English language learners* learning introductory statistics:

Research, recommendations, and resources

Supported in part by Project LEAP-UP (2008-12 US Dept. of Ed. grant T195N070132; PI: J. Tinajero)

LARRY LESSER (joint work with AMY WAGLERMATTHEW WINSORGUADALUPE VALENZUELA)

*An ELL speaks English “with enough limitations that he or she cannot fully participate in mainstream English instruction.”

Motivations

• English is world’s modal 2nd language (Rollnick, 1998)

• ELL issues increasingly present: fraction of US public K-12 students who are ELLs went from 1/20 (in 1990) to 1/9 (now), and projected to be 1/4 in 20 years (Goldenberg, 2008)

• Need to model “ELPS pedagogy” for future teachers I teach

• Equity (e.g., position statement of NCTM, 2008)

• Dearth of research on ELLs learning statistics, compared to math.

This matters because….

statistics ≠ math! (Cobb & Moore, 1997; Moore, 1988)

Math: context obscures structure

Stats: context provides meaning• math-wise, observational & experimental data have same theoretical model, but different interpretation• math-wise, Χ2 test for homogeneity or independence has same table, but different interpretation• 2 sample t-test of means and

F-test of variances have equal theory standing (as likelihood ratio tests), but F is less robust in practice to nonnormality

This work is relevant for teaching non-ELLs, too!

Even students whose first language is English have difficulty navigating the technical language of statistics (Nolan, 2002; Ortiz, Cañizares, Batanero, & Serrano, 2002; Rangecroft, 2002).

ELLs & non-ELLs are affected by the differences between the everyday language use of terms and the academic use of those terms (Garaway 1994)

from TX K-12 English Language Proficiency Standards “intro”

In order for ELLs to be successful, they must acquire both social and academic language proficiency in English.

Social language proficiency in English consists of the English needed for daily social interactions.

Academic language proficiency consists of the English needed to think critically, understand and learn new concepts, process complex academic material, and interact and communicate in English academic settings.

Note: BICS(social) CALP(academic)

Texas’ K-12 ELPS• Adopted 12/25/0 (Texas Administrative Code: Chapter 74, Rule 4, Title 19, Part 2, Subchapter A)

• 4 levels of students’ language proficiency (not grade specific): beginning, intermediate, advanced, advanced high

• Gradually increasing complexity, give opportunities at current level in 4 modes: listening, speaking, reading, writing

• “School districts shall implement… as an integral part of each subject in the required curriculum.” CHALLENGE: most teachers lack substantial training in teaching their content to ELLs (Batt, 2008; Esch et al., 2005)

Some models

Register: a subset of language used for a particular purpose.

• Just because someone knows statistics and is conversationally fluent in a particular language, it does not mean they can communicate about statistics in that particular language.

• ELLs learn by making multiple meanings for words, not just acquiring word lists. “The notion of register…includes phonology, morphology, syntax, and semantics as well as nonlinguistic behavior.” (Moschkovich, 2002)

CHALLENGE: Statistics adds another register to prior mathematical and everyday registers

the academic meaning of a statistics term may…

• not have an everyday counterpart

• be the same as the everyday meaning [least challenging]

• be different from the everyday meaning [most challenging]

(Lavy & Mashiach-Eizenberg, 2009; Kaplan et al., 2009, 2010)

ASA (2005): being statistically educated in an intro course includes:

“recognize that words such as ‘normal’, ‘random’ and ‘correlation’ have specific meanings in statistics that may differ from common usage.”

Additional CHALLENGE: when statistics usage ≠ math usage!• in HS algebra, direct or inverse

or joint variation (e.g., A = πr2, I = k/d2, or V = πr2h, respectively)

are deterministic relationships (with NO statistical variation)! (Lesser & Winsor, 2009)

• “to estimate” means “to approximate” in math, but means “precise calculation of the ‘prediction’ of the value of a parameter” in statistics class. (Rangecroft, 2002)

• range of a function is not the same as range of a dataset

Interdisciplinary research team

scholarly backgrounds include collective expertise in:

•Statistics content

•Statistics education

•Mathematics education

•English as a second language

L. Lesser

• experience living/teaching in EP, where I am still reclaiming 3 yrs of HS Spanish!• interests in equity, culture, and statistics education; LEAP-UP grant

M. Winsor (now Illinois State U.)

• Bilingual (2 years in Argentina)• 1995-99: at a southern CA HS,

assigned to teach ELLs math “in English” (Prop. 227 passed in 1998)

• school had no ELL materials (told him: “can’t you just translate the book?”)

