Title: Linking Mathematics Achievement to Successful Mathematics Learning throughout Elementary School: Cri
1Linking Mathematics Achievement to
Successful Mathematics Learning throughout
Elementary School Critical Big Ideas and Their
Instructional Applications Ben Clarke,
Ph.DPacific Institutes for ResearchFebruary 23,
2005
2Contact Information
- Email
- clarkeb_at_uoregon.edu
- Phone
- (541) 342-8471
- Special thanks to David Chard, Scott Baker,
Russell Gersten, and Bethel School District
3Warm up Two machines one job
- Rons Recycle Shop was started when Ron bought a
used paper-shredding machine. Business was good,
so Ron bought a new shredding machine. The old
machine could shred a truckload of paper in 4
hours. The new machine could shred the same
truckload in only 2 hours. How long will it take
to shred a truckload of paper if Ron runs both
shredders at the same time?
4A primary goal of schools is the development of
students with skills in mathematics
- Mathematics is a language that is used to express
relations between and among objects, events, and
times. The language of mathematics employs a set
of symbols and rules to express these relations.
(Howell, Fox, Morehead, 1993)
5Numbers are abstractions
- To criticize mathematics for its abstraction is
to miss the point entirely. Abstraction is what
makes mathematics work. If you concentrate too
closely on too limited an application of a
mathematical idea, you rob the mathematician of
his or her most important tools analogy,
generality, and simplicity (Stewart, 1989, p.
291) - The difficulty in teaching math is to make an
abstract idea concrete but not to make the
concrete interpretation the only understanding
the child has (i.e. generalization must be
incorporated).
6The Number 7
- Could be used to describe
- Time
- Temperature
- Length
- Count/Quantity
- Position
- Versatility makes number fundamental to how we
interact with the world
7Mathematical knowledge is fundamental to function
in society
- 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)
8Proficiency in mathematics is a vital skill in
todays changing global economy
- Many fields with the greatest rate of growth will
require workers skilled in mathematics.
(Bureau of Labor Statistics,1997) - Companies place a premium on basic mathematics
skill even in jobs not typically associated with
mathematics. - Individuals who are proficient in mathematics
earn 38 more than individuals who are not.
(Riley, 1997)
9Despite the efforts of educators many students
are not developing basic proficiency in
mathematics
- Only 21 of fourth grade students were classified
as at or above proficiency in mathematics, while
36 were classified as below basic. This pattern
was repeated for 8th and 12th grade (NAEP, 1996). - According to the TIMS (1998), US students perform
poorly compared to students in other countries.
United States 12th graders ranked 19th out 21
countries. - The result Students lack both the skills and
desire to do well in mathematics (MLSC, 2001)
10Achievement stability over time
- The inability to identify mathematics problems
early and use formative evaluation is problematic
given the stability of academic performance. - In reading, the probability of a poor reader in
Grade 1 being a poor reader in Grade 4 is .88
(Juel, 1988). - The stability of reading achievement over time
has led to the development of DIBELS.
11Trajectories The Predictions
- Students on a poor reading trajectory are at
risk for poor academic and behavioral outcomes in
school and beyond. - Students who start out on the right track tend to
stay on it.
(Good, Simmons, Smith, 1998)
12Developmental math research
- Acquisition of early mathematics serves as the
foundation for later math acquisitions (Ginsburg
Allardice, 1984) - Success or failure in early mathematics can
fundamentally alter a mathematics education
(Jordan, 1985)
13Discussion Point Trajectories
- Do math trajectories and reading trajectories
develop in the same way? - How could they be similar?
- How could they be different?
- Are they the same for different types of learners
(e.g. at-risk)?
14Discussion Point Number Sense
- What is number sense?
- What does number sense look like for the
grade/students you work with? - How does number sense change over time and what
differentiates those with and without number
sense over time?
15The Ghost in the MachineNumber Sense
- Number sense is difficult to define but easy to
recognize - (Case, 1998)
- Nonetheless he defined it!
16Case (1998) Definition
- Fluent, accurate estimation and judgment of
magnitude comparisons. - Flexibility when mentally computing.
