Code-Switching Explorations in Teaching Early Number Sense
2. Theoretical Perspectives
2.1. Language and Mathematics Teaching
2.2. Code Switching and the Language of Mathematics
Data Collection and Analysis
4.1. Educators’ Understandings And Views About Language in Mathematics Teaching
4.2. Code Switching Between the Language of Instruction and the Mathematics Register
- they talked about numbers and used expressions such as “how many,” “how many more,” “count,” “counting,” “counting on,” “amount,” and “number combinations” (used only by E1);
- they talked about number words such as “one,” “two” “ten;”
- they compared sizes, amounts, and weight and used terms such as “more than,” “less,” big,” “small,” “more” “heavier,” lighter,” “tallest,” and “shortest;”
- they talked about regularities and used the term “pattern;”
- they talked about “groups;”
- they brought a lesson to closure (i.e., “That’s a pattern!” “This is called counting forwards”); and
- when E1 (the grade 1 teacher) talked about sets and changes in a set, and used terms like “division,” “addition,” “subtraction,” “equal,” “double,” “take away,” “plus,” “number sentence,” “equation,” “equal to,” “two numbers that make,” “combination for what number?”
- Re-voiced students’ ideas (i.e., “when I am counting this way, I am counting forwards”);
- Explained and then checked students’ actions and thinking (i.e., “you are thinking-ten”);
- Explained their own actions (i.e., “Should I cut this one in half?”);
- Used contextual clues (i.e., “Daniel is saying that he saw a square to count four”;
- Provided new math information (i.e., “This is called making ten”); and
- Chose between terms that belonged to the math register (i.e., “When you were making the patterns—the groups with your counters, were some groups easier to make than others?).
4.3. Language Choices
- protoquantitative terms  (i.e., “a little more”);
- everyday language (i.e., How many we will need to borrow from that five to make that nine a ten?”); and
- the educators’ explanation of how mathematics happens (i.e., “a pattern repeats over and over again”).
Conflicts of Interest
- Chapman, A.P. Language Practices in School Mathematics: A Social Semiotic Approach; Edwin Mellen Press: Lewiston, NY, USA, 2003; ISBN 0773469672. [Google Scholar]
- Moschkovich, J.N. Examining mathematical discourse practices. Learn. Math. 2007, 27, 24–30. [Google Scholar]
- Moschkovich, J.N. Language and Mathematics Education: Multiple Perspectives and Directions for Research; Information Age Publishing: Charlotte, NC, USA, 2010; ISBN 1617351598. [Google Scholar]
- Iannacci, L. Beyond the pragmatic and the liminal: Culturally and linguistically diverse students code-switching in early-years classrooms. TESL Can. J. 2008, 25, 103–123. [Google Scholar] [CrossRef]
- Macaro, E. Analysing student teachers’ codeswitching in foreign language classrooms: Theories and decision-making. Mod. Lang. J. 2001, 85, 531–548. [Google Scholar] [CrossRef]
- Turnbull, M.; Dailey-O’Cain, J. First Language Use in Second and Foreign Language Learning: Intersection of Theory, Practice, Curriculum and Policy; Multilingual Matters: Clevedon, UK, 2009; ISBN 9781847691965. [Google Scholar]
- Turnbull, M.; McMillan, B. Teachers’ use of the first language in French immersion: Revisiting a core principle. In First Language Use in Second and Foreign Language Learning: Intersection of Theory, Practice, Curriculum and Policy; Turnbull, M., Dailey-O’Cain, J., Eds.; Multilingual Matters: Clevedon, UK, 2009; pp. 15–34. [Google Scholar]
- National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics; NCTM: Reston, VA, USA, 2000. [Google Scholar]
- Organization for Economic Co-Operation and Development (OECD). OECD Skills Outlook; OECD: Paris, France, 2013. [Google Scholar]
- Barton, D.; Hamilton, M.; Ivanic, R. Situated Literacies; Routledge: London, UK, 2000; ISBN 978-0415206716. [Google Scholar]
- Vygotsky, L.S. Thinking and Speech; Plenum Press: New York, NY, USA, 1987. [Google Scholar]
- Vygotsky, L.S. Mind and Society: The Development of Higher Mental Processes; Harvard University Press: Cambridge, MA, USA, 1978. [Google Scholar]
