What’s so improper about fractions?

1. What’s so improper about fractions? Stephen Chrisomalis Dept. of Anthropology, Wayne State University chrisomalis@wayne.edu Humanities Center Brownbag Series Tuesday, Jan. 14, 2014 Mathematical prescriptivism at Math Corps

2. A Prescriptive Episode Mrs. Davis: Why does this make sense, Rahim? Rahim: Because you have to break it … cause you have to make the eight into an improper fraction. TAs: Huh? What’s that? TA-1: What? What? What? What the heck is that? Mrs. Davis: Ohhh … Improper fraction? What’s so improper about fractions? Rahim: Well, a fraction greater than one. TA-1: All right. TA-2: Thought they was behaving badly! TA-3: That sounds kind of familiar. (2009/07/26)

3. Math Corps • An enrichment mathematical community at Wayne State University since 1992 • Serves Detroit public and charter school students from 7th grade onwards (~400 participants per year) • Commitment to curricular excellence in mathematics, personal growth, and social change • Long-term participation common (many students return for 7+ years in various roles)

4. Math Corps: a community of practice • ‘Community of practice’ (Wenger 1998; Lave and Wenger 1991): a social group sharing knowledge and skills in a particular domain of activity • Characterized by legitimate peripheral participation: newcomers become members by participating in simple tasks that help them learn its tasks, principles, and vocabulary

5. ‘The Math Corps Way’ • Set of mathematical, social, and linguistic practices transmitted both formally and informally that are explicitly divergent from those outside the community • Mathematical practices: algorithms, procedures for solving problems that contrast with public school curriculum • Social/moral practices: Emphasis on kindness, compassion, integrity, courage, and a central ideal of ‘greatness’ • Linguistic practices: Prescriptive linguistic norms and divergent lexicon

6. Research question • How and why does the Math Corps community of practice use prescriptive language as a socialization practice?

7. The study • Participant observation of Math Corps Site 1 from 2009 – 2011 • 136 semi-structured interviews with 56 individuals including students and staff • Discourse analysis of linguistic events occurring in the program • Focus on prescriptive language as an element of social life in the community

8. Prescriptivism • - Cultural norms, standards, and beliefs governing the (in)appropriateness of particular linguistic forms • ‘Verbal hygiene’ (Cameron 1995) • Contrast with descriptivism: objective analysis of how language is spoken in a community (is vs. ought) • Often viewed as an undesirable exercise of social power, but is inevitable and even sometimes useful • Domain-specific prescriptivism: Aviation English (Tajima 2004)

9. Prescriptions at Math Corps • Procedural (algorithms) • Improper fraction (fraction greater than one) • Reduce (simplify) • Cancel (factor out a one) • Cross (factor out a one) • Lexical (number reading) • ‘and’ (one hundred andone) • ‘point’ (six and three tenths, not six point three) • ‘over’ (five sixths, not five over six)

10. Mathematical prescriptivism • Domain-specific prescriptions in language relating to mathematics and numbers • Not exclusive to Math Corps: • “One solution is to call such a case an “improper” fraction, but this carries an unfortunate connotation, while also begging the question. We should not attempt to evade the issue by saying a certain ‘part of a whole’ is an improper part.” (Pace 1978: 155) • Characterized by a discourse of rationality: a belief (implicit or explicit) that verbal hygiene (improved language) can lead to cognitive hygiene (improved thought)

11. Reading numbers like a mathematician Mr. Jamieson: Percent means hundredths, so … if I have thirty-seven percent, what is that equal to as a fraction? What would that be equal to as a fraction? Yes, Bashira? Bashira: Thirty-seven over a hundred. Mr. J: Thirty-seven over a hundred. How can we say this number like a mathematician? You’re close. Oh, when I had this number, how would you say this, Malik? Malik: One fifth. Mr. J: One fifth. How do you think you’re gonna say … and when we had this? We had, we said seven twentieths. So what do you think this would be? Bashira: Thirty-seven hundredths. Mr. J: Yes, thirty-seven hundredths. (2011/07/07)