- Ability to recognize unreasonable results.
- Ability to move among different representations
and to use the most appropriate representation.
17Cases Definition (cont.)
- Regarding fluent estimation and judgment
- of magnitude (i.e. rate and accuracy).
- Recent empirical support for this insight
- Landerl, Bevan, Butterworth
- (2004), 3rd grade
- Passolunghi Siegel (2004),
- 5th grade
18Number Sense
- A key aspect of various definitions of number
sense is flexibility. - a childs fluidity and flexibility with numbers,
the sense of what numbers mean, and an ability to
perform mental mathematics and to look at the
world and make comparisons - Students with number sense can use numbers in
multiple contexts in multiple ways to make
multiple mathematics decisions.
(Gerston Chard, 1999)
19Math LD
- 5 to 8 percent of students
- Basic numerical competencies are intact but
delayed - Number id
- Magnitude comparison
- Difficulty in fact retrieval
- Proposed as a basis for RTI and LD diagnosis
20Math LD (cont.)
- Working memory deficits hypothesized to underlie
fact retrieval difficulty - Students use less efficient strategies in
solving math problems due to memory deficits - Procedural deficits often combine with conceptual
misunderstanding to make solving more complex
problems difficult.
21Initial Comments about Mathematics Research
The knowledge base on documented effective
instructional practices in mathematics
is less developed than reading.
Mathematics instruction has been a concern to
U.S. educators since the 1950s, however,
research has lagged behind that of
reading.
Efforts to study mathematics and mathematics
disabilities has enjoyed increased interests
recently.
22Purpose
- To analyze findings from experimental research
that was conducted in school settings to improve
mathematics achievement for students with
learning disabilities.
23Identifying High Quality Instructional Research
24Method
- Included only studies using experimental or
quasi-experimental group designs. - Included only studies with LD or LD/ADHD samples
OR studies where LD was analyzed separately. - Only 26 studies met the criteria in a 20-year
period (Through 1998).
25RESULTSFeedback to Teachers on Student
Performance
- Seems much more effective for special educators
than general educators, though there is less
research for general educators. - May be that general education curriculum is often
too hard.
26Feedback to Teachers on Student Performance
(cont.)
- 3. Always better to provide data and
suggestions rather than only data profiles (e.g.,
textbook pages, examples, packets, ideas on
alternate strategies).
27Feedback to Students on Their Math Performance
- Just telling students they are right or wrong
without follow-up strategy is ineffective (2
studies). - Item-by-item feedback had a small effect (1
study). - Feedback on effort expended while students do
hard work (e.g., I notice how hard you are
working on this mathematics) has a moderate
effect on student performance.
28Goal Setting
- Studies that have used goal setting as an
independent variable, however, show effects that
have not been promising. - Fear of failure?
- Requires too much organizational skill?
29Peer Assisted Learning
- Largest effects for well-trained, older students
providing mathematics instruction to younger
students - Modest effect sizes (.12 and .29) were documented
for LD kids in PALS studies. These were
implemented by a wide range of elementary
teachers (general ed) with peer tutors who didnt
receive any specialized training (More recent
PALS data not included)
30Curriculum and Instruction
- Explicit teacher modeling, often accompanied by
student verbal rehearsal of steps and
applications - -moderately large effects
- Teaching students how to use visual
representations for problem solving - -moderate effects
31Key Aspects of Curriculum Findings
- The research, to date, shows that these
techniques work whether students do a lot of
independent generation of the think alouds or the
graphics or whether students are explicitly
taught specific strategies
32Overview of Findings
- Teacher modeling and student verbal rehearsal
remains phenomenally promising and tends to be
effective. - Feedback on effort is underutilized and the
effects are underestimated. - Cross-age tutoring seems to hold a lot of promise
as long as tutors are well trained. - Teaching students how to use visuals to solve
problems is beneficial. - Suggesting multiple representations would be
good.
