Mathematics:

1
Mathematics A Diagnostic Testing/ Prescriptive
Instruction Model Pennsylvania Department of
Education Conference Hershey Lodge and Convention
Center February 22-24, 2005
Richard C. Miller REALMIGHT
2

• Workshop Objectives
• To Explore
• Characteristics and Behaviors of Mathematically
  Gifted Students
• The Assessment of High Reasoning Ability in
  Mathematics
• Differentiated Mathematics Curriculum and
  Instruction for Mathematically Gifted Students

3
(No Transcript)
4
Characteristics and Behaviors
of the Mathematically Gifted

• Keen Awareness, Intense Curiosity
• Quickness in Learning, Understanding, and
  Applying
• Sees Mathematical Patterns and Relationships
• Thinks and Works Abstractly
• Uses Analytical, Deductive, and Inductive
  Reasoning
• Flexible and Creative in Mathematics Problem
  Solving
• Generalizes and Transfers Concepts

Such Lists Do NOT include Computational
Proficiency as a Characteristic of the
Mathematically Gifted!
5
What Is the DT PI Model ?

• The DT PI Model is
• An individualized approach to developing
  instructional programs for students with a high
  degree of mathematical ability
• Flexibly paced differentiated mathematics program
• Based on the JHU CTY (used by C-MITES, IU8 MMP)
• - Student Identification
• - Instructional Program
• - Student/Program Evaluation

6
Is DT PI Research Based? Data Driven?
YES
Yes

• Pioneered by Dr. Julian C. Stanley
• Conducted first talent search at Johns Hopkins
  University in 1972
• Significant research and tracking of results from
  JHU, C-MITES, IU8 MMP
• Objective data used to place students and guide
  mathematics instruction

7
How Does DT-PI Model Work?

• In-level testing data is used to identify
  students with mathematical ability
• Students math ability and achievement are
  further measured with diagnostic tests such as
  SCAT, STEP, Explore, Orleans Hanna Algebra
  Prognosis
• Diagnostic testing provides data for placement,
  rate of instruction and prescriptive instruction

8
Assessing Mathematics Reasoning Ability

• Step One Talent Screening

• Limitations
• Insufficient Ceiling
• Few Difficult Items

Based on Johns Hopkins, CTY Model
9
Assessing Mathematics Reasoning Ability

• Step Two Out-Of-Level Assessment

Based on Johns Hopkins, CTY Model
10
Matching Instruction with Student Ability
11
Mathematics Reasoning Tests
12
(No Transcript)
13
Out-of-Level Ability Tests (Model Mathematics
Project)  
 Out-of-Level Ability Tests (Model Mathematics
Project) 
 SAT I Identifying Scores (Model Mathematics
Project) 
14
Out-of-Level Ability Tests (Model Mathematics
Project)  
Sample Test Item
J and M are the heights in centimeters of John
and Mark, respectively. J M 2 centimeters
Column A
Column B
The height of John
The height of Mark
A if the part in Column A is greater, B if
the part in Column B is greater, C if the two
parts are equal, D if not enough information is
given
15
Why Use the DT PI Model?

• Students with high mathematics ability think
  about mathematics very differently from the
  average student and benefit from an approach that
  addresses their ability.
• The DT PI Model helps a school district to
  better meet the educational needs of students
  having very high ability in mathematics.

16
Differentiated Mathematics Curriculum and
Instruction
17
Ways that mathematically gifted students differ
from their classmates

• Pace of learning
• Depth of understanding
• See relationships
• Solve complex problems
• Originality flexibility
• Generalize and transfer
• High interest

18
Curriculum

• Mathematics Curriculum must be Comprehensive
  and Rigorous
• Support Future Mathematicians
• Problem Solving Based
• Internationally Competitive (Algebra I- Eighth
  Grade)
• Present Mathematics as a Body of Knowledge
  Defined by Axioms and Derived Theorems

In USA Schools, Mathematics Curriculum is Often
a Mile Wide and Taught at One Inch Depth.
-Third International Mathematics and Science
Study The National Center of Educational
Statistics
19
Instruction

• Traditional (Common in most US Classrooms)
• Teacher Instructs Students in Concept or Skill
• Teacher Solves Example Problems With Class
• Students Practice On Their Own (Teacher Assists
  Individual Students)
• Critical Thinking and Problem Solving
• Teacher Introduces Complex Problem
• Students Work With Problem/ Present to Class
• Various Solutions Discussed
• Teachers Assists in Summarizing/ Symbolic
  Representation
• Student Practice and Internalization

20
Kristinas Problem
What numbers do the box and triangle represent?
Come up with your own puzzle! 1. There should
be two unknowns. 2. You must use
multiplication, division, addition, subtraction,
and exponentiation at least once. 3. There
should be four equations.
21
Sum Stumpers
What is the sum of the first 100 positive odd
whole numbers?
22
Diagnostic/Prescriptive Instruction

• Diagnostic Pre-Assessment
• CBA/Standardized
• At Student Instructional Level (50)
• Power Testing
• Mastery Levels (80-85, 90)
• Certification
• Item Analysis
• Post-Assessment

23
Diagnostic/Prescriptive Instruction

• Prescriptive Instruction
• Placement in Course Sequences
• Focus on Un-mastered Skills Concepts
• Emphasis on Review
• Instructional Pacing Rate (Flexible Pacing)
• Linearization of Curriculum
• Role of Enrichment
• Mastery Levels

24
Differentiating Mathematics Content

• Comprehensive and Rigorous
• May Use Traditional Materials
• Multiple Levels of Text
• Multiple Resources
• High Quality Enrichment
  (Accelerative Enrichment)

25
Other Differentiating Strategies
Instruction Methods

• Differentiated Assignments
• Learning Contracts, Learning Centers
• Emphasis on Open-ended Problems and Higher Level
  Thinking Skills
• Encourage Original, Creative Thought
• Reduce Repetitions and Cyclical Review

26
Other Differentiating Strategies
Learning Environment

• Interaction with Ability Peers
• Flexible Grouping
• Foster Independence/Provide Instruction
• Foster Original and Creative Thought
• Use of Technology and Manipulatives

27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30

• Meeting the Needs of Gifted Students
• Differentiating Mathematics and Science
  Instruction
• (www.Nwrel.org/msec/images/resources/justgood/12.9
  9.pdf)
• Northwest Regional Education Laboratory

• Developing Mathematical Talent
• A Guide for Challenging and Educating Gifted
  Students
• Susan Assouline Ann Lupkowsky-Shoplik

• A Nation Deceived
• How Schools Hold Back Americas Brightest
  Students
• The Templeton National Report on Acceleration