Kimberly Cardimona, PhDBloomsburg University

Pennsylvania State University

QuestionWhat type of instructional strategies can we use

to promote……Active engagement?Independent problem solving?Academic vocabulary?

http://www.youtube.com/watch?v=eU2v8hUypTQ

http://www.youtube.com/watch?v=iVUAGhJih8I

A Mathematical ProblemCommon misconception

Polled teachers, administrators, students:Believe math is universalRequires minimal reading/literacy skillsCan be accomplished with minimal English

language proficiency skills

REALITYMathematics requires reading and literacy

skills: directions, word problems, operationsMathematics requires knowledge of arithmetic

vocabularyMathematics is culturally loaded/contain many

cultural references (measurement, weight, time)

Problem solving processes in math vary from country to country: not universal!

Background for the studyObservations gleaned from personal experience as co-teacher for ELLs in a mainstream mathematics classroomInstructional strategies

Traditional lock-step method (all learners proceed at the same pace).

Demonstration-mimicry-practiceOne size fits all strategy that did not include the

language needs or experiences of ELLs

ResultsNo opportunity to practice arithmetic

vocabulary.Does not foster additional language development

or literacy skills.No opportunity to participate or co-construct the

problem solving process.No opportunity to collaborate with classmates

and create a community of learners.Compromises affective variables. No opportunity to discuss, justify, or explain the

problem solving processes.

Observation of additional content classrooms and instructional delivery led to similar results more presentation than interaction.

Teacher attitudes regarding social interaction in the classroom were similar across content areas: specifically that it results in an ill-managed classroom and promotes cheating

Research Purpose• Deeper investigation revealed:• Increase in number of ELLs • Increase in the drop out rate for ELLs (Fry,

2008; Menken, 2008).• ELL scores of high school exit exams in math

30-40% lower than non-ELLs (Center on Education Policy, 2005).

• Teacher preparation• Paucity of research

Social InteractionWhy social interaction?

Reviewed literature to develop a conceptual frameworkSociocultural Approaches to Learning and

Development

Conceptual FrameworkImportant characteristics of interaction

Questioning and wait time are important tools:to engage students in thinking increase active participationprovide students an opportunity to

demonstrate problem solving processes and vocabulary under the guidance of a peer or more capable other .

Study: Research Questions How do advanced ELL tutors interact with

ELL tutees to develop ownership of mathematical problem solving activity? How do effective tutors interact differently from ineffective tutors?What types of tutor questions support active

participation, independent problem solving, and accurate completion of problem sets as tutees engage in small group activities (ownership)?

In what way does tutors’ wait time contribute to tutees’ use of arithmetic vocabulary?

Study: Research Procedure18 dyads

15 ELL (expert)student/ELL (novice) student 3 teacher/ELL (novice) student

Five math problems PSSA grade 11 mathematics test item sampler Distributed to tutees to indicate difficulty in

problem solutionProblem sets were divided among tutors

Thirty minutes for problem solvingDyads were allowed no more than 30 minutes to complete a sample set of 2-5 problems.

Research: Analytic MethodsOpen line by line coding/selective coding

Data from recordings were transcribed via longhand in conjunction with field notes.

Questions were highlighted.Pauses were highlighted as possible wait time.Arithmetic vocabulary highlighted.Compared and contrasted data within and

across dyads to identify patterns and themes that emerged by focusing on tutee response , questions , and use of wait time.

Procedural questions:Provided the least amount of instructional support.Provided the least amount of opportunity for the

tutee to actively participate or demonstrate independent problem solving .

Consisted primarily of yes/no answers .Used by tutors as comprehension checks, prompts

for acknowledgement, or agreement, and to direct attention to the problem solving process.

Often posed after the introduction of an activity, explanation of procedures, or elicit tutee confidence in reproducing the activity.

Often associated with moving the activity forward

Guiding questions:Provide a high level of instructional support.Provided opportunity for tutee to actively

participate in co-constructing the problem solving process.