33Big Ideas in Math Instruction
- Math instruction should build competency within
and across different strands of mathematics
proficiency - Math instruction should link informal
understanding to formal mathematics - Math instruction should be based on effective
instructional practices research
34Big Idea Five Strands of Mathematical Proficiency
- 1. Conceptual Understanding-comprehension of
mathematical concepts, operation, and relations - 2. Procedural Fluency-skill in carrying out
procedures flexibly, accurately, efficiently, and
appropriately - 3. Strategic Competence-ability to formulate,
represent, and solve mathematical problems
35Five Strands (cont.)
- 4. Adaptive Reasoning-capacity for logical
thought, reflection, explanation, and
justification - 5. Productive Disposition-habitual inclination to
see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence
and one's own efficacy.
36Big Idea Informal to Formal Mathematics
- Early math concepts are linked to informal
knowledge that a student brings to school
(Jordan, 1995) - Linking informal to formal math knowledge has
been a persistent theme in the mathematics
literature (Baroody, 1987)
37Counting
- Sequence words w/out reference to objects
- 1) through 20 is unstructured learned through
rote memorization - 2) students learn 1-9 repeated structure
- 3) students learn decade transitions
38Counting (cont.)
- Counting occurs when sequences words are assigned
to objects on a one to one basis - Counting first step in making quantitative
judgments about the world exact
39Counting (cont.)
- 5 Principles of Counting
- One to one correspondence
- Stable order principle
- Cardinal principle (critical)
- Item indifference
- Order indifference
- (Gelman Gallistel, 1978)
40Cardinality
- Developed around the age of 4 (Ginsburg Russel,
1981) - All or nothing phenomena(Permangent, 1982)
- Can be taught and focus children on seeing
individual items in terms or being part of a
larger unit (Fuson Hall, 1983)
41From counting to addition
- Addition makes counting abstract
- Addition is counting sets
- 2 apples and 3 apples
42The link to addition
- Count All starting with First addend (CAF)
- Count All starting with Larger addend (CAL)
- Count On from First addend (COF)
- Count on from Larger addend (COL)
43Development of early addition (cont.)
- Students first use CAF and COF supporting a
uniary view of addition (e.g. changing one
number) - CAL and COL supports a binary (I.e. combining two
number) view of addition - Based on principle of commutativity
- Students who understand communtativity can use
the COL strategy
44Addition Strategies
- COL strategy has been termed the Min strategy
because it requires the minimal amount of
counting steps to solve a problem - Recognized as the most cognitively efficient
45Addition (cont.)
- Some problems were solved quicker than expected
- Based on patterns such as doubles, tens
- Indicate development of number sense
- (Groen Parkman, 1972)
- Siegler (1982) hypothesized that use of the min
effect was the critical variable in 1st grade
math and failure to do so was predictive of later
failure in mathematics
46Big Idea Effective Instruction
- Five key components of effective instruction are
- Big Ideas
- Conspicous strategies
- Review/Reteaching
- Scaffolding
- Integration
47Mathematical concepts, operations, and strategies
that
- form the basis for further mathematical
learning.
- map to the content standards outlined by your
state.
- are sufficiently powerful to allow for broad
application.
48Mathematics Big Ideas
- Place Value Place a number holds gives
information about its value - Expanded Notation A number can be reduced to its
parts (e.g. 432 is 400 30 and 2) - Commutative property a b b a
- Equivalence Quantity to the left and right of
equal sign are equivalent 32 15 47 - Rate of composition/decomposition Rate is base
10 system is 10.
(Kamenui et al, 1998)
49Conspicuous Strategies
Expert actions for problem solving that are made
overt through teacher and peer modeling.
In selecting exemplars consider teaching
- the general case for which the strategy works.
- both how and when to apply a strategy.
- when the strategy doesnt work.
50Time
51Instructional Scaffolding includes
Sequencing instruction to avoid confusion of
similar concepts.
Carefully selecting and sequencing of examples.
Pre-teaching of prerequisite knowledge.
Ensuring mathematical proficiency when necessary.
Providing specific feedback on students efforts.
Offering ample opportunities for students to
discuss their approaches to problem solving.
