differentiating math instruction: project equal small group instruction responding to learners’...
Post on 13-Dec-2015
220 Views
Preview:
TRANSCRIPT
Differentiating Math Instruction:
Project EQUAL
Small Group InstructionResponding to Learners’ Needs
Big Ideas of Mathematics~Number & Operations~Algebra~Geometry~Measurement~Data analysis & probability
Processes for Doing Mathematics~Problem Solving~Reasoning & Proof~Connections~Communications~Representation
Responsive Teaching Framework for Differentiating Mathematics Instruction
How programs are designed is critical!
-Spiral vs. Strand
-Traditional vs. Explicit
-Scaffolding to increase mastery & generalization of skills/strategies vs. demonstrate & replicate
-Prior knowledge: Instruction vs. Assumption
-Examples & non-examples
-Sequencing of skills
-Progress monitoring vs. “wait and see”
Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007
Making mathematics
accessible through responsive teaching
Understanding & teachingThe big ideas in math ANDThe big ideas for DOING math
Understanding learning characteristics/ barriers
for students with difficulties In mathematics
Continuously assessing learningTo make informed instructional
decisions
Model for Meaningful Mathematics Instruction
Pre-Assessme
nt
Formative
Assessment
Summative Assessment
Making mathematics
accessible through responsive teaching
Educators must create meaningful learning experiences for students with persistent math difficulties.
This is accomplished by considering learner profile/characteristics and creating a match for the student through careful selection/use of:
MethodsPracticesProcedures
Making Mathematics Accessible Through Responsive Teaching
Readiness Interest Learning preferences
Differentiation of Instruction
based on students’
teachers can differentiate
Tomlinson, The Common Sense of Differentiation, ASCD, 2005 OPTIONS, FDLRS Action Resource Center
Differentiated Instructionis
A teacher’s response to a learner’s needs
clearlearning goals
respectful tasks
flexible grouping
ongoing assessment and
adjustment
positive learning environment
Content Process Product
guided by general principles of differentiation, such as
Forming Flexible Small Groups
Teachers may group students in order to:
o Provide more explicit, intensive, well-scaffolded instruction
o Increase a students’ stage of learning from initial acquisition to advanced acquisition
o Provide independent learner experiences to build proficiency
o Provide learning extension experiences
Using data to flexibly group students
What math data do you currently gather on:
ALL students?Some students?Few students?
Small Group Responsive Teaching FrameworkStep 1: Revisit Summary of MDA Results
SAZDJDADRFFJ
RJ ?SKNMJMXMTRJT
TW
FFFMIMMFIIIFMM
IIIMMMMIMMMIMM
IIIIMIIIMMMIII
MMMMMMMMMMMIMM
IIIMMMMIMMMIMM
MMMMMMMMMMMMMM
NameAbstract
Expressive Receptive
Representational
ExpressiveExpressive Receptive Receptive
Concrete
Key: M=Mastery 95%, I=Instructional 70/75%-95% F=Frustrational below 70%
Step 2: Consider additional Flexible Student Interviews
Who might we want to have an additional conversation with?
What would we want to know?
SA, ZD, JD, TR
Ask them to look at & create concrete & representational examples of 2 fractions, and explain how they know which one is greater…
What evidence do they show re: fractions = area? Do they understand that fractions represent area between 0 and 1 on the number line?
Step 2: Consider additionalFlexible Student Interviews
Ideas for your “Conversation”:
o “Listen” for student’s mathematical thinking
o Ask them to describe to you how they solved the problem
o Ask them to “teach” you how to solve the problem
o Ask them to watch you solve the problem and generate question(s)
o Use concrete objects or drawingsto further explore student understandings
So…What if I have a small group problem or an individual student problem… What do I do to increase student achievement?
How do I meet the needs of studentswho are struggling with mathematics?
Make In
structi
on
Explic
it!
Step 3: Develop Hypothesis for Small Group(s)
A group of my students can…demonstrate receptive concrete and representational
understanding when comparing fractions with like denominators
Using… manipulatives and drawings
However, they do not demonstrate an understanding of this concept…
At the expressive concrete or representational levels
I think this is because…They do not have conceptual understanding what
fractions represent (area) or the part : whole relationship.
Step 4: Form Flexible Small Group(s) for Mathematics Instruction
Who would be in our small, skill-based group?