- Scheleppegrell, M. Language in mathematics teaching and learning. In Language and Mathematics Education: Multiple Perspectives and Directions for Research; Moschkovich, J.N., Ed.; Information Age Publishing: Charlotte, NC, USA, 2010; pp. 75–112. [Google Scholar]
- Moschkovich, J.N. Academic Literacy in Mathematics for English Learners. J. Math. Behav. 2015, 40, 43–62. [Google Scholar] [CrossRef]
- Gee, J.P. An Introduction to Discourse Analysis, 4th ed.; Routledge: New York, NY, USA, 2014; ISBN 978-0415725569. [Google Scholar]
- Barton, B. The Language of Mathematics; Mathematics Education Library; Springer: New York, NY, USA, 2008; ISBN 978-0-387-72859-9. [Google Scholar]
- Mercer, N. The Guided Construction of Knowledge; Multilingual Matters: Cleveland, UK, 1995; ISBN 9781853592621. [Google Scholar]
- Adler, J. The Dilemma of Transparency: Seeing and Seeing through Talk in the Mathematics Classroom. J. Res. Math. Educ. 1999, 30, 47–64. [Google Scholar] [CrossRef]
- Pimm, D. Speaking Mathematically; Routledge & Kegan Paul: London, UK, 1987; ISBN 978-0710211330. [Google Scholar]
- Kotsopoulos, D. Mathematics discourse: “It’s like hearing a foreign language.”. Math. Teach. 2007, 101, 301–305. [Google Scholar]
- Resnick, L.B. Developing mathematical knowledge. Am. Psychol. 1989, 44, 162–169. [Google Scholar] [CrossRef]
- Macaro, E. Codeswitching in the L2 classroom: A communication and learning strategy. In Non-Native Language Teachers: Perceptions, Challenges and Contributions to the Profession; Llurda, E., Ed.; Springer: Boston, MA, USA, 2005; pp. 63–84. [Google Scholar]
- Adler, J. A language of teaching dilemmas: Unlocking the complex multilingual secondary mathematics classroom. Learn. Math. 1998, 18, 24–33. [Google Scholar]
- Setati, M.; Adler, J. Between languages and discourses: Language practices in primary multilingual mathematics classrooms in South Africa. Educ. Stud. Math. 2000, 43, 243–269. [Google Scholar] [CrossRef]
- Setati, M.; Adler, J.; Reed, Y.; Bapoo, A. Incomplete journeys: Code-switching and other language practices in mathematics, science and English language classrooms in South Africa. Lang. Educ. 2002, 16, 128–149. [Google Scholar] [CrossRef]
- Zazkis, R. Using code-switching as a tool for learning mathematical language. Learn. Math. 2000, 20, 38–43. [Google Scholar]
- Patton, M.Q. Qualitative Research & Evaluation Methods, 4th ed.; Sage: Los Angeles, CA, USA, 2014. [Google Scholar]
- Yin, R.K. Case studies. In International Encyclopedia of the Social & Behavioral Sciences, 2nd ed.; Wright, J.D., Ed.; Elsevier: Oxford, UK, 2015; pp. 194–201. [Google Scholar]
- Merriam, S.B. Qualitative Research and Case Study Applications in Education; Jossey-Bass: San Francisco, CA, USA, 1998; ISBN 978-0787910099. [Google Scholar]
- Stake, R.E. The Art of Case Study Research; Sage: Thousand Oaks, CA, USA, 2006. [Google Scholar]
- Fontana, A.; Frey, J. The interview: From structured questions to negotiated text. In Handbook of Qualitative Research, 2nd ed.; Denzin, N.K., Lincoln, Y.S., Eds.; Sage: Los Angeles, CA, USA, 2000; pp. 645–672. [Google Scholar]
- Moore, D. Case Study Code-switching and Learning in the Classroom. Int. J. Biling. Educ. Biling. 2002, 5, 279–293. [Google Scholar] [CrossRef]
- Anderson, A.; Anderson, J.; Shapiro, J. Supporting multiple literacies: Parents’ and children’s mathematical talk within storybook reading. Math. Educ. Res. J. 2005, 16, 5–26. [Google Scholar] [CrossRef]
- Anderson, A.; Kim, J.E.; McLellan, S. Culture and mathematical learning: A case study of South Asian parents and children playing a board game. In Language, Learning, and Culture in Early Childhood. Home, School, and Community Contexts; Anderson, A., Anderson, J., Hare, J., MacTavish, M., Eds.; Routledge: New York, NY, USA, 2016; pp. 142–167. [Google Scholar]
|Stanza 8 (in math class)|
Mike had to count how many people
were in class. Mary had previously
counted clothes pins with the
children’s names; Mary had counted
EP1: Ok, we need to count thirteen
EP2: Let’s count…
Ch3: one, twooo, three, four, fiiiive,
six, seven, eight, nine ten, eleeeeven,
EP4: Thirteen!! Is it the same?
|Stanza 8a (educator’s video recall)|
R: Could you have use the term equal instead of “the same”?
