Cutting to the Common Core: Mathematically Speaking

Developing mathematics instruction for English Language Learners (ELLs) that is aligned with the Common Core State Standards (CCSS) can be achieved through research-based teaching practices that often run counter to commonsense notions of language and mathematics.

This article outlines recommendations that are motivated by a commitment to improving mathematics learning through language for all students and especially for students who are learning English; they are not intended as recipes or quick fixes, but rather as principles to help guide teachers, curriculum developers, and teacher educators in developing their own approaches to supporting mathematical reasoning and sense-making for students who are learning English.

According to research (please see https://languagemagazine.com/?page_id=123600 for a list of references), teachers who have demonstrated success with students from non-dominant communities share some characteristics: a) a high commitment to students’ academic success, b) high expectations for all students, c) the autonomy to change curriculum and instruction to meet the specific needs of students, and d) a rejection of models of their students as intellectually disadvantaged. Furthermore, researchers recommend that mathematics instruction for ELLs: 1) treat language as a resource, not a deficit; 2) address much more than vocabulary and support ELLs’ participation in mathematical discussions as they learn English; and 3) draw on multiple resources available in classrooms, such as objects, drawings, graphs, and gestures as well as home languages and experiences outside of school. This research shows that ELLs, even as they are learning English, can participate in discussions where they grapple with important mathematical content. Instruction for this population should not emphasize low-level language skills over opportunities to actively communicate about mathematical ideas. One of the goals of mathematics instruction for ELLs should be to support all students, regardless of their proficiency in English, in participating in discussions that focus on important mathematical concepts and reasoning, rather than on pronunciation, vocabulary, or low-level linguistic skills. By learning to recognize how ELLs express their mathematical ideas as they are learning English, teachers can maintain a focus on mathematical reasoning as well as on language development.

I use the phrase “the language of mathematics” not to mean a list of vocabulary or technical words with precise meanings but the communicative competence necessary and sufficient for competent participation in mathematical discourse and mathematical practices. Research describes how learning to communicate mathematically is not merely or primarily a matter of learning vocabulary. During discussions in mathematics classrooms, students are also learning to describe patterns, make generalizations, and use representations to support their claims. The question is not whether ELLs should learn vocabulary but rather how instruction can best support them as they learn both vocabulary and mathematics. In sum, instruction should provide opportunities for students to actively use mathematical language to communicate about and negotiate meaning for mathematical situations.

The recommendations focus on teaching that simultaneously: a) is aligned with the CCSS for mathematics, b) supports students in learning English, and c) supports students in learning important mathematical content. Overall, the recommendations address the following questions: How can instruction provide opportunities for mathematical reasoning and sense making for students who are learning English? What instructional strategies support ELLs’ mathematical reasoning and sense making? How can instruction help ELL students communicate their reasoning effectively in multiple ways?

Alignment with Common Core State Standards
The CCSS provide guidelines on how to teach mathematics for understanding by focusing on students’ mathematical reasoning and sense-making. Here, I summarize three emphases in the CCSS to describe how mathematics instruction for ELLs needs to take these three emphases seriously.

1. Balancing conceptual understanding and procedural fluency
Instruction should balance student activities that address both important conceptual understanding and procedural fluency.

2. Maintaining high cognitive demand
Instruction should: i) use high-cognitive-demand math tasks and ii) maintain the rigor of mathematical tasks throughout lessons and units, for example by asking students to explain and justify their reasoning.

3. Engaging students in mathematical practices
Instruction should provide opportunities for students to engage in eight mathematical practices described in the CCSS: i) Make sense of problems and persevere in solving them, ii) reason abstractly and quantitatively, iii) construct viable arguments and critique the reasoning of others, iv) model with mathematics, v) use appropriate tools strategically, vi) attend to precision, vii) look for and make use of structure, and viii) look for and express regularity in repeated reasoning.

We can see from these areas of emphasis that students should be focusing on making connections, understanding multiple representations of mathematical concepts, communicating their thinking, and justifying their reasoning. Several of the mathematical practices involve language and discourse (in the sense of talking, listening, reading, and writing), in particular practices #iii and #viii. In order to engage students in these mathematical practices, instruction needs to include time and support for mathematical discussions and use a variety of participation structures (including teacher-led, small group, pairs, student presentations) that support students in learning to participate in such discussions.

According to a review of the research on effective mathematics teaching, teaching that makes a difference in student achievement and promotes conceptual understanding in mathematics has two central features: one is that teachers and students attend explicitly to concepts, and the other is that teachers give students the time to wrestle with important mathematics. Mathematics instruction for ELLs should follow these two recommendations for high-quality mathematics instruction.

