematical discourse. Equity in Discourse for Mathematics Education (pp. 89-105). Netherlands: Springer.

Schoenfeld, A. H. (1983). Beyond the purely cognitive: Belief systems, social cognitions, and metacognitions as driving forces in intellectual performance. Cognitive Science, 7(4), 329-363.

Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.

Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355.

Schoenfeld, A. H. (1989). Ideas in the air: Speculations on small group learning, environmental and cultural influences on cognition, and epistemology. International Journal of Educational Research, 13(1), 71-88.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). New York, NY: Macmillan.

Schoenfeld, A. H. (2002). A highly interactive discourse structure. Advances in Research on Teaching, 9, 131-170.

Schoenfeld, A. H. (2006). Mathematics teaching and learning. In P. A. Alexander & P. H. Winne (Eds.), Handbook of Educational Psychology (2nd edition) (pp. 479-510). Mahwah, NJ: Erlbaum.

Schoenfeld, A. H., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching mathematics. In D. Tirosh & T. Wood (Eds.), International Handbook of Mathematics Teacher Education, Volume 2: Tools and Processes in Mathematics Teacher Education (pp. 321-354). Rotterdam, Netherlands: Sense Publishers.

Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.

Schoenfeld, A. H. (2012). Problematizing the didactic triangle. ZDM, the International Journal of Mathematics Education, 44(5), 587-599.

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A hallmark of the understanding prioritized by the CCSS-M is the ability to use mathematical reasoning to construct and defend an argument (this is what I did and why it makes sense). Brief, single-sentence student utterances are generally insufficient for a viable argument.

Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. New York: Granada Learning.

Michaels, S., O’Connor, C., & Resnick, L. B. (2008). Deliberative discourse idealized and realized: Accountable talk in the classroom and in civic life. Studies in Philosophy and Education, 27(4), 283-297.

Osborne, J. (2010). Arguing to learn in science: The role of collaborative, critical discourse. Science, 328(5977), 463-466.

Resnick, L. B., Michaels, S., & O’Connor, C. (2010). How (well-structured) talk builds the mind. In R. J. Sternberg & D. D. Preiss (Eds.) Innovations in educational psychology: Perspectives on learning, teaching and human development (pp. 163-194). New York: Springer Publishing.

Swan, M. (2006). Learning GCSE mathematics through discussion: What are the effects on students? Journal of Further and Higher education, 30(3), 229-241.

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Understanding each other’s reasoning develops reasoning proficiency: Students learn about mathematics by exploring their own and others' reasoning in problem-solving situations. Actively listening to peers increases the time focused on mathematical thinking and promotes the cognitive flexibility that is highly valued in college and career.

Chapin, S. H., O'Connor, C., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6. Sausalito, CA: Math Solutions.

Chin, C., & Osborne, J. (2008). Students' questions: A potential resource for teaching and learning science. Studies in Science Education, 44(1), 1-39.

Michaels, S., O’Connor, C., & Resnick, L. B. (2008). Deliberative discourse idealized and realized: Accountable talk in the classroom and in civic life. Studies in Philosophy and Education, 27(4), 283-297.

Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24-30.

O'Connor, M. C. (1998). Language socialization in the mathematics classroom: Discourse practices and mathematical thinking. In M. Lampert & M. L. Blunk (Eds), Talking mathematics in school: Studies of teaching and learning in school (pp. 17-55). New York, NY: Cambridge University Press.

Resnick, L. B., Michaels, S., & O’Connor, C. (2010). How (well-structured) talk builds the mind. In R. J. Sternberg & D. D. Preiss (Eds.), Innovations in educational psychology: Perspectives on learning, teaching and human development (pp. 163-194). New York: Springer Publishing.

Zwiers, J., & Crawford, M. (2011). Academic conversations: Classroom talk that fosters critical thinking and content understandings. Portland, ME: Stenhouse Publishers.

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As students become more mathematically proficient and their reasoning improves, they should be able to identify flaws in their own and others' thinking. Revising work as a routine matter leads to better problem solving.

Dweck, C. (2006). Mindset: The new psychology of success. New York, NY: Random House LLC.

