achieving instructional equity zenaida aguirre-muñoz, ph. d. west texas middle school math science...

Teaching Math to Diverse Adolescent

Learners:Achieving Instructional Equity

Zenaida Aguirre-Muñoz, Ph. D.West Texas Middle School Math Science PartnershipTexas Tech University

June – July, 2010

Expectations Design Instruction Around Big Ideas Maximize Growth Potential Plan to Scaffold & Differentiate Help Students Reason Mathematically Draw on Students’ Language & Culture

◦ If time permits

Workshop Overview

Self-Monitoring Activities Form Submissions Blogging Conference Presentations

Expectations

‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

Instructional Equity Premise

6th grade student explanation of the relationship between area, volume, and distance.

I think they are a chain, so if you know your volume, you will be able to find your area, so like a chain if you know one you’ll know the other. So I think the relationship is that if you know one you’ll know the other. If you know your calculations of volume, you’d be able to find your area. If you use what volume is which is length and width, area, and perimeter. With volume you’ll be able to find area, and with area you’ll be able to find out your distance.

Getting Started

Measurement of Geometric Shapes

Unit:uUnit:u

Unit:u2Unit:u2

Unit:u3Unit:u3

Unit:Name

Unit:Name

Areaformulas

S I Z E

M e a s u r e

0-dimensionNumber

0-dimensionNumber

1-dimensionLength

1-dimensionLength

2-dimensionsArea

2-dimensionsArea

3-dimensionsVolume

3-dimensionsVolume

Counting Distanceformula

Volumeformulas

U n i t A n a l y s i s

Design Instruction Around Big Ideas

Defining & Identifying Big Ideas

Teaching for ‘exposure’ Teaching without objectives, with ‘fun’

activities Neither empowers students to solve complex

problems

The Challenge

Emphasize Big Ideas◦ Highly selective concepts and principles◦ Clarify connections between smaller concepts ◦ Facilitate links to new concepts and problem

solving situations Build students’ understanding and use of

conceptual knowledge

Development of Conceptual Knowledge

Earth Science Example

Reveals how different natural phenomena follow the same flow of matter/energy that represents a rectangular figure.

Convection: a specific pattern of cause-and-effect relations involving phenomena that range from a pot of boiling water to ocean currents to earthquakes.

Links several smaller ideas (Density, heating and cooling, force, and pressure) and strategies together: to demonstrate how they operate in similar ways.

Middle School Math Example

Unit:uUnit:u

Unit:u2Unit:u2

Unit:u3Unit:u3

Unit:NameUnit:Name

Areaformulas

S I Z EM e a s u r e

0-dimensionNumber

0-dimensionNumber

1-dimensionLength

1-dimensionLength

2-dimensionsArea

2-dimensionsArea

3-dimensionsVolume

3-dimensionsVolume

CountingDistanceformula

Volumeformulas

U n i t A n a l y s i s

Size: measurement of an object which is based on its dimensionality.

Links smaller ideas (dimen-sion, distance, area, volume) and strategies (formulas & unit analysis) to demonstrate how they are related.Reveals how the process of translation is similar across objects of different dimen-sions.

Used to teach a variety of math content and strategies

Provide referential starting points for new math concepts and strategies◦ Include, size, proportion, estimation, etc

Explicitly described and modeled by the teacher

Summary of Big Ideas

Unwrap Standards1. Underline content nouns

Represent concepts (what students need to know)2. Circle verbs

Represent skills (what students need to be able to do)

3. Examine verbs to determine the intended level of thinking/reasoning Correspond to Bloom’s Taxonomy

4. Determine organizing/’power’ concepts (big ideas)

Identifying Big Ideas

Sample Standard Analysis

Standard Expectations

(5.10) Measurement. The student

applies measurement concepts

involving length (including

perimeter), area,

capacity/volume, and

weight/mass to solve problems.

