Instructional Demands of Teaching Mathematics with English-Language Learners

1. Instructional Demands of Teaching Mathematics with English-Language Learners Jenny T Sealy University of Michigan

2. Rationale • There is growing linguistic diversity in the world at large. • There are increasing numbers of students who are English-language learners (ELLs) in mathematics classrooms.

3. Rationale • Our current teaching force is under-prepared to support the learning of English-language learners. • There is not enough research addressing specifically the teaching of mathematics with English-language learners.

4. Overview • The Language of Mathematics • Review of Research • Methods • Highlighted Studies • Instructional Demands & Research Recommendations • Implications for Teacher Education • Discussion

5. English-Language Learners

6. English-language Learners • Bring an additional linguistic resource to the classroom • Are in the process of learning the English language used in the mathematics classroom

7. The Language of Mathematics • Vocabulary • Words borrowed from everyday English and redefined (e.g. line) • Created out of everyday English • Locutions (e.g. least common multiple) • Combinations (e.g. output) • Created by combining Greek and Latin words (e.g. triangle, parabola) (Halliday, 1978; Han & Ginsburg, 2001; Usiskin, 1996)

8. The Language of Mathematics “A register is a set of meaning that is appropriate to a particular function of language, together with the words and structures which express those meanings.”(Halliday, 1978, p. 195) • In the mathematics register words are arranged and used differently than everyday English

9. Mathematics Register • Numbers are treated differently both grammatically and conceptually. (Nesher & Katriel, 1986) • In everyday English they act as predicates (e.g. Three dogs) • In mathematics they are objects (e.g. Three is a prime number)

10. The Language of Mathematics • Sentences are generally more dense and complex than everyday English and contain fewer contextual cues • Words are used to represent symbolic expressions that embody a set of processes (e.g. x3+(x+1)3 = 35 is translated into ‘the sum of the cubes of two consecutive numbers is 35’)

11. The Language of Mathematics • Is multi-semiotic (Lemke, 2003) and “exploits the meaning potential of linguistic, symbolic, and visual systems of representation” (O’Halloran, 2003, p.189) • Symbolic and visual representation are intertwined (Cuoco, 2001) • Multiple representations form one integrating meaning-making system

12. The Language of Mathematics

13. Consider 2/3 A truck uses 2 litres of gas every 3 kms.

14. The Language of Mathematics • Symbols • Special rules for interpretation of meaning based on order, orientation, and position (Laborde, 1990) • x9, 915, 9x, f(9), 10019, (9, 13)…etc • Visual representations • Graphs, diagrams..etc are powerful means to display trends and patterns • Gestures are a powerful means to add contextual cues to meaning (Roth, 2001)

15. Review of Research • How has the teaching and learning of mathematics in classes containing English-language learners been conceptualized? • What key issues do teachers need to negotiate in this instructional space?

16. Review of Research • Methods: • Cross-disciplinary, integrative research review • Studies published 1984-2008 • Findings: • 104 pieces of relevant research - 63 articles, 37 books, 3 statistical reports & 1 dissertation • Studies fell under 3 main perspectives or conceptual stances

17. Representation Perspective • Studies using this perspective focused upon the components of mathematical language • Highlighted studies: • Role of language comprehension in problem solving • Comparison of mathematical vocabulary across languages • Flexible use of multiple representations

18. Language Comprehension • Language acts as a mediator in mathematics problem solving (Ellerton & Clarkston, 1996; Mestre, 1988) • Language in tasks affect their level of cognitive demand (Campbell, Adams & Davis, 2007) • Recommendation - Make language more comprehensible by: • Explicitly teaching math vocabulary • Providing contextual cues (e.g. relia) • Simplify language in problem statements

19. Comparing Languages • Mathematical language is a cognitive tool • The cognitive representation of numbers differ by language (Miura, Okamoto, Kim, Steere, 1993) • It is easier to learn to count in Chinese than in English (Miller & Stigler, 1987; Miura, 1987) • Commonly used mathematical terms have greater clarity in Chinese than English (Han & Ginsburg, 2001) • E.g. ‘Four-side shape’ versus Quadrilateral