• search for strategies--> Dec2007 Mathematics Teacher paper

Lesser with thesis advisees

R. Gonzalez and G. Valenzuela

A. Wagler

• Statistician (foci include education research)

• Committee member for 2009 thesis that analyzed 1st CLASS

• as faculty on LEAP-UP grant, launched spring 2011 ELL-friendly online stat. course (JSM 2011 roundtable?)

CASE STUDY RESEARCH QUESTIONS:

• How do ELLs encounter language in statistics? (how might this relate to or differ from math?)

• What factors may help or interfere with ELLs distinguishing between the everyday sense of a word and the academic sense?

Respondents S1 & S2(both Latinas)

• SR. math major with 2ndary minor in ed

• Took stats; taking* capstone

• Pre-service HS teacher

• taught in Spanish through 8th gr. (not including algebra)

• Self-rated English proficiency:

2+ & Adv.+

* = when study began

• JR. education major

• Taking* stats

• Pre-service ES teacher

• taught in Spanish through end of HS

• Self-rated English proficiency:

2 & Adv.

S2

S1

Semi-structured interview protocol of

two Latina ELLs(with Spanish & calculator available)

• background (e.g., self-rate language proficiency)

• use each “word card” in a sentence (& classify usage)

• find descriptive statistics for {1,2,3,4,6,6,13}

[an ‘optimal data set’ from Lesser (2011)]

• assess independence

(of various pairs of events)

Rigor of Case Study• triangulation of data source: taped

interviews, field notes, artifacts• researcher doing interview not

respondent’s teacher at the time• reverse/back/double translation of interview protocol• transcription of tapes by former

professional transcriber• member checking interviews of

respondents; no attrition• researchers independently coded and

made cross-case display • all 11 graduate students in a sp08

research course independently coded (and did peer debriefing

on) blinded transcripts

(for more detail, see paper in Nov. 2009 Statistics Ed. Research J.)

Context/CultureM: The first event is ‘dime lands on heads.’ The second event is ‘quarter lands on tails.’

S2: What is tails on the quarter?…

M: cara [face] is heads, cruz [cross] is tails.

[Mexican coins: seal (or sun) and eagle;

other Latin America: shield, crown, face]

=============================

M: “If you had an i-Pod Shuffle that chooses a song at random from those on your i-Pod, what would you think if the same track were chosen twice in a row?”

============================

(Utts 2005, 11.12b) At a ski resort, suppose Y = ‘number of ski accidents’ and X = ‘average wait time for the ski lift.’ If X and Y have a positive relationship, which is the best interpretation?

confusion between and within registers (Lesser & Winsor, 2009)

ENGLISH SPANISH

mode moda

median mediana

mean promedio

average media

on average por término medio

average (ordinary) mediano

medium (i.e., size) medio

the middle one el de en medio

How many values in {1,2,3,4,6,6,13} are at least 6?

[note: a prerequisite to find Pr(≥ 6 successes)] S2: Four.M: Okay, and how did you get that?S2: …the numbers in the set that are

lower than 6.

M: How many values are at most 6?S2: One.M: Okay. How did you get that?S2: The only number that is greater than 6 is 13.

Note: Less than = menos de At least = por lo menos More than = más At most = a lo más

“word card” example: range (rango, which also

means rank)

M: What is the range of that set? [{1, 2, 3, 4, 6, 6, 13}]S1: 7M: Okay, how did you get that?S1: Just the number of the elements.***********************************************S2: Range. [Could I have] the word in

Spanish?M: It is rango.S2: Rango? Okay. Range is the

approximately number in…Well, I don’t know very well, but I think that it is the number that is not most common in some—when you have in your---numbers for something in research that you do and the, like, all the numbers that are –they are the range.

….S2: Okay. For example, the range of the

research was about ten to twenty people.

Why might range be confused with

sample size?

•Probably not everyday Spanish, because the translation of “prices range from $10 to $20” into Spanish and back would return as “the prices are between $10 and $20.”

•Probably not mathematical Spanish, since that word is a cognate (rango), though rango can also mean “rank”.

•Possibly mathematical English, since the range (rango) of a function tends to be an interval of numbers rather than a single number.

•Probably everyday English, since the everyday usage of “range” evokes scanning (or “ranging”) through all elements of the dataset (the “range of possibilities”), which may suggest the sample size. S2 referred to “all the numbers” and then gave the endpoints of an interval. 

“word card” example: parameter (parámetro)

• S2: It’s like a measure.• M: Okay. Can you use it in a

sentence?• S2: The parameter-- oh, I don’t

know.• M: Can you use it in a Spanish

sentence?• S2: …. like know the measure

of something.…in Spanish, it will be like ‘la regla mide treinta centímetros’ [the ruler measures 30 cm]

What might cause parameter to be associated with linear measure?