The student is expected to:

(A) perform simple conversions within

the same measurement system (SI

(metric) or customary);

(B) connect models for

perimeter, area, and volume with

their respective formulas; and (C)

select and use appropriate units

and formulas to measure length,

perimeter, area, and volume.

Concepts (nouns) Skills (verbs) Potential Big Ideas

MeasurementLengthAreaWeight/massCapacity/volumePerimeterUnitsFormulasModels

Applies Solve Perform Connect SelectUse Measure

Organizing Content –Handout 1

Size/measurement, dimensionality

Skill K C Ap An S E Type of “Thinking Question”

Applies

Solve

Perform

Connect

Select

Use

Measure

Organizing Content –Handouts 2&3

X

X

X

X

X

X

X

X

Design Instruction Around Big Ideas

Elements of Conceptually-Based Instruction

C-Scope identifies concepts that can be used as starting points

Teacher should identify “power” concepts and develop students understanding of the relationships between concepts

Should be foundational to the lesson Should be applicable across lessons (e.g.,

size)

Identify Math Key Concepts

Should Focus Attention on Big Ideas Should Generate higher-level thinking Instruction focused on deep conceptual

knowledge results in higher achievement Instruction focused on lower level skills

leads to smaller gains over time

Design and/or Selection of Tasks

Strategy:◦ Examine verbs in question prompt (instruction)◦ Think about the steps involved in the expected

solution strategies Review page 9 in the instructional guide

and discuss why each task is categorized the way it is.

Share your findings with the class.

Try it!

Review the case study you received on Wednesday.

As you review think about the following:◦ Why do you think Kevin and Fran selected the tasks

they did? Where the tasks capable of bringing out the ideas they thought were important? Explain.

◦ How was Kevin’s approach to students different than Fran’s approach?

Examine the tasks presented to students and determine the level of reasoning involved.◦ Was the task selection related to Kevin and Fran’s

success? Explain.

Task Design Case Study

Strategy refers to a routine that leads to both the acquisition and use of knowledge

The ultimate purpose of a strategy is meaningful application, HOWEVER

For diverse learners, acquisition is most reliable when instruction focuses on stretegy first

The purpose of strategy instruction is to illuminate expert cognitive processes (mathematical reasoning) so that they are visible to the novice learner

Teach Strategies Before Introducing Authentic Tasks

Volume Strategy Instruction1. Link to prior knowledge2. Introduce new strategy3. Compute area4. Compute volume5. Write complete answer

Strategy Example Case (Handout 4)

How would you modify the instruction to reflect what you have learned this week about size?

Compare the volume strategy with that which is described for proportion. How can strategy instruction be implemented for proportion?

Apply It!

Use visual maps, models to present big ideas

Visual aids should make obvious the connections that are important

Refer back to links during instruction and in feedback to students

Feedback to students should draw attention to big idea and links among concepts

Use and emphasize words to call attention to big ideas

Demonstrate Links between Big Ideas, Prior Knowledge, & Strategies

Diverse learners benefit from good strategy instruction if and only if the strategies are designed to result in transferable knowledge of their application.

Remember…

Use Handout 5 to brainstorm and outline how strategy instruction could be done on a unit focused on the measurement of geometric shapes.

Be prepared to share your outline.

Try It!

Instruction should introduce and combine information in ways that result in new or more complex knowledge. ◦ What concepts need to be integrated for size?◦ In what sequence should these concepts be

taught?

Apply Conceptual Understanding to New Content

Provide multiple meaningful practice opportunities using big idea with new strategy

Apply big idea to the math strategy using a variety of problem solving situations

Pair a visual cue with each math big idea Post visual cue along with one sentence

describing why the big idea is important

Groundwork for Conceptual Understanding

After a recent review of your math TAKS scores, you notice that 30% of your students scored significantly lower on the measurement items of the test. Design a higher-level reasoning task involving the big idea of size. Include the following information in the description of the design:

◦ What visual aids would you provide?◦ What strategy would you introduce?◦ How would you model the strategy and its

connection to the big idea?