12. Repair in reading numerals SC: Okay, so I know that fractions are numbers and I know that that’s in your notes. But then what if somebody is still like, “Okay, that’s fine but not every number is a fraction.” They might be like, what kind of a number is a fraction? Malik: Like a half, like half, like a hundreds, like thirty-one over —no, thirty-one hundredths, or they just, half of a number, or like half of what you eat. Like if I had half of a sandwich. (2011/07/11)

13. Linking mathematical and non-mathematical prescriptivism If you had assigned a group of people to develop a language – we’re gonna build a society, we’re not cave people anymore, now we need a written language – okay, Joe Schmoe and Mary whatever, and you guys were making a committee, come up with a language for us. Spoken and written language. The people who would come back to you and say, here it is, and present you with English, we would’ve given an F as a grade, and that would have been kind. We’d be like, are you guys idiots? Are you complete morons? Every single sentence in the English language, just about, has an atrocity … So in the Math Corps, that’s essentially what we did. We looked at math and recognized, this is a stinkin’ mess, just like English. It’s a mess. It’s being taught in ridiculous ways.

14. Prescriptions and procedures • Taken as linguistic prescriptions alone, mathematical prescriptivism at Math Corps resembles ideas of linguistic relativity (Sapir-Whorf hypothesis • BUT these ideas are linked, in many cases, to specific algorithms and procedures that also form part of the Math Corps Way

15. Violating the Math Corps Way SC: So, now I’d like you to multiply two fractions together in your head. What is nine elevenths times eleven twelfths? Charlene: [0:04] Three fourths. SC: Alright, you got that answer pretty fast. That’s correct. How did you do that? Charlene: [smiling] I cancelled out the elevenths. SC: Ah. Charlene: [laughing] And then I was left with nine over twelve, then I reduced it. SC: So, why, why are you laughing? Charlene: 'Cause that’s not the Math Corps way. (2011/07/26)

16. Doing arithmetic the Math Corps Way Rather than ‘cancelling’ the 11s in the fractions at left, or ‘reducing’ 9/12 to 3/4 by dividing, each step involves finding a form of 1 (11/11, 3/3) and, drawing a large 1 through it, finding the next step in the calculation Charlene’s linguistic violations parallel her arithmetical ones – she is highlighting her resistance to these norms

17. Cross-multiplying vs. Anti-simplifying

18. The prescriptive melodrama • Mrs. Davis: Why does this make sense, Rahim? • Rahim: Because you have to break it … cause you have to make the eight into an improper fraction. • TAs: Huh? What’s that? • TA-1: What? What? What? What the heck is that? • Mrs. Davis: Ohhh… (disgusted) Improper fraction? What’s so improper about fractions? • Rahim: Well, a fraction greater than one. • TA-1: All right. • TA-2: Thought they was behaving badly! • TA-3: That sounds kind of familiar.

19. Prescriptive melodramas: a discursive frame • Informal and playful classroom discussion full of melodramatic expression regarding a stigmatized word or phrase • Involves students, teaching assistants, and instructors in a mutually-understood frame (Goffman 1959, 1974) • Orients participants’ attention to the linguistic norms of the Math Corps Way while avoiding personal criticism of students • 15 prescriptive melodramas observed during study (20 seconds – 2 minutes in length)

20. Prescriptive melodramas: features and structure • Student says a stigmatized word • OR: instructor gently broaches the subject to introduce students to a new prohibition • Teaching assistants react with negative affect • Overdramatic and playful expressions of disgust, disease, or anger from instructor directed at the word • Discussion of the semantics of the prohibited word (often playful or polysemous) • Discourse of rationality: attempt to outline a rationale for the prescription (uncommon) • Word/phrase is discussed as taboo ‘the C word’ • Order restored by the original ‘offender’ or others in the class offering the preferred term