52Integration
it is not necessary that students master place
value before they learn a multi-digit algorithm
the two can be developed in tandem. --(Mathemati
cs Learning Study Committee, 2002)
53Review/Reteaching
Review must be
- sufficient to enable a student to perform the
task - without hesitation,
- cumulative with less complex information
integrated - into more complex tasks, and
- varied to illustrate the wide application of a
students - understanding of the information.
54Discussion Point Effective Instructional
Principles
- How are effective instructional principles likely
to vary by student skill level - Big Ideas
- Conspicuous Strategies
- Scaffolding
- Integration
- Review/Reteach
55Big Idea - Addition
Plan and design instruction that
- Develops student understanding from concrete to
conceptual,
- Applies the research in effective math
instruction, and
- Scaffolds support from teacher ? peer ?
independent application.
56Sequencing Skills and Strategies
Concrete/ conceptual
Adding w/ manipulatives/fingers Adding w/
semi-concrete objects (lines or dots) Adding
using a number line Min strategy Missing addend
addition Addition number family facts Mental
addition (1, 2, 0) Addition fact memorization
Abstract
Abstract
57Sequence of Instruction
1.
2.
4.
3.
581.
2.
4.
3.
1. Teach prerequisite skills thoroughly.
6 3 ?
What are the prerequisite skills students need to
master to introduce adding single digit numbers?
591.
2.
4.
3.
2. Teach easier skills and strategies before
more difficult ones.
Vertical, horizontal, or mix?
Sums to 5, 10, or 18?
Adding any number 0-9 or any number 1-9?
601.
2.
4.
3.
3. Introduce strategies one at a time until
mastered. Separate strategies that are
potentially confusing.
As you teach students to add, when should you
introduce subtraction?
611.
2.
4.
3.
3.
621.
2.
4.
3.
4. Introduce new skills, strategies, and
applications over a period of time through a
series of lessons beginning with lots of
teacher modeling, guided practice,
integration, independent practice, and review.
63Scaffold the Instruction
Time
64Lesson Planning Addition with Manipulatives
Scaffolding
Teacher Monitored Independent
Independent (no teacher monitoring)
Guide Strategy
Strategy Integration
Model
Day 1 2 problems
Day 3 4 problems
Day 5 6 problems
Day 7 8 problems
Day 9 until accurate
Until fluent
65Selection of examples
Selection of Examples
Which problems would be appropriate for
introducing students to addition?
What misrules might students make?
66Selection of examples
2 5
6 2
1. Choose examples with single digit addends
and sums up to 10.
3 6 7 0
2. Write problems with a mix of larger and
smaller addends first.
3. Start with horizontal alignment, then
introduce vertical alignment, then mix.
4. Introduce with addends 1-9, then introduce
addends of 0.
67Introduction to the Concept of Addition
68Addition of Semi-concrete Representational Models
5 3 ?
69The Min Strategy
5 3 ?
8
70Missing Addend Addition
4 ? 6
71Mental Math
5 1
1 2 3 4 5 6 7 8 9 10
72Mental Math
5 1
1 2 3 4 5 6 7 8 9 10
73Mental Math
5 2
1 2 3 4 5 6 7 8 9 10
74Mental Math
5 2
1 2 3 4 5 6 7 8 9 10
75Number Families
4 3
7
76Fact Memorization
4 3
1 8
5 2
6 0
77Lesson Planning Addition with Manipulatives
Scaffolding
Teacher Monitored Independent
Independent (no teacher monitoring)
Guide Strategy
Strategy Integration
Model
Day 1 2 problems
Day 3 4 problems
Day 5 6 problems
Day 7 8 problems
Day 9 until accurate
Until fluent
78Steps in Building Computation Fluency (Van de
Walle 2004)
- Direct Modeling
- Counts by Ones ---- Ten Frames
- Invented Strategies
- Written Records
- Mental Math
- Traditional Algorithms
- Guided development
-
79Invented Strategies/Traditional Algorithms
- Van de Walle states
- Invented strategies are number orientated rather
than digit orientated - 45 32 focus is on 40 30 rather than 4 3
- Emphasis on place value
- Invented Strategies are left handed vs. right
handed - 26x47 start with 20x40
80Invented Strategies (cont.)