Apply It!—Classroom Scenario 1

Maximize Growth Potential

Theoretical Foundations

Less than 3% growth of K-12 US population 56% growth of ELLs between1995 and

2005 Greatest increases in areas with

traditionally little to no ELL populations Providing equal opportunity to learn

content and skills continues to be a critical issue

Teacher training and curricular materials in short supply

The Challenge

Students develop higher-order functions through language use.◦ Mental processes involved in higher-order thinking

(e.g., math reasoning) From the socio-cultural/situative

perspective, language mediates the development of higher mental processes (synthesis, evaluation)

Thus, the basic argument in education is that language plays a critical role in the development of conceptual understanding.

Use Language to “Think Together”

Language is the main vehicle of thought and all language use is based on social interaction

Language supports thinking and is evident when inner speech is overt:

◦ “Oops, that can’t be right…Maybe I should start by making a function table…Ah, good! I see why that relationship is off.”

Thought and Language

Language develops almost exclusively from interaction.

Thought is essentially internalized speech (age 2+), and speech emerged in social interaction.

Learning occurs first thru social interaction-on the inter-psychological plane, then is internalized in the intra-psychological plane.

Interaction, Language, & Thought—Vygotsky

Cognitive Development

Thought

Language

Age 2

The child does not merely “copy and paste” what they see and hear.

Internalization is a process of transformation involving appropriation and reconstruction. ◦ All learning is co-constructed◦ Learner transforms the social learning into individual

learning over time◦ Takes place in the zone of proximal development

(ZPD) Can occur between peers

◦ Joint construction of knowledge◦ Must foster active involvement, initiative, and

autonomy--AGENCY

Learning as Transformation

Many students do not exercise their agency. Participation moves from apprenticeship

(marginal participation) to appropriation (doing math)◦ Qualitative changes in participation

Over time, students appropriate the ways of thinking, acting, and interacting that is valued in school.

“It is more revealing to observe students’ participation in academic activity over time, to see how their potential is gradually realized” (Walqui & van Lier, 2010, pp. 12)

Learning as Change in Participation Over Time

Interaction that fosters appropriate support and leads to higher level functioning (not too much and not too little)

Requires explicit planning and incorporating supports or scaffolds to enable learners to take advantage of learning opportunities

It is NOT simply helping students complete tasks they cannot do independently.◦ The teacher would be doing all the (talking and)

thinking Scaffolds allow students to interact in their ZPD Every ZPD is unique AND constantly changing

Getting in the Zone

In order for teachers to maximize a child’s growth potential, scaffolding entails routinely differentiating the scaffolds provided to individual students across topics and tasks and to continue to do so over time.

Summary of Growth Potential

Plan to Scaffold & DifferentiateStructure & Processes

Scaffolding is used imprecisely. is often conceived of a structure, ignoring

the process. enables differentiation to occur. Is a structural instructional element AND an

instructional process Is how the ZPD is established and

learning takes place.

The Challenge

Involves both the predictable unpredictable aspects of the instructional context

Predictable◦ The structure of instruction (task design)◦ Planning and nature of task/activity

Unpredictable◦ Process of carrying out instructional

events/activities◦ Moment-by-moment words and actions◦ Teacher’s responsiveness to students unexpected

actions (feedback to students)

Scaffolding De-Mystified

Allows teachers to identify signs of an emerging skill, such as a word, behavior, or expression, and use it to engage the student in higher level functioning

Allows the student to take increasing control of the thinking

Control of thinking is shared Entices the student to take as much

initiative as possible

Successful Scaffolding

Allow for learner autonomy and initiative Neither stifling of development nor lead to

chaos Facilitate the process (lead to the identification

of signs emerging skill) Consider the following description:

◦ The builders put a scaffold around a building that needs to be renovated, but the scaffold itself is only useful to the extent that it facilitates the work to be done. The scaffold is constantly changed, dismantled, extended, and adapted in accordance with the needs of the workers. In itself, it has no value.