A word of caution: concepts can often be interpreted to mean definitions. However, paying explicit attention to concepts does not mean that teachers should focus on providing definitions or stating general principles. Instead the CCSS and the National Council of Teachers of Mathematics (NCTM) Standards provide examples of how instruction can focus on important mathematical concepts (e.g. equivalent fractions or the meaning of fraction multiplication, etc.). Similarly, they also provide examples of how students can show their understanding of concepts (conceptual understanding) not by giving a definition or describing a procedure, but by using multiple representations. For example, students can show conceptual understanding by using a picture of a rectangle as an area model to show that two fractions are equivalent or that multiplication by a positive fraction smaller than one makes the result smaller, and pictures can be accompanied by oral or written explanations.

These examples point to several challenges that students face in mathematics classrooms focused on conceptual understanding. Since conceptual understanding is most often made visible by showing a solution, describing reasoning, or explaining “why,” instead of simply providing an answer, the CCSS shifts expectation for students from carrying out procedures to communicating their reasoning. Since the CCSS documents already provide descriptions of how to teach mathematics for understanding and use multiple representations, the recommendations outlined below will focus on how to connect mathematical content to language, in particular through “engaging students in mathematical practices” (Emphasis #3).

Recommendations for Connecting Mathematical Content to Language
1. Focus on students’ mathematical reasoning, not accuracy in using language
Instruction should focus on uncovering, hearing, and supporting students’ mathematical reasoning, not on accuracy in using language (either English or a student’s first language). When the goal is to engage students in mathematical practices, student contributions are likely to first appear in imperfect language. Teachers should not be sidetracked by expressions of mathematical ideas or practices expressed in imperfect language. Instead, teachers should first focus on promoting meaning. Eventually, after students have ample time to engage in mathematical practices both orally and in writing, instruction can then carefully consider how to move students toward more formal ways of communicating mathematical ideas.

As a teacher, it can be difficult to understand the mathematical ideas in students’ talk in the moment. It is only possible to uncover the mathematical ideas in what students say if students have the opportunity to participate in a discussion and if this discussion is focused on mathematics. Understanding and re-phrasing student contributions can be a challenge, perhaps especially when working with students who are learning English. It may not be easy (or even possible) to sort out which aspects of what a student says are due to the student’s conceptual understanding or the student’s English language proficiency. However, if the goal is to support student participation in a mathematical discussions and in mathematical practices, determining the origin of an error is not as important as listening to the students and uncovering the mathematical content in what they are saying.

2. Shift to a focus on mathematical discourse practices, move away from simplified views of language
In keeping with the CCSS focus on mathematical practices (Emphasis #3) and research in mathematics education, the focus of classroom activity should be on student participation in mathematical discourse practices (explaining, conjecturing, justifying, etc.). Instruction should move away from simplified views of language as words, phrases, vocabulary, or a list of definitions. In particular, teaching needs to move away from oversimplified views of language as vocabulary and leave behind an overemphasis on correct vocabulary and formal language, which limits the linguistic resources teachers and students can use in the classroom to learn mathematics with understanding. Instruction needs to move beyond interpretations of the mathematics register as merely a set of words and phrases that are particular to mathematics. The mathematics register includes styles of meaning, modes of argument, and mathematical practices, and has several dimensions such as the concepts involved, how mathematical discourse positions students, and how mathematics texts are organized.

Another simplified view of language is the belief that precision lies primarily in individual word meaning. For example, we could imagine that attending to precision (mathematical practice #6) means using two different words for the set of symbols “x+3” and the set of symbols “x+3 =10.” If we are being precise at the level of individual word meaning, the first is an “expression” while the second is an “equation.” However, attending to precision is not so much about using the perfect word; a more significant aspect of this mathematical practice is making claims about precise situations. We can contrast the claim “Multiplication makes bigger,” which is not precise, with the question and claim “When does multiplication make the result bigger? Multiplication makes the result bigger when you multiply by a positive number greater than 1.” Notice that when contrasting these two claims, precision does not lie in the individual words nor are the words used in the more precise claim fancy math words, rather in the mathematical practice of specifying when the claim is true. In sum, instruction should move away from interpreting precision to mean using the precise word, and instead focus on how precisions works within a claim.

3. Recognize and support students to engage with the complexity of language in math classrooms
Language in mathematics classrooms is complex and involves a) multiple modes (oral, written, receptive, expressive, etc.), b) multiple representations (including objects, pictures, words, symbols, tables, graphs, etc.), c) different types of written texts (textbooks, word problems, student explanations, teacher explanations, etc.), d) different types of talk (exploratory and expository), and e) different audiences (presentations to the teacher, to peers, by the teacher, by peers, etc.). “Language” needs to expand beyond talk to consider the interaction of the three semiotic systems involved in mathematical discourse: natural language, mathematics symbol systems, and visual displays. Instruction should recognize and strategically support ELL students’ opportunity to engage with this linguistic complexity.

Instruction needs to distinguish among multiple modalities (written and oral) as well as between receptive and productive skills. Other important distinctions are between listening and oral comprehension, comprehending and producing oral contributions, and comprehending and producing written text. There are also distinctions among different mathematical domains, genres of mathematical texts (for example word problems and textbooks). Instruction should support movement between and among different types of texts, spoken and written, such as homework, blackboard diagrams, textbooks, interactions between teacher and students, and interactions among students. Instruction should: a) recognize the multimodal and multi-semiotic nature of mathematical communication, b) move from viewing language as autonomous and instead recognize language as a complex meaning-making system, and c) embrace the nature of mathematical activity as multimodal and multi-semiotic.