• academic Spanish: parámetro similar to perímetro [perimeter]

• everyday Spanish:

para = for metro = meter

• everyday English: a situation’s parameters are its boundaries

• Greek suffix “meter” means “measure”

“word card” example: bias (errores de sesgos)

M: I think in Spanish it’s errores de sesgos…

S1: bias?

M: Yeah

S1: Yeah, it’s something about area.

• Sesgos can also mean “slant” in English (i.e., person’s inclinación) or pyramid’s slant height)

• Personal prejudice vs. systematically mismeasured

Everyday vs. academic uses of independence

S2: “they[independent events] are different, so they have no connection….they are separate”

S2: “México is an independent country”

NOTE: confusion with everyday use of “independent” is not limited to ELLs or novices; e.g., Mishra & Koehler (2006) refer to a view of “teacher knowledge as two circles independent of each other”:

Pitfalls of “deficit model”

• ELLs’ resources include their first language, gestures, objects, everyday experiences, code switching and mathematical representations (e.g., “unimodal, right-skewed distribution”)

• ELLs used to having to go back and forth between English and Spanish may be more primed than monolinguals to navigate between academic registers; bilinguals may have advantage of selective attention – enhanced ability to choose which pieces of information or aspects of context to emphasize (Moschkovich, 2005)

• Bilinguals (including college-age students) do better in math when taught bilingually than monolingual English-speaking Hispanics or ELLs do when taught monolingually. (Valverde, 1984)

• among Alzheimer’s patients, bilingual patients had symptoms that appeared 4-5 years later than monolinguals (reported by Assoc. Press 2/19/11)

ELL resources in statistics (Lesser & Winsor, 2009)

• Bilingual math terms handbooks (e.g., Velázquez or COMAP) or math glossaries (COMAP)

• multilingual (29 languages!) handbook of statistics terms at http://isi.cbs.nl/glossary/

• Spanish counterparts of collections of applets (http://nlvm.usu.edu/es/nav/ & www.eduteka.org/MI/master/interactivate)

• Statistics analogies (Martin, 2003)

• sentence frames, word squares, word walls, etc.

• materials by TODOS, MELL, etc.

Word square

Sentence frame

“The p-value obtained was ___, which is [less / greater] than our preset significance level of ___, and therefore we [reject / fail to reject] the null hypothesis that________.”

SOME RECOMMENDATIONS:

• Identify key words highly similar in sound/appearance (e.g., mean, median, mode, etc.)

• Identify key words/symbols whose meaning may differ in everyday or other registers (e.g., random, confidence, population, bias, independent, normal, significant; p)

• Phrase > knowing each “key” word (e.g., “in the long run”; “expected value”)

• Identify content resources in other language (e.g., applets, glossary)

• Assess recognition of multiple terms? (e.g,. reg. line; z-score)

SOME RECOMMENDATIONS:

• Streamlined language

• Use everyday analogy/context (e.g., median of road)

• Be aware of cultural pitfalls (e.g., courtroom analogy for testing Ho)

• Scaffold vocabulary learning (e.g., sentence frames, word squares)

• Group work, active learning, wait time• Use multiple representations

CURRICULUM:ELL research informed my work on FAPP 8e (& 9e):

Real, everyday connections:

“just as a median divides a road into two halves (with opposite directions of travel),

a median divides a dataset into two halves” (p. 159)

Everyday language & syntax: calculating the first quartile

“Arrange the observations in increasing order and locate the median M in the ordered list of observations. The first quartile is the median of the observations whose position in the ordered list is to the left of the location of the overall median.”

versus

“Use the median to split the data set into two halves – an upper half and a lower half. The first quartile is the median of the lower half.”

Ongoing Timeline

• 2006-07: lit. review, IRB, pilot by grad student on his HS students

• 2007-08: IRB protocol revised; case study interviews conducted, transcribed, analyzed

• 2008-09: analysis completed, Lesser & Winsor paper accepted for Nov. ‘09 Stat. Ed. Research J.

• 2009-10: followup survey research (joined by Wagler & Valenzuela)

• 2010-11: exploratory factor analysis informs revision of CLASS for spring 2011 administration

Quantitative followup: CLASS (Communications, Language

And Statistics Survey)

• Motivation: to see what plays out differently for ELLs and non-ELLs

• Developed and piloted in summer 2009 from findings in SERJ paper; 53 or 72 items; some MC content, mostly Likert level of agreement

• First day fall 2009 term, CLASS given to all 5 Stat 1380 sections (n = 137, 38% identify as Spanish-speaking ELLs)

• Exploratory factor analysis(EFA) and reliability analysis conducted and being further reviewed (e.g., to revise and streamline instrument for future use)

[limitations: small sample, multiple scales]

ELL v. NO ELL Comparisonson 1-7 (VSD-SD-D-N-A-SA-VSA) scale:

(5 of 31 numerical CLASS items were significant)

