Planning Effective Mathematics Instruction in a Variety of Educational Environments
David Allsopp, Ph.D.University of South Florida
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Essential Questions
• What is the target mathematics concept?• What did I learn from my mathematics dynamic
assessment (or other pre-unit assessment)• What is my instructional hypothesis (what students
know, don’t know, and why)• How do students think about these ideas differently
from adults and how can I use this information to inform instruction?
• How will I differentiate the instructional needs of my students?
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Essential Questions
• What authentic contexts will I use?• How will I introduce/model the target concept to the
whole class?• How will I differentiate the
instructional/scaffolding/extension (generalization and adaption) needs of my students?
• How will I provide practice opportunities that promote proficiency/maintenance?
• How will I evaluate my students’ learning and determine the effectiveness of my instruction?
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics Instruction
• Consider what students know, don’t know, and why• Consider stages of learning• Determine differentiated instruction needs/objectives• Select authentic contexts• Plan whole class instruction• Plan differentiated instruction• Plan practice opportunities• Plan how you will evaluate learning and your
instruction
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
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
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
Core Instruction:• The areas to be studied in mathematics from pre-kindergarten through
eighth grade should be streamlined
• Proficiency with whole numbers, fractions, and certain aspects of geometry and measurement are the foundations for algebra.
• Conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other.
• Students should develop immediate recall of arithmetic facts to free the "working memory" for solving more complex problems.
What we know…National Math Panel
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.
How programs are designed is critical!
-Spiral vs. Strand design
-Traditional vs. Explicit
-Use of scaffolding to increase mastery & generalization of skills/strategies vs. demonstrate & replicate
-Prior knowledge: Instruction in related vocabulary and review of mastered prerequisite skills vs. assumption of prior knowledge
-Examples & non-examples
-Sequencing of skills (macro/micro)
-Progress monitoring vs. “wait and see”
Time to Reflect…
Individual Think-WriteWhat are some of the characteristics of my core math program?
Mix and Match 2
Summarizing CRA Results
Based on C-R-A Assessment Mrs. Carsen concludes students:
o Have difficulty representing fractions that are >, <, or = using unlike denominators
o Have difficulty determining >, <, or = using symbols between fractions with unlike denominators (abstract, representational)
o Have some ability to do this with fractions that have natural relationships (1/2 and 2/4, 4/6 and 2/3-abstract)
o Have difficulty relating written fractions to drawings(Lack concept of the meaning of a fraction)
o Difficulty with concept of “equivalent area” of whole to part when drawing
MDAResults
Hypothesis
SA
ZD
JD
AD
RF
FJ
RJ
SK
NM
JM
XM
TR
JT
TW
F
F
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Instructional HypothesisWhat’s it all about?
Provides you with a focused approach to teaching that specifically addresses the needs of your students based on the results of a mathematics dynamic assessment.
Instructional Hypothesis
Context:
What Students Can Do:
What Students Can’t Do:
Reason:
“Given _____________________,
Students can _______________
Students cannot _____________
Because ____________________”Keep It Simple!
Given two fractions…
Students are able to…determine >, <, and = when fractions have like denominators at concrete, representational and abstract levels.
Students are unable to…determine >, <, and = when fractions have unlike denominators at concrete, representational and abstract levels.
…becausethey lack understanding of the area that fractions represent
Instructional Hypothesis
MDAResults
Hypothesis
SA
ZD
JD
AD
RF
FJ
RJ
SK
NM
JM
XM
TR
JT
TW
F
F
F
M
I
M
M
F
I
I
I
F
M
M
I
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Given a set of 2 fractions, the majority of my students can determine >, <, and = when fractions have like denominators at the C R & A levels; however they cannotdetermine >, <, and = when fractions have unlike denominators because they lackunderstanding of the area that fractions represent.
Consider Stages of Learning:A Framework for Understanding How Struggling Learners Learn
Entrylevel
Acquisition Proficiency Maintenance Generalization AdaptionInitial Advanced
Accuracy
Rate
Retention
Stages of Learning
ExtensionTo focus instruction, it is important to know at what stage of learning students are with respect to the target math concept/skill…
Planning for Responsive Mathematics Instruction Determine
differentiated instructional
needs/objectives
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics Instruction
Determine differentiated instructional
needs/objectives
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics InstructionSelect Authentic
ContextPlan whole class
instruction
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics InstructionPlan differentiated instruction
Plan practice opportunities
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics Instruction
Plan evaluation of learning and instruction
Plan practice opportunities
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics Instruction
Plan evaluation of learning and instruction
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Responsive Mathematics Instruction
Plan differentiated instruction
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
Planning for Specific Learning Barriers of Students
Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.
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