4. Treat everyday language and experiences as resources, not as obstacles
Everyday language and experiences are not necessarily obstacles to developing academic ways of communicating in mathematics. It is not useful to dichotomize everyday and academic language. Instead, instruction needs to consider how to support students in connecting the two ways of communicating, building on everyday communication, and contrasting the two when necessary. In looking for mathematical practices, we need to consider the spectrum of mathematical activity as a continuum rather than reifying the separation between practices in out-of-school settings and the practices in school. Rather than debating whether an utterance, lesson, or discussion is or is not mathematical discourse, teachers should instead explore what practices, inscriptions, and talk mean to the participants and how they use these to accomplish their goals. Instruction needs to a) shift from monolithic views of mathematical discourse and dichotomized views of discourse practices, and b) consider everyday and scientific discourses as interdependent, dialectical, and related rather than assume they are mutually exclusive.

The ambiguity and multiplicity of meanings in everyday language should be recognized and treated not as a failure to be mathematically precise but as fundamental to making sense of mathematical meanings and to learning mathematics with understanding. Mathematical language may not be as precise as mathematicians or mathematics instructors imagine it to be. Although many of us may be deeply attached to the precision we imagine mathematics provides, ambiguity and vagueness have been reported as common in mathematical conversations and have been documented as resources in teaching and learning mathematics.

5. Uncover the mathematics in what students say and do
Teachers need to learn how to recognize the emerging mathematical reasoning learners construct in, through, and with emerging language. In order to focus on the mathematical meanings learners construct rather than the mistakes they make or the obstacles they face, curriculum materials and professional development will need to support teachers in learning to recognize the emerging mathematical reasoning that learners are constructing in, through, and with emerging language (and as they learn to use multiple representations). Materials and professional development should support teachers so that they are better prepared to deal with the tensions around language and mathematical content, in particular a) how to uncover the mathematics in student contributions, b) when to move from everyday to more mathematical ways of communicating, and c) when and how to approach and develop “mathematical precision.” Mathematical precision seems particularly important to consider because it is one of the mathematical practices in the CCSS that can be interpreted in multiple ways (see Recommendations #2 and #4 for examples).

In sum, materials and professional development should raise teachers’ awareness about language, provide teachers with ways to talk explicitly about language, and model ways to respond to students. Teachers need support in developing the following competencies: using talk to effectively build on students’ everyday language as well as developing their academic literacy in mathematics; providing interaction, scaffolding, and other supports for academic literacy in mathematics; making judgments about defining terms and allowing students to use informal language in mathematics classrooms, and deciding when imprecise or ambiguous language might be pedagogically preferable and when not.

Notes
This piece is based on the paper “Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for English Learners Aligned with the Common Core. Commissioned Papers on Language and Literacy Issues in the Common Core State Standards and Next Generation Science Standards. pp. 17-31. Proceedings of “Understanding Language” Conference. Palo Alto, CA: Stanford University. Available online at http://ell.stanford.edu/papers/practice
For references, please visit this page and download the full paper:
http://ell.stanford.edu/sites/default/files/pdf/academic-papers/02-JMoschkovich%20Math%20FINAL_bound%20with%20appendix.pdf
There is also a short video at http://ell.stanford.edu/papers/practice or at https://www.youtube.com/watch?v=gUfpnIbq4TA

Dr. Judit Moschkovich is Professor of Mathematics Education in the Education Department at the University of California, Santa Cruz. Her research uses socio-cultural approaches to examine mathematical thinking and learning in three areas: algebraic thinking, mathematical discourse practices, and mathematics learners who are bilingual, learning English, and/or Latino/a.

She has conducted classroom research in secondary mathematics classrooms with a large number of Latino/a students, analyzed mathematical discussions, and examined the relationship between language(s) and learning mathematics. In addition to published articles and book chapters, she is the editor of the book “Language and mathematics education: Multiple perspectives and directions for research” (2010). She was the Principal Investigator of the NSF research project “Mathematical discourse in bilingual settings: Learning mathematics in two languages” (1998-2003), and Co-PI for CEMELA (Center for the Mathematics Education of Latinos/as) a Center for Learning and Teaching funded by NSF from 2004 to 2011. She serves on the review board for the Journal of the Learning Sciences and on the International Program Committee of the International Council for Mathematics Instruction (ICMI) Study #21 titled “Mathematics education and language diversity.” Dr. Moschkovich is a founding partner of “Understanding Language” (http://ell.stanford.edu) (an initiative focusing on the role of language in subject-area learning and ways to support English Language Learners to meet the Common Core State Standards) and she was Co-Chair of the initiative’s Mathematics Work Group. For more publications by Dr. Moschkovich, see http://people.ucsc.edu/~jmoschko/publications.html