Emig, J. (1977). Writing as a mode of learning. College Composition and Communication, 28(2), 122-128.

Graham, S., & Hebert, M. (2010). Writing to read: Evidence for how writing can improve reading: A report from Carnegie Corporation of New York. Carnegie Corporation of New York.

Hillocks, G. (1984). What works in teaching composition: A meta-analysis of experimental treatment studies. American Journal of Education, 93(1), 133-170.

Hillocks Jr, G. (1986). Research on written composition: New directions for teaching. Urbana, IL: ERIC Clearinghouse on Reading and Communication Skills.

Klein, P. D., & Kirkpatrick, L. C. (2010). A framework for content area writing: Mediators and moderators. Journal of Writing Research, 2(1), 1-46.

Langer, J. A., & Applebee, A. N. (1987). How writing shapes thinking: A study of teaching and learning. NCTE Research Report No. 22. Urbana, IL: National Council of Teachers of English.

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Mathematically proficient students comprehend and produce mathematical representations (symbolic expressions, graphs, tables, number lines, etc.) that are embedded in ordinary and academic explanations and justifications. Students comprehend and produce the paragraphs, sentences, phrases and words characteristic of justifications, explanations and word problems typical for their grade level.

Chapin, S. H., O'Connor, C., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6. Sausalito, CA: Math Solutions.

Fillmore, L. W., & Snow, C. E. (2000). What teachers need to know about language. Washington, DC: Center for Applied Linguistics.

Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, (27)1, 24-30.

Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23(2), 139-159.

Snow, C. E., & Uccelli, P. (2009). The challenge of academic language. In D. R. Olson & N. Torrance (Eds.), The Cambridge handbook of literacy (pp 112-133). New York, NY: Cambridge University Press.

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English learners may hesitate to speak in class precisely because their control of English is limited. But practice speaking allows them to become more proficient. Bridging the language barrier is important for ELLs to thrive in the types of classrooms the CCSS-M promotes.

Barwell, R. (2005). Ambiguity in the mathematics classroom. Language and Education, 19(2), 117-125.

Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11-19.

Moschkovich, J.N. (2000) Learning mathematics in two languages: Moving from obstacles to resources. In W. Secada (Ed.), Changing faces of mathematics (Vol. 1): Perspectives on multiculturalism and gender equity. Reston, VA: NCTM.

Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical thinking and learning, 4(2-3), 189-212.

Moschkovich, J. N. (2007). Beyond words to mathematical content: Assessing English learners in the mathematics classroom. In A. Schoenfeld (Ed.), Assessing Mathematical Proficiency, (53), 345-352. New York, NY: Cambridge University Press.

Moschkovich, J. N. (2007) Bilingual Mathematics Learners: How views of language, bilingual learners, and mathematical communication impact instruction. In N. Nassir and P. Cobb (Eds.), Diversity, equity, and access to mathematical ideas. New York, NY: Teachers College Press.

Moschkovich, J. (2007). Using two languages when learning mathematics. Educational Studies in Mathematics, 64(2), 121-144.

Moschkovich, J. (2012). Mathematics, the Common Core, and language: Recommendations for mathematics instruction for ELs aligned with the Common Core. Understanding language: Commissioned papers on language and literacy issues in the Common Core State Standards and Next Generation Science Standards, 17-31.

Savignon, S. J. (1991). Communicative language teaching: State of the art. TESOL Quarterly, 25(2), 261-278.

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When students persist in making sense of a challenging problem and trying different strategies for solution, they are more likely to learn the mathematics than students who give up quickly or avoid challenge to the greatest extent possible.

Boaler, J. (2013). Ability and mathematics: The mindset revolution that is reshaping education. FORUM (55)1, 143-152.

Dweck, C. (2006). Mindset: The new psychology of success. New York, NY: Random House

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5x8 Card Team

Contributors

Harold Asturias

Patrick Callahan

Phil Daro, Lead Developer

Mickey Davis

Suzanne Donovan

David Dudley

Matt Ellinger

Karen Tran

Oakland Unified School District

San Francisco Unified School District

The 5x8 Card Team

Strategic Education Research Partnership

1100 Connecticut Ave NW, Suite 650 • Washington, DC 20036

(202) 223-8555