SA, ZD, JD, TR
A group of my students can…demonstrate receptive concrete and representational understanding when comparing fractions with like denominatorsUsing… manipulatives and drawings However, they do not demonstrate an understanding of this concept…At the expressive concrete or representational levelsI think this is because…They do not have conceptual understanding what fractions represent (area) or the part : whole relationship.
How can teachers reach & teach
these students?Differentiating Mathematics Instruction:
An Explicit, Intensive Small Group Lesson Sequence
Variables that influence students’ acquisition of
mathematics:
Instructional Design
Instructional Delivery
Classroom Organization & Management
Instructional DesignDecisions affecting What to Teach
Sequence of skills and concepts
Explicit instructional strategies
Pre-skills
Example selection
Practice and Review
Sequence of Skills and Concepts:
The order of which skills and strategies are introduced affects the difficulty students have in learning them.
Pre-skills of a strategy are taught before the strategy.Easy skills are taught before more difficult ones.Strategies and information that are likely to be confused are not introduced consecutively.Stein, Kinder, Silbert & Carnine, Designing Effective Mathematics Instruction: A
Direct Instruction Approach, 2006
Instructional Strategies:
Clear, accurate & unambiguous (Gersten, 2002)
Strategies must be well-designed and generalizable
Should draw focused attention to the relationship between and among math concepts and skills
Pre-skills:
Component skills of a strategy are taught before the strategy itself is introduced
Ensure students have mastered the pre-skills before introducing a new
instructional strategy
Assess preskills! This helps in determining where instruction should begin!
Example Selection:
Include only problems that students can solve by using a strategy that has been previously taught
Include both examples of the currently introduced type of problem as well as previously introduced problem types that are similar (discrimination)
CAUTION! Many commercial programs do not contain sufficient numbers of examples in their initial teaching to reach mastery, and rarely employ and adequate number of discrimination problems.
Systematic Practice
is necessary for those with low performance rates; without it, students may confuse or forget earlier taught strategies.
Facilitates retention over time
As students reach their goal, gradually decrease the amount of practice of that skill
Practice should never entirely disappear!Kinder, Silbert, Carnine, 2006
Instructional Delivery:How to Teach
Initial assessment and progress monitoring
Presentation techniques
Error-correction procedures
Diagnosis & Remediation
Instructional Delivery:Initial Assessment & Progress Monitoring
Initial (Pre) Assessment:
Answers the Question:
Which skills must be taught?
Dynamic Mathematics AssessmentFocus: Accuracy
Progress Monitoring:Answers the Question:
How are my students
responding to instruction?
On-going / continuous Progress Monitoring
Focus: Rate of Learning + Accuracy
Instructional Delivery:Presentation Techniques
Adequate Time
Appropriate Pacing
Frequent Student Responding
Responsive Scaffolding
Precise Monitoring
Positive, Corrective, & Immediate Feedback
Instructional Delivery:Diagnosis & Remediation
Diagnosis: Determining the cause of a pattern of errors
“Can’t do… or … won’t do?”
If you need deeper diagnostic informationregarding a student’s knowledge of fractions, consider administering a Fraction Inventory
Remediation: Process of re-teaching a skill
Provide Instruction! Address Motivation!
Classroom Organization & Management
Elements of Daily Math Lessons
Teacher-Directed Instruction-Introduction of new skills-Remediation of previously taught skills
Independent Work -Exercises students complete without assistance
-Never assign independent work involving problem types/skills that have NOT been
instructed
Workcheck-Designed to correct errors students make on independent work-Provides insights into nature of student errors -Enables teacher to make necessary instructional adjustments
I. Teacher explains taskII. Teacher models task “I do…”III. Teacher & students practice task
together—guided practice “We do…”
IV. Students (independently) practice tasks “You do...”
Step 6: Plan & Deliver Instructional Routine
Guided Practice!
Math Lab:Explicit Instructional Routine
Step 7: Develop & implement
student monitoring plan
Determine:
…who will be monitored?
…on what skill?
…using which tool?
…frequency (how often)?
Precision Teaching or Progress Monitoring?
Indicator: Progress Monitoring (Uses Equal Interval Chart)
Precision Teaching (Uses semi-logarithmic chart)
Uses one-minute timings to analyze data to determine student progress toward specific skills
Yes Yes
Helpful in making instructional decisions based on the review and analysis of student data
Yes Yes
One-minute timings always on grade level skills
Yes Not necessarily. If a student is working below grade level, the timing would be on the student’s instructional level.