21. A prescriptive melodrama on Twitter @Erika: Lol shut up please... you can't be "bad" and a hoodrat! The hoodrat cancels out everything! @Philip: We don't say "cancel" @Janine: lol yes we can just not in math @Erika: lol I knew someone from MC was going to say something! @Philip: lol no canceling ever! And I am independent! @Janine: no you're not! Lol @Erika: CANCEL CANCEL CANCEL CANCEL CANCEL!!!!!!!!!! lol @Erika: I wonder did @Philip see my cancel tweet lol @Philip: "Canceling" a certain someone's employment with a certain place... @Erika: I remember when I didn't talk to my kids for a whole day because they kept saying cross cancel... I was so mad lol

22. Prescription for social solidarity • Prescriptive melodrama highlights social and linguistic norms without attacking students • The word, not the speaker, is the target • Use of a discursive frame highlights playfulness –although humor directed at people always carries danger (Basso 1979) • Highlights opposition (and superiority) of Math Corps Way to other norms • Reinforces social ties among community members through common linguistic usage

23. Prescription as anti-language • Anti-language (Halliday 1976): a lexically elaborate vocabulary used by a group to highlight social distinctions (prisoners, military, etc.) • Learning an anti-language is part of socializing individuals into a group and establishing their membership • Linguistic oppositions index social oppositions • At Math Corps, contrast is with Detroit and especially Detroit schools

24. The anti-language of Math Corps SC: What is cross-multiplying? Gary: I don't know what the hell that is. Is that appropriate language for this? [laughs] SC: Hell, yeah! Gary: [laughs] What is cross-multiplying? It’s actually a way of getting a common denominator without actually doing all of that. That’s what it is. SC: Alright. Gary: Mm-hmm. SC: So when you said, “I don't know what the hell that is,” why’d you say that? Gary: [laughs] Because here we don’t use that, we don’t use that language here. SC: Alright. Gary: So. SC: Hell? Or [laughs] … Gary: [laughs] SC: Alright. So do you ever cross-multiply? Gary: Of course. I do it all the time. I do it all the time. SC: But not in class? Gary: But not in class. Right. (2011/07/21)

25. African American English at Math Corps Mr. Jamieson: Let’s count, let’s count, everybody. We’re gonna start at ninety-… Peter: Oh, I mean, three million one hunnid. Mr. J: Three million one hunnid. I love it. You must be from the East Side? You from the East Side? That’s what I’m talking about. All: [laughter] Mr. J: Three million one hunnid. Three million one hundred. Very good, Peter. Very good comeback. Very good. Three million one hundred. All right, we’re moving along, moving along. Mr. J: Add, we love addition, cause we wanna add beautiful and good things to this world, we don’t wanna take away so in this case I want you to add three hundred twenty one plus ninety- two, three hundred twenty one plus ninety-two. (2011/07/14)

26. Prescription and AAE • African American students are subject to prescriptive language frequently in classroom settings, particularly directed at AAE • Fordham (1999) argues that this isolates students and builds an oppositional identity towards schooling • At Math Corps, in contrast, prescriptive language is shared by everyone (regardless of race) and learned through daily practice, and becomes a positive source of belonging in the community • AAE is never stigmatized prescriptively

27. (2013/08/08) … but not six hunnit AND fiddydollas

28. Conclusions • Mathematical prescriptivism at Math Corps serves several interrelated functions • As part of a discourse of rationality, to highlight a presumed direct linkage between ‘good’ language and improved mathematical cognition • To focus attention on a set of locally preferred algorithms and notations used in daily activities • To build student identification with a locally preferred mathematical lexicon in opposition to ordinary school math • To promote social solidarity within an enduring community of practice that is not only mathematical, but also social and moral

29. Thanks to: • The staff and students of Math Corps for their ongoing support of my work • Monica Rodriguez for research assistance • Merrill Palmer Skillman Institute • Wayne State University, OVPR Questions? chrisomalis@wayne.edu