Consist of known answer/display questions.Used by tutors during the problem solving task to

effectively model a problem solving process, assess background knowledge, elicit a response, or focus attention on an activity.

Often encouraged remediation or tutor demonstration of proper techniques or steps to collaboratively arrive at a correct solution to the activity.

Reflection questions:Provide a mid-range level of instructional support as

guidance if the tutee requested assistance or arrived at an incorrect solution.

High level of opportunity for tutee to demonstrate independent problem solving .

Questions are open, implicit, or vague.Often used at the close of a task to determine

internalization or comprehension of the problem solving activity.

Offered by the tutor as a means for the tutee to physically demonstrate, explain, or justify the problem solving process, reflection questions offered tutors a means to assess what the tutee had learned.

Pause betweenQuestions/response sequencesResponse/reflection sequences

totaled three seconds or more.Instances of wait time promoting arithmetic vocabulary

Tina In each of the three dyadic interactions between

Tina and a tutee, there was at least one episode of interaction in which wait time increased the use of arithmetic vocabulary.

Jean In two of the dyadic interactions between Jean and

the tutee, wait time contributed at least once in the tutee’s use of arithmetic vocabulary.

Jean: most effectiveInteraction StyleIntroduces the lesson and gives the tutee a visual and verbal preview of what they will do in order to solve the problemFractures the problem into manageable fragments.Uses a variety of question types (procedural and primarily guiding) to engage tutee in active participation of co-constructing the problem solving process as they solve each piece of the fractured problem to produce a unified problem set.

Uses guiding questions and wait time to prompt tutees use of arithmetic vocabulary .

Uses reflection questions to offer the tutee an opportunity to physically demonstrate independent problem solving of a similar problem and often accompanies this with a request to demonstrate or justify the process the tutee had completed.

Tutee usually demonstrated ownership of the mathematic problem solving activity in as little as 5 minutes.

Cindy: least effective Interaction Style

Seldom offers a demonstration, explanation, or preview of the problem solving process necessary to complete the mathematic activity.

Occasionally uses guiding questions to encourage active participation, however it is often unclear what the tutee is solving for.

Often fractures the problem similar to Jean, however, waters down the content and vocabulary with simplified versions that the tutees would not encounter in a grade appropriate mathematics assessment.

Does not use reflection questions, therefore it is difficult to determine if tutee ownership of the mathematics activity did occur.

Key Benefits of the StudyEvidence that interaction can be used in the

secondary mathematics classroom (and across the curriculum)to develop ELL ownership of mathematics problem solving activityThe importance of tutor questions and wait time

The combination of the three types of questions, procedural, guiding, and reflection in conjunction with episodes of wait time contributed to developing tutee ownership of the mathematic activity.

Key Benefits ContinuedSocial interaction

This provides evidence of the positive effects of social interaction in mathematics learning which dispels the myth that dyadic or small group work results in bad habits (cheating, copying) and provides no evidence of independent problem solving.

Practical solutions for secondary mathematics teachers of ELLsExample: Visual representation of the problem accompanied by explanation of the vocabulary and/or problem solving processPoses a procedural question as a comprehension check or to direct attention to a particular area of the problem.Fractures the problem.Allows students to work in small groups and then actively engage (through guiding questions) in co-constructing the problem solving process as a whole class.

Pose reflection questions to allow students an opportunity to reflect, debate, discuss one another's answers

Pair students in dyads to work on a set of problems similar to the one previously demonstrated.

Circulate room and offer extra help where neededUpon completion of the tutoring episode, teacher

can revisit problem sets, fracture them, and once again actively engage whole class in reassembling one or two activities.

Assign individual problems sets for independent practice.

ImplicationsProfessional development for teachers and

pre-service educationRemember: it is not only about interaction,

but the type of strategies we use to interact in the classroom:

ConclusionIt is extremely important that all teachers

understand how to implement effective problem solving strategies for ELLs (and all students) through social interaction.

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