EP1: I wouldn’t use “equal.” I wouldn’t,
EP2: I just would never use it because you have to go with the language they understand
EP3: Later on I am not sure …
EP4: it’s certainly something to think about…
|Stanza 4 (in math class)|
The teacher and the children were working in small groups. Children were creating masks and gluing different number of paper noses, eyes, and teeth depending on how many they got when they rolled the die.
EK1: how many teeth do you have? (pointing at the mask that Sean had created)
EK3: Danny has two teeth
EK4: and you have two teeth
EK5: NOW you have something that is THE SAME!
Joanna6: I have ONE tooth
EK7: just like Emma,
EK8: you have one tooth and she has one tooth
EK9: .. that’s the same!
|Stanza 4a (educator’s video recall)|
R: Could you explain the use of the term “the same” in the clips?
EK5: So I think it is
EK6: one of those words
EK7: that I’m always trying
EK8: to have the visual to pair with so…so…
EK9: because it is really hard to explain
EK10: what “the same” meeeeans …
R11: Is it “the same,” or is it “equal?”
EK17: I will use the word “the same”
EK18: rather than “equal,”
EK21: “same” is…is a word
EK22: that they get, they understand…
EK36: when we are making groups
EK37: that are numerically equivalent
EK38: I will use “the same” rather than “equal..”…
EK44: what I probably should have said was
EK45: “make the same NUMBER in your group.”
|Stanza 4 (in class)|
During morning meeting the educator brought a scale and manipulatives.
EG31: Ok, I have five here and I have four here (pointing at each side of the scale)
EG32: it’s almost balanced…(she dropped one more block on the side that had 4)
EG33: Five and five.
EG36: are they balanced now?
Mary37: pretty much.
EG38: Ok, if it is balanced,
EG39: what do we know about those numbers?
EG40: They are the.Claire41: Equal.
EG42: They are equal, what’s another WORD for equal?
EG44: SAME! Exactly!
EG45: THAT’S THE WORD I am thinking about!
EG46: If it’s equal IT IS the same.
|Stanza 3a (educator’s video recall) |
R: Was that the first time you were showing them the relationship between equations?
R7: What do you think about the language you use?
EG17: like in my mind,
EG18: the word “same”
EG19: it’s an easier term
EG20: because they can transfer that term “same” in so many different ways, you know?
EG21: Not just in math but also in other ways
EG22: I think lots of them DO understand what “same” means,
EG23: and could make that connection
EG24: between “same” and the term “equal” in math.
|R3: Is it important that at some point a teacher starts saying “sets?”|
|EK5: I think at some point|
|EK6: children will have to be able to understand|
|EK7: that sets and groups mean the same thing…you know?|
|EK8: ..I think that’s important..|
|EK9: that children need to learn that..|
|EK10: WHEN children need to learn that?|
|EK11: I don’t know..|
|EK12: But for me right now...|
|EK13: I want them to understand these dots we are looking at|
|EK14: so, “groups” is a familiar word, so I can use it|
|EK15: and I don’t have to do a lot of teaching before I teach about the dots.|
|Stanza 14 (in math class)|
Five children were asked to stand up in a row. Each child held a poster with a number (from 1 to 5) written on it.
EK1: ah…I want to ask you,
EK2: what number comes after one?
EK4: what number comes after four?
EK6: ffffive…that’s right...
EK7: When I say after I mean…
EK8: If I am talking about what number comes after four (touching the head of girl that had number 4)
EK9: it means what number comes next
(touching the head of the girl that had number 5)
EK16: Now, here is a new word,
EK17: what number comes BEFORE three?
(touching the head of the girl that had number 3)
Sam and Lucy18: ttwooo
EK19: Two, that’s right,
EK20: that means what number do I say first;
EK21: I say two and THEENN I say three.