20. Multiple Representations • The flexible use of multiple representations, representational competency, is important for student learning • ELLs taught with multiple representations were better able to represent the math in word problems (Brenner et al, 1997) • Representational competence is positively linked to student achievement (Brenner, Herman, Ho, & Zimmer, 1999)

21. Communication Perspective • Studies using this perspective focused upon the use of mathematical language • Highlighted studies: • Model of mathematical language development • Word walking • Selection of mathematical tasks

22. Model of Mathematical Language Development

23. Word Walking • Students make substitutions in word problems that keep the meaning in everyday English but change the mathematical meaning • Such substitutions that affect the mathematical model of the problem are called word walking (Mitchell, 2001) • “through the last half of the trip”  “in the last part of the trip”

24. Task Selection • Tasks set in familiar contexts and designed to engage students in the action described in the problem can facilitate the expression of ELLs reasoning through word and gesture (Dominguez, 2005) • Tasks that require students to talk and produce output are crucial for language acquisition (Cummins & Swain, 1986)

25. Participation Perspective • Studies using this perspective focused on participation in a mathematical language community • Highlighted studies: • Teacher’s role in establishing norms that support ELLs • Role of code-switching in the classroom • Teacher’s role as expert participant

26. Norms that Support Participation • The structuring and use of small group discussion can support ELLs participation in mathematical discussions • It can also prepare students for participation in whole class discussions(Brenner, 1998) “The teacher plays a central role in establishing norms for mathematical aspects of students’ activity” (Yackel & Cobb, 1996, p.475)

27. The Role of Code-Switching • A study in South Africa (Setati & Adler, 2001), with 11 official languages, found code-switching to be a useful classroom strategy • Students freely explored mathematical ideas in their preferred language and then formally express conjectures to the class in English

28. Expert Participant • In the mathematical language community of the classroom the teacher acts as the expert participant who models what it means to communicate mathematically in the wider mathematics community • The teacher supports ELLs by: encouraging language production, providing feedback and modeling the target language (Adler, 1999; Aljaafreh & Lantolf, 1994; Cummins & Swain, 1986; Doughty & Long, 2003; Ellis, Loewen & Erlam, 2006; Moschkovich, 1999)

29. Gesture & Assessment • Gesture is a powerful means of communication and has a significant role in teaching and learning • See Roth’s (2001) review • There are several difficulties in accurately assessing ELLs • See Abedi, Hofstetter & Lord’s (2004) review

30. Instructional Demands • Shift in the work of the mathematics teachers • Teach both mathematics & language (Bay-Williams & Herrera, 2007; Freeman, 2004) • Selection of appropriate tasks and activities

31. Instructional Demands • Making language comprehensible to students • Managing the overlap and interplay between everyday English and mathematical English in the classroom • Guide ELLs learning while being aware that their verbalizations may not reliably display their mathematical thinking

32. Common Recommendations • Use multiple representations • Maximize students’ opportunities to communicate • Select tasks that facilitate student learning of mathematics and language • Allow code-switching where feasible to make use of students’ additional linguistic resources

33. Implications for Teacher Education • Attention to the development of teachers’ representational competence • Course and professional development (PD) offerings that provide training on language learning • A change in both teachers’ and our conception of the work of mathematics teachers to include attention to language learning

34. Discussion • What other instructional demands have you identified from your own work with mathematics teachers who work with English-language learners? • What recommendations can we suggest for managing these demands? • What implications do they have for teacher education?

35. Discussion “A vision of reform aimed at the academic achievement of ELLs requires integrating knowledge of academic disciplines with knowledge of English language and literacy development.” (Lee, 2005, p.492) • What does this mean for us in mathematics education? • Given the unique nature of mathematical language • Given the Communication Standard (NCTM, 2000)

36. Questions? Please note the handouts of research references at the front.

37. Thank you for your participation! Jenny T Sealy University of Michigan SealyJ@umich.edu