Mathematics Instruction
Best Practices:
A Formula for Success
How Many Triangles?
Pair off with another person, count the
number of triangles, explain the process, and
record the number.
How Many Triangles?
Math Inventory
What are our beliefs about Math?
• Results from Survey Monkey:
Why Math?
Increasing recognition of the importance of
mathematical knowledge
“For people to participate fully in society, they
must know basic mathematics. Citizens who
cannot reason mathematically are cut off from
whole realms of human endeavor.
Innumeracy deprives them not only of
opportunity but also of competence in
everyday tasks.” (Adding it Up, 2001)
State of Mathematics
• Achievement on the NAEP trending upward for
4th/8th grade and steady for 12th grade
– Large numbers of students still lacking proficient
skills
– Persistent income and ethnicity gaps
– Drop in achievement at the time algebra
instruction begins
• TIMSS data indicate significant lower levels of
achievement between the US and other nations
• Jobs requiring intensive mathematics and science
knowledge will outpace job growth 3:1 and everyday
work will require greater mathematical
understanding (STEM)
Meaningful Differences in
Math Readiness
• Long term trajectories are established as early as
kindergarten (Morgan & Farkas, 2009)
– 70% of students exiting K below the 10th %ile remain below
the 10th %ile at the end of 5th grade
• Middle and high SES children come to school with
much more informal instruction in numbers and
quantitative concepts (Griffin, 1994)
• Children lacking these opportunities require formal
explicit instruction to develop early numeracy skills
The National Mathematics Advisory Panel report
offers recommendations for how we can best
prepare elementary and middle school
students for success in algebra, a gateway to
mathematics in high school and beyond.
Critical Benchmarks for Algebra Success
Fuzzy Math?
We need BALANCE in mathematics…
CONCEPT/APPLICATION +
COMPUTATION
= STUDENT ACHIEVEMENT
At the Elementary Level,
Most Students that Struggle in Math Have
Difficulty with:
• Solving problems (Montague, 1997; Xin Yan & Jitendra,
1999)
•
•
•
•
•
Visually representing problems (Montague, 2005)
Processing problem information (Montague, 2005)
Memory (Kroesbergen & Van Luit, 2003)
Self-monitoring (Montague, 2005)
Fluency of math facts (Fuchs, 2005)
What should we do for these students?
What to do?
• Implement an effective core CURRICULUM based on
– Critical mathematics content (Common Core State Standards)
– Research-based instructional design principles (EnVision Math)
•
•
Ensure student understanding through high-quality instruction using both student-centered and
teacher-centered strategies.
–
Procedural Understanding/Skill Acquisition (Instructional Focus Continuum)
–
Conceptual Understanding (CRA model)
–
Scaffolded Instruction (I do, We do, Ya’all do, You do)
Use reliable assessment tools
–
Regular formative assessment (District CFA’s)
–
Benchmark screening and progress monitoring (M-CBM’s)
~ Foundations for Success: The Final Report of
the National Mathematics Advisory Panel (2008).
Types of CURRICULUM
• Recommended Curriculum = recommended by
experts in the field (NCTM, National Standards,
IRA)
• Written Curriculum = state, district, school, and
teacher documents specifying what is to be
taught (math maps)
• Supported Curriculum = what is in instructional
materials (textbooks, media)
• Tested Curriculum = what is embodied in state
tests, school tests, and teacher tests
Taught Curriculum = what teachers actually deliver
Learned Curriculum = what students learn
Hidden Curriculum = unintended content learned
from school culture and climate
Excluded Curriculum = what has been left out,
intentionally or not
Common Core State Standards
• All students need to develop mathematical practices such as
solving problems, making connections, understanding
multiple representations of mathematical ideas,
communicating their thought processes, and justifying their
reasoning.
• All students need both conceptual and procedural knowledge
related to a mathematical topic, and they need to understand
how the two types of knowledge are connected.
• Curriculum documents should organize learning expectations
in ways that reflect research on how children learn
mathematics.
• All students need opportunities for reasoning and sense
making across the mathematics curriculum—and they need to
believe that mathematics is sensible, worthwhile, and doable.
EnVision Instructional Content
EnVision focuses on key strands rather than a
broad array of mathematical content
– Numbers and Operations
– Geometry
– Measurement
– Data Analysis
– Algebra
NCTM Curriculum Focal
Points (2006)
What to do?