EK22: What number comes before two?
(touching the head of the boy that had number 2)
EK23: Anybody knows?
EK25: yeah, one..
EK26: I say one first and THEENN I say two.
|Stanza 14a (educator’s video recall)|
R: Could you tell me more about this scenario? How does your talking help children to understand the idea of the “number after?”
EK1: Yeah… “before” and “after”
EK2: it’s like...(laughing)
EK3: Because now, I have changed that terminology
EK4: because that didn’t stick
EK5: …they are like “Ah? Ah? We don’t know what she is talking about”
EK6: So I thought,
EK7: “Ok, using what comes first, what comes next..”
|Stanza 1 (in math class)|
The educator gathered the group and
explained that Julie brought a treat (a rice crispy square) that she would like to share with all
EP21: How many people can have a piece
if I just cut it in two?
EP23: Only two people
EP24: Oh… Is there more than two people?
EP26: What do you think I should do now?
EP27: We’ll cut this side on half and…
should we cut this one on half too?
EP29: How many are we going to have now?
EP31: How many friends do I have?
Samuel32: One, two, three, four, five, six,
seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, fifteen! (he touched each child)
EP33: Would these pieces fill fifteen people?
EP35: What should I do?
Ch36: Cut it!
EP37: Cut them in half?
EP39: Is this half?
(Making a knife mark through half)
|Stanza 1a (educator’s video recall)|
R: How does mathematics language unfold through this clip?
EP1: I also said “half,”
EP2: but I didn’t say “quarter,”
EP3: because I didn’t feel…
EP4: because if they got “half,” that’s good
EP5: because some kids would have experience with “half,”
EP6: like “do you want half an apple?”
EP7: With “quarter?” mmm…
EP8: If I put it in, I will…consciously put it in.
R: How do you make those decisions?
EP1: I think I am aware of the language
that I am using…
EP2: and I try not to go above their level;
EP3: I try to be sure I am down at the level
where they are.
|Stanza 1 (in math class)|
The teacher placed a printed worksheet on the white board entitled “Fact family cones.” The worksheet displayed four columns and four rows of empty squares and a picture of an ice cream cone below each column. In the worksheet, each ice cream cone had 3 numbers. At the bottom of the sheet there were 16 equations for addition and subtraction that matched up with the numbers of the four ice cream cones.
EG1: Ok grade one,
EG2: you are going to take a look,
EG3: there are all kinds of number sentences or equations that are down here,
EG4: and you need to figure out
EG5: what four equations or number sentences
EG6: are going to fit and match up with each cone ok?
|Stanza 1b (educator’s video recall)|
R: As you know, I will be investigating how teachers switch between the language of instruction and the language of mathematics. How do you see that switching happening in this video clip?
EG1: That was a conscious choice..
EG2: I was consciously trying
EG3: to make that connection
EG4: between the terms “number sentence,” and the term “equation.”
R5: Do you think that the term equation…
EG6: I feel it’s a higher level…
R7: Could you explain please?
EG8: I don’t know…(pause, 0.3)
EG9: I think…
EG10: they understand the term “sentence,”
EG11: and we’ve used the simple terms like “number sentence,”
EG12: we‘ve used it soooo much…
EG13: now, I think that the term “equation,”
EG14: is just a bigger word, you know?
EG15: Just even to get your tongue around
EG16: to even say “e-qua-tion,”
EG17: that articulation of the word equation…
EG18: for them to say it, and hear it,
EG19: and remember it, and understand it…
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Arias de Sanchez, G.; Gabriel, M.A.; Anderson, A.; Turnbull, M. Code-Switching Explorations in Teaching Early Number Sense. Educ. Sci. 2018, 8, 38. https://doi.org/10.3390/educsci8010038
Arias de Sanchez G, Gabriel MA, Anderson A, Turnbull M. Code-Switching Explorations in Teaching Early Number Sense. Education Sciences. 2018; 8(1):38. https://doi.org/10.3390/educsci8010038Chicago/Turabian Style
Arias de Sanchez, Gabriela, Martha A. Gabriel, Ann Anderson, and Miles Turnbull. 2018. "Code-Switching Explorations in Teaching Early Number Sense" Education Sciences 8, no. 1: 38. https://doi.org/10.3390/educsci8010038