Questions

ELL Mean

(n)

NO ELL Mean

(n) t Unadjust.

p Sidak

Adjusted pDECODE        

Q49: Confusing with similar words (mean, median, mode)

4.12(51)

3.43(82) 2.98 0.003* .017*

STUDENT PRACTICES         

Q25: To have time to answer

4.43(53)

3.61(82) 3.17 0.002* .009*

Q52:To pretend  to understand

3.89(53)

3.36(83) 2.19 0.030* .144

TEACHING STRATEGIES 

       

Q22: Professor does not wait enough time

4.87(53)

4.32(82) 2.42 0.017* .081

CONTEXT        

Q51: Knowing the context helps me

5.00(53)

5.39(83) -2.32 0.022* .104

Q35: What is the range of the dataset

{1,2,3,4,6,6,13}?

a) 2 b) 3 c) 4 d) 5 e) 6 f) 7 g) 12 h) 13 i) I have no idea

Chi-square 19.175; p = .004

e.g., ELL’s more likely than non-ELLs to say 13

Q41: culture? (chi-sq = 12.6; p = .006)

In statistics, the “null hypothesis” is what we assume is true until there is significant evidence found against it. What would you say is the null hypothesis for a trial in a court of law?a)the defendant is innocentb)the defendant is guiltyc)It could be either of the above, depending on what culture you are fromd)I do not understand the question

a b c D

ELL 20 9 13 10

Non-ELL 46 11 24 2

ELL v. NO ELL Comparisons(answer patterns on 20 nominal items)

Questions Chi-square exact p-valueQ4 (everyday usage of ‘confounded’)  10.517 0.024*Q6 1.474 0.453Q7 0.819 0.838Q8 0.803 0.860Q9 6.790 0.109Q10 3.804 0.248Q5 0.747 0.634Q1 4.652 0.313Q2 7.118 0.073Q34 4.416 0.726Q35 (find range of dataset) 19.175 0.004*Q36 7.751 0.214Q37 3.381 0.727Q38 5.197 0.507Q39 (everyday  usage of words:          normal, random, correlation) 16.897 0.009*

Q40 7.664 0.094Q41 (null hypothesis in a courtroom) 12.591 0.006*Q42 1.097 0.817Q43 0.952 0.579Q44 0.045 0.860

Construct Validation & Internal Consistency of

CLASS instrument

• Internal consistency measured using Cronbach’s alpha &

ordinal coefficient theta

• Construct validity assessed via exploratory factor analysis (EFA), separate EFAs run for each numerical underlying construct measured by Likert scale items:– Decoding Strategies– Student Practices– Teaching Strategies– Context

Lesser@utep.edu

BICS (3-5 years)Basic Interpersonal Communicative Skills•being able to communicate in everyday situations•more “cues”•more visible skills (pronunciation, vocabulary knowledge, grammar)•knowledge, comprehension, and application.

CALP (4-7 years)Cognitive Academic Language Proficiency•being able to communicate in decontextualized academic situations•semantics and functional meaning in an academic, specialized context•analysis, synthesis, and evaluation.•More years to acquire than BICS

THEORETICAL CONSIDERATIONS

Possible parallel: ambiguity of symbols

Testing if the proportion p of success significantly exceeded po = 0.5, we obtained 8 of 10 successes, yielding p < .05, meaning p =

Culturally relevant examples in prob/stat

Setting

• University: Doct. research intensive of 20K located in a southwestern US large city on México border

• university mission: give regional population access to quality higher education

• Roughly 77% of student (and city) population is Hispanic. About 10% is Mexican nationals who commute.

• Call for volunteers (9/07): ELLs whose first (and stronger) language is Spanish now taking undergrad math or statistics classes at this university

Limitations of Study

• Only Spanish-speaking ELLs

• Level of English proficiency was self-rated

• # of participants/interviews

• Individual/Interview setting only

Directions for Future Research

• Counterpart to science ed randomized experiment [n = 49 minority 5th graders]: Brown & Ryoo (2008) found benefit to using everyday language prior to scientific language

• Study how ELLs use certain features from language-sensitive curricula

(e.g., Sullivan, 2010; COMAP, 2009)• Other perspectives: situated-

sociocultural approach (Moschkovich, 2002), 3rd-level ethnostatistics (Gephart, 1988), functional language analysis (Fang & Schleppegrell, 2008)

• Incorporate observations in classroom settings

Deliberate “bad” problemslightly adapted from Lesser (2011)

As reported in the mainstream Maine media, a psychic stood on the Main St. esplanade near the intersection and insulted drivers 70 times over an 8-hour period. This was his normal mode since failing to live within his means. By any means, determine what we mean by the medium’s mean rate of mean comments by the mowed Main median.