Tasks that Promote Autonomy

Read dialogues found on Table 2, page 19 of the instructional guide◦ Identify instances of scaffolding.◦ Identify who has the control of the direction of the

interaction? Compare the interactions captured on page

20 of the instructional guide. ◦ Who has control of the direction of the

interaction?◦ What are the instances of scaffolding?

Teacher Interactions that Promote Autonomy

Summary Features of Scaffolding

More Planned

More Improvis

ed

Continuity and Coherencetask repetition with variation; connecting tasks and activities; project-based or action-based learning

Supportive Environmentenvironment of safety and trust; experiential links and bridges

Intersubjectivitymutual engagement; being ‘in tune’ with each other

Flowstudent skills and learning challenges in balance; students fully engaged

Contingencytask procedures and task progress dependent on actions of learners

Emergence, or Handover/Takeoverincreasing importance of learner agency

Directly describe and model the skill. Perform the skill/task while thinking aloud (asking and answering

questions aloud). Provide immediate and specific feedback.

◦ Incorrect response: praise the student for effort while also describing and modeling the correct process/response; ASK QUESTIONS!

◦ Correct response: provide positive reinforcement by specifically stating what it is they did correctly; ASK QUESTIONS!

As students demonstrate success, ask for an increased number of student responses or ask more complex questions.

Continue to fade your direction, prompting students to complete more and more of the problem solving process: Relinquish CONTROL

When students understand the problem-solving process, invite them to actively problem-solve with you ◦ Let STUDENTS ‘TAKE OVER’◦ students direct problem-solving, students ask questions

Let student accuracy of responses guide your decisions about when to continue fading your direction.

Implementation

Listen carefully to the scaffolding demonstration video.

What are the instances of scaffolding? Who is in control of the interaction? What would you do differently? Why?

View and Analyze

Help Students Reason

MathematicallyThinking Questions

Focusing instruction on big ideas is necessary but insufficient

Requires ongoing monitoring of student understanding of those big ideas

The prevailing form of questioning is low-level fill in the blank questions

Instruction is focused on getting students to say the right things

Challenge

Typical Classroom Interaction

Dialogue 1 (IRE)* Dialogue 2 (IRF)**

Teacher: Excuse me.

Student: Yes?

Teacher: Can you tell me how I can get to Highway 1 from here?

Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it.

Teacher: Okay. Listen. Go straight TO the traffic light, turn left, and go straight ahead UNTIL you see the sign for Highway 1.

Student: Ehm…go

straight TO traffic light…(etc)

Teacher: Excuse me.

Student: Yes?

Teacher: Can you tell me how I can get to Highway 1 from here?

Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it.

Teacher: Thanks!

Student: You welcome!

Using different kinds of questions for different purposes can help differentiate instruction

Help monitoring conceptual understanding Increase the proportion of students who

remain engaged in conversations about important math ideas◦ Engaging questions◦ Refocusing questions◦ Clarifying questions

Pose “Thinking” Questions

Open-ended with multiple acceptable answers

Invite students into discussion Keep them engaged in conversation Re-engage students who have “tuned-out.” Students with low math self-efficacy benefit

most by being invited into discussions that reward multiple solutions based on alternative, accurate math reasoning.

Engaging Questions

Get students back on track or to move away from a dead-end strategy◦ Used instead of simply telling students what to do

differently Remind students about some important

aspect of a problem they may be overlooking

Refocusing Questions

Help students explain their thinking or help the teacher understand their thinking

Can be used when: A. A student understands an idea but the language

used to explain that thinking is not clear or precisea.“What does ‘it’ refer to?”

B. More needs to be revealed about a student’s thinking to make sense of ita.“How did you get that answer?”

Clarifying Questions

Math Example

Two Similar Rectangles

Engaging Question: How can we decide what value the question mark stands for?

