achieving instructional equity zenaida aguirre-muñoz, ph. d. west texas middle school math science...
TRANSCRIPT
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
‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).
Instructional Equity Premise
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
‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).
Instructional Equity Premise
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!
‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).
Instructional Equity Premise
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?
Classroom Scenario 3
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.
Language Moves in Classrooms
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)
Language Moves in Classrooms
Refer to page 31 in teacher guide.
Classroom Scenario 4
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
‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).
Instructional Equity Premise