•
Implement an effective core curriculum based on
–
Critical mathematics content (Common Core State Standards)
–
Research-based instructional design principles (EnVision Math)
• Ensure student understanding through high-quality
INSTRUCTION using both student-centered and teachercentered strategies.
– Procedural Understanding/Skill Acquisition (Instructional
Focus Continuum)
– Conceptual Understanding (CRA model)
– Scaffolded Instruction (I do, We do, Ya’all do, You do)
•
Use reliable assessment tools
–
Regular formative assessment (District CFA’s)
–
Benchmark screening and progress monitoring (M-CBM’s)
~ Foundations for Success: The Final Report of
the National Mathematics Advisory Panel (2008).
INSTRUCTION
HOW do I teach . . .
Research on Effective Instruction indicates:
Quality of Instruction - reflects quality of curriculum, lesson
preparation, and teaching skill
Appropriate Level - lesson is neither too easy nor too
difficult
Effective Pacing – time is used efficiently, the pace is
“perky”
Incentive - students are engaged and motivated to learn
Instructional Focus Continuum
Instructional Strategies for Building Skill
Accuracy
• Explicit Teaching
– Teacher modeling, guided practice, independent
practice
• Teacher Feedback
– Specific positive confirmations
– Corrective feedback on errors
• “Cover, copy, compare”
Explicit Instruction
• What is Explicit?
– Precise and consistent language
– Clear, accurate, and unambiguous teaching
– I do it, we do it, Ya’all do it, You do it
– Why is it important?
• Often when students encounter improper fractions (e.g. 5/4)
the strategies they were taught using a single unit don’t
work.
• Many commercially developed programs suggest that
students generate a number of alternative problem solving
strategies. Teachers need to select only the most
generalizable, useful, and explicit strategies (Stein, 2006).
Explicit Instruction
• High Achieving countries all implement
connections problems as connections
problems
• U.S. implements connection problems as a
set of procedures
Exponents and Geometry?
What is 42 ?
Why is it 4 x 4 when it
• looks like 4 x 2 and
• sounds like a geometry term?
It means ‘make a square out of your 4 unit side’
Exponents and Geometry?
What is 42 ?
--4 units-1
1
1
1
You’d get how many little
1 by 1 inch squares?
42 = 16
Precise and Consistent Language
Kids are
Sponges!
Ambiguous Language Examples
4<7
8>2
What NOT to teach…
• Math strategies that do not support
– Future learning
• EX: “When subtracting, the larger number ALWAYS goes
on top.”
– Accurate conceptual understanding
• EX: “When subtracting, borrow a 1 from the tens
place.”
Treatment Integrity for
Cover, Copy, & Compare
_____ Provided worksheet with problems and solutions
on the lefts side of the page and the same problems
without answers to the left of the page.
_____ Provided checklist of the steps to complete.
_____ Watched each student to be sure they are doing
steps correctly.
_____ Provided error correction and praise as needed.
Student Directions for
Cover, Copy, Compare
_____ Look at the problem and the answer.
_____ Cover the problem and answer.
_____ Write the answer.
_____ Uncover the problem and check if you wrote the
answer correctly.
_____ If your answer is not the same, try again until it is.
Cover, Copy, Compare
Skill Probe Generator:
http://www.lefthandlogic.com/mathprobe_ol
d/allmult.php
Share with your partner any
additional ideas you have for
building accuracy in a skill.
Instructional Focus Continuum
Instructional Strategies for Building Skill
Fluency
•
•
•
•
Multiple exposures to skill or concept
Goal setting for increased automaticity
Computer games
Peer games
Computer Resources for Math Skill
Fluency
• http://www.internet4classrooms.com/grade_level_h
elp.htm
• http://nlvm.usu.edu/en/nav/index.html
• http://www.gamequarium.org/dir/Gamequarium/M
ath/
• http://cte.jhu.edu/techacademy/web/2000/heal/ma
thsites.htm
Peer Math Games
• Math Fact WAR w/Flash cards
• Dot Game w/Flash cards or Dice
• Math Fact Bingo
Share with your partner any
additional ideas you have for
building fluency in a skill.
Instructional Focus Continuum
Instructional Strategies for Building Skill
Application
• Word Problem Solving
• Use questioning strategies that
require learners to go deeper
• Peer tutoring
Share with your partner any
additional ideas you have for
building application in a skill.