Refocusing Question: What does it mean for two rectangles to be similar?

Clarifying Question: (In response to a student who says that the answer is 5) How did you get 5?

4 3

6 ?

Suppose the learning target for a lesson is to distinguish area from perimeter. In a class discussion a student says the area is 50 centimeters.

If the teacher wants to refocus the student to the general math idea (unit analysis), what question can be posed? If the teacher wants to call attention to (clarify) the student’s response what question can be posed?

Classroom Scenario 2

The conversation in Handout 6 occurs when the teacher stops to talk with two students who have been playing a game (based on Fraction Tracks). Considering how the questions are phrased, what do you think the purposes of the following questions are? How could you move your pieces across to

the other side if your card was ? Could you go and then ? Can I move the whole now?

Analyzing Questions

Suppose a teacher asks students what number goes in the box (below) to make a true number sentence.

152 + 230 = + 240

A student replies “382.” The teacher then asks a clarifying question “How did you get 382?” To which the student replies “I added 152 and 230.” As the student replies, she notices that another student wrote “= 622” after 240. What fundamental misconception do these responses represent?


Turn to page 27 in guide.

Go to Page 25 in Instructional Guide for process instructions.

Investigate Your Questioning to Support Conceptual Understanding

Help Students Reason

MathematicallyUse ‘Teacher Talk’ to Model Ways of Thinking about Mathematics

There is value in helping students conceptualize math as more than a set of procedures.

Students need to understand that math is a thinking and reasoning process rather than a set of steps to go through to get the right answer.

A focus on language use enables teachers to develop and reinforce norms for talking mathematics in valued ways which, in turn, affects students’ math beliefs and self-efficacy

Model Ways of Thinking About Math

Stepping out More explicit language moves that include

reflection on math actions; talking about math.

That’s a great example of the kind of explanation I’m looking for. It’s important that you not only give your answer but that you also explain what you did and why you did it. I want you to explain the process you went through, not just give an answer.

Language Moves in Classrooms

Revoicing Less explicit language moves that allow the

teacher to reformulate a student’s response by clarifying or extending what a student has said in an effort to help other students understand the math significance of the contribution

Recast student’s verbal contributions in more technical terminology with slight changes so as to move the discussion forward, leading to more conceptually-based explanations that originated with the student’s contribution.


Revoicing is used to clarify statements, make connections, or fill in missing elements of an explanation.

“By helping students articulate their understanding, teachers provide opportunities for students to agree or disagree with the reformulated version, teaching them to explain their reasoning.” (emphasis added, pp. 25)


Refer to page 31 in teacher guide.


Engage Students in a Math Discourse Community

Class Norms

Students positioned to be thinkers and explainers within math community…

Transmitted Message

Math:·Is flexible·Is about meaning·Makes sense

· Has reasons for its procedure

· Requires particular ways of reasoning and explaining

Language MovesUsing language moves, the teacher is able to: · Request multiple solutions· Amplify solutions· Revoice to make math

processes clearer and more precise

· Make students aware of math thinking and relationships

· Help students develop ways of thinking and talking about math

Planning for classroom interaction is a way to offer all students opportunities to observe math reasoning in action and to develop their own abilities with math reasoning.

Attending to language moves in the classroom that both reveals your own thinking processes and clarifies those of your students is a step toward constructing more meaningful math learning for all students,

Remember…

Developing reasoning about math takes time

Teachers at all levels can help students begin to do so by modelling ways to:◦ talk about math, ◦ reason about the activities they are engaged in.

Refer to page 32 for assignment.

Investigate your language moves in the classroom

achieving instructional equity zenaida aguirre-muñoz, ph. d. west texas middle school math science...

Documents

u unit

dimensions area

dimensions volume

area youll

volume youll

teacher slide

calculations of volume

dimension length