One minute timings conducted three-four times per year on grade level skills
Yes No. The precision teaching fluency-building timings are conducted daily or three to four times per week.
Adding it Up, National Research Council, p. 117, 2007
In order for students tobe successful in Mathematics, each of these intertwined strands must work together, so…
What do we measure to determine if they are making progress?
Continuously Assessing LearningEducators should evaluate what students know
and can do before, during and after instruction
Before Instruction: evaluate student prerequisite knowledge, skills, experiences and interests that relate to the target concept. Know where to begin instruction.
During Instruction: Are students gaining understanding? Are they able to use knowledge and skills proficiently? Results allow for immediate adjustment, if necessary.
After Instruction: Where are students in terms of their conceptual learning and skill proficiency? Assessing for the purpose of answering this question will help in planning future instruction.
Types of Assessment:Mathematics
Before: Screening
During: Formative
Diagnostic
(if necessary)
After: Summative
Pre-Assessment
Formative
Assessment
Summative
Assessment
Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007
Making mathematics
accessible through responsive teaching
Understanding & teachingThe big ideas in math ANDThe big ideas for DOING math
Understanding learning characteristics/ barriers
for students with difficulties In mathematics
Continuously assessing learningTo make informed instructional
decisions
Model for Meaningful Mathematics Instruction
Pre-Assessme
nt
Formative
Assessment
Summative Assessment
Big Ideas of Mathematics~Number & Operations~Algebra~Geometry~Measurement~Data analysis & probability
Processes for Doing Mathematics~Problem Solving~Reasoning & Proof~Connections~Communications~Representation
Responsive Teaching Framework for Differentiating Mathematics Instruction
Most Intensive
Few
Some
ALL
• Daily or Weekly, precise
• Graphically represented
EX. : Precision Teaching Great Leaps CBM probes
Least Intensive
• Weekly or Bi-weekly, more precise• Graphically
representedEX: CBM Probes
The more intensive the instruction…
the more intensive the monitoringshould be!
• District math benchmark assessments (as indicated)• Curriculum embedded assessments• Flexible student
interviews• Error Analysis• Student Work Samples• Rubrics for Problem
Solving (Applied Math)
MonitoringProgressorProgress Monitoring?
ProgressMonitoring
Monitoring Progress for
How can I measure student knowledge levels?Thinking Differently…
CRA Level of Understandin
gMethod Criterion
Abstract 1-5 minute timings(depends on nature of
target concept)
Fluency (Rate & Accuracy)
Representational(Drawing) 8-10 tasks Accuracy
90-100% 3 times
Concrete 3 tasks Accuracy100% 3 times
Start
INVESTIGATE!Review the continuous, formative assessment tools you have at your table.
As a Table Group, prepare to explain what it is, & how it might be used.
Think in terms of the intensity equalizer. Is it more intensive, or less intensive?
Progress Monitoring Probe
Abstract Level
Procedural:
See/Write 2 digit addition without
regrouping (sums < 20)
Measure:# of digits correct
Examples of Concrete and Representational/Drawing Probe Tasks
Use circle pieces and string to solve the following equations.
1. 3 x 4 = 12
Concrete
Representational/ Drawing
Making Instructional Decisions
Create a visual display of student performance data• Chart• Graph• Think of this visual display as a
“picture” of your students’ learning
Evaluate what the learning picture reveals about student learning
“goal line”
Visual Display
“corrects”
“incorrects”
What does this learning picture show?
Allsopp, 2008
Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs, verbal rules, and standard
algebraic notation.
NCTM Process: Representation
NCTM Process: Communication
0 1 2 3
Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs,
verbal rules, and standard algebraic notation. Allsopp, 2008
Some General Guidelines for PM
Incorporate at concrete, drawing & abstract levels
Use short, easy to evaluate “probes”
Pinpoint key concepts for monitoring
Teach students to chart their learning
Use as a way to engage students in setting learning goals
At least 2-3 times weekly, more often if needed
CELEBRATE SUCCESS!!
Learn More About Continuous Progress Monitoring
at the MathVIDS Website:
http://fcit.usf.edu/mathvids/
Other Ongoing Monitoring Resources include:www.interventioncentral.org www.aimsweb.com
A Word About Follow Up
Choose an instructional practice you learned about in Project EQUAL
Implement in your Classroom
Join our Project Equal Wiki
Post a reflection about your action research.
top related