What to do?
•
Implement an effective core curriculum based on
–
Critical mathematics content (Common Core State Standards)
–
Research-based instructional design principles (EnVision Math)
• Ensure student understanding through high-quality
instruction using both student-centered and teachercentered strategies.
– Procedural Understanding/Skill Acquisition (Instructional
Focus Continuum)
– Conceptual Understanding (CRA model)
– Scaffolded Instruction (I do, We do, Ya’all do, You do)
•
Use reliable assessment tools
–
Regular formative assessment (District CFA’s)
–
Benchmark screening and progress monitoring (M-CBM’s)
~ Foundations for Success: The Final Report of
the National Mathematics Advisory Panel (2008).
Concrete-Representational-Abstract
Instructional Approach (C-R-A)
• CONCRETE: Uses hands-on physical
models or manipulatives to represent
numbers and unknowns.
• REPRESENTATIONAL: Draws or uses
pictorial representations of the models.
• ABSTRACT: Involves numbers as abstract
symbols of pictorial displays.
49
Concrete Level
• Definition: A teaching method that uses
actual objects such as people, shoes, toys,
fruits, cubes, base-ten blocks, or fraction
tiles.
• What concrete items have you used in your
classroom to teach math concepts?
Representational
Level
• Definition: A teaching method that uses
pictures, tally marks, diagrams, and drawings.
These pictorial representations relate directly
to the manipulatives and set up the student to
solve numeric problems without pictures.
• From your experiences, what have you used
that is representational in your math
classroom?
Abstract Level
• Definition: A teaching method that uses
written words, symbols (such as variables or
numerals), or verbal expressions.
Amber has 3 toy cars. If there are 4
wheels on each car, how many wheels
are there on her toy cars?
Concrete
Representational
Abstract
Amber has 3 toy cars. If there are 4
wheels on each car, how many wheels
are there on her toy cars?
Concrete
Representational
Abstract
With your
partner, come
up with 2- 3
different ways
you would
teach this
problem using
CONCRETE
objects
With your
partner, come up
with 2- 3
different ways
you would teach
this problem
using
REPRESENTATION
With your
partner, come
up with 2- 3
different ways
you would
teach this
problem using
ABSTRACT
symbols
What to do?
•
Implement an effective core curriculum based on
–
Critical mathematics content (Common Core State Standards)
–
Research-based instructional design principles (EnVision Math)
• Ensure student understanding through high-quality
instruction using both student-centered and teachercentered strategies.
– Procedural Understanding/Skill Acquisition (Instructional
Focus Continuum)
– Conceptual Understanding (CRA model)
– Scaffolded Instruction (I do, We do, Ya’all do, You do)
•
Use reliable assessment tools
–
Regular formative assessment (District CFA’s)
–
Benchmark screening and progress monitoring (M-CBM’s)
~ Foundations for Success: The Final Report of
the National Mathematics Advisory Panel (2008).
Scaffolded Instruction
Gradual Release of Responsibility
“I do - We do –Ya’all do- You do”
Explicit Instruction and Modeling
Guided Practice
Independent Practice
A Day in the Life of EnVision Math
The CORE and More
Lesson Checklist
• INSERT Core and more document here
What to do?
•
Implement an effective core curriculum based on
–
Critical mathematics content (Common Core State Standards)
–
Research-based instructional design principles (EnVision Math)
• Ensure student understanding through high-quality instruction using both
student-centered and teacher-centered strategies.
–
Procedural Understanding/Skill Acquisition (Instructional Focus Continuum)
–
Conceptual Understanding (CRA model)
–
Scaffolded Instruction (I do, We do, Ya’all do, You do)
• Use reliable ASSESSMENT tools
– Regular formative assessment (District CFA’s)
– Benchmark screening and progress monitoring
(M-CBM’s)
~ Foundations for Success: The Final Report of
the National Mathematics Advisory Panel (2008).
ASSESSMENT
Where do I START instruction? How do I EVALUATE student learning?
Screening Assessment:
Foundational skills
Identify children at risk
Diagnostic Assessment:
What deficits are impeding the development of proficiency?
Multi-faceted approach
Progress Monitoring (Formative) Assessment:
Curriculum-based
On-going (students at risk assessment more frequently)
Outcome (Summative) Assessment