- Invented Strategies are flexible rather than
rigid - 7000 -25 traditional algorithm requires complex
steps
81Exercise
- Work with a partner to come up with three ways
students could solve - 46 38 ?
82Invented Strategies Addition examples 4638 ?
- Add Tens, Add Ones, Combine
- 40 and 30 is 70
- 6 and 8 is 14
- 70 and 14 is 84
- Add on Tens, then Add Ones
- 40 and 36 is 76
- Then add on 8, 76 and 4 is 80 and another 4 is 84
83Invented Strategies Addition examples 4638 ?
- Add Tens, Add Ones, Combine
- 40 and 30 is 70
- 6 and 8 is 14
- 70 and 14 is 84
- Add on Tens, then Add Ones
- 40 and 36 is 76
- Then add on 8, 76 and 4 is 80 and another 4 is 84
84Invented Strategies Addition examples 4638 ?
- Move some to make tens
- 2 from 46 put with 38 to make 40
- You have 44 and 40 more is 84
- Use a nice number and compensate
- 46 and 40 is 86
- I used 2 extra so 84
85Discussion Point Invented Strategies
- With a partner discuss
- What works about invented strategies
- How would you incorporate invented strategies
into your classroom - Would invented strategies work equally well for
all students.
86Big Idea - Functions
Plan and design instruction that
- Develops student understanding from concrete to
conceptual,
- Applies the research in effective math
instruction, and
- Scaffolds support from teacher ? peer ?
independent application.
87Sequencing Skills and Strategies
Concrete/ conceptual
Identify examples of simple functions (temperature
conversions, rate-kph) Distinguishing linear
from non-linear functions in graphs Determine
and graph ordered pairs from a given an
algebraic function Develop an algebraic function
given a table of ordered pairs Given a relevant
authentic problem develop and an graph an
algebraic function
Abstract
Abstract
88Sequence of Instruction
1.
2.
4.
3.
891.
2.
4.
3.
1. Teach prerequisite skills thoroughly.
f(x)2.54x
What are the prerequisite skills students need to
master to introduce functions and coordinate
geometry?
901.
2.
4.
3.
2. Teach easier skills and strategies before
more difficult ones.
Simple functions e.g. y3x
Ordered pairs that do not require complex
operations
Using numbers 0-9 then extending to gt10
911.
2.
4.
3.
3. Introduce strategies one at a time until
mastered. Separate strategies that are
potentially confusing.
As you teach students to solve functions, when
should you introduce graphing?
921.
2.
4.
3.
3.
931.
2.
4.
3.
4. Introduce new skills, strategies, and
applications over a period of time through a
series of lessons beginning with lots of
teacher modeling, guided practice,
integration, independent practice, and review.
94Introduction to the Concept of Functions
Input 2
Output 6
95Functions with increasingly complex operations
y x12
96Functions to Ordered Pairs Ordered Pairs
to Functions
97Math Lesson Planning for Graphing Functions
Day 3 problems
2 focus
1 focus
3 focus, 3 ord. pairs
Strategy Integration
Model Strategy
Guide Strategy
Ex (3 - focus 3 discrim.) y x f(x) 2
x f(x) 3x 3 (graphing sets of ordered
pairs)
Examples (1 - focus only) y 2x
3
Ex (2 - focus only) y 7 x f(x)
4x - 1
98Problem Solving
Problem solving is the selection and application
of known concepts or skills in a new or different
setting. For example When measuring a board of
lumber, what concepts and skills are involved?
99Reasoning/Problem Solving
Commutative Property of Addition
Knowledge Forms
Equality
Number families
100Complex Strategies
Divergent
Rule Relationships
Knowledge Forms
Basic Concepts
Convergent (Conventions)
Facts/Associations
101Solving Problems or Teaching through Problem
Solving?
- Problem Solving
- Places the focus on students sense making.
- Develops the belief in students that they are
mathematically capable. - Provides ongoing assessment data that can be
used for instructional decisions. - Develops mathematical power.
- Allows an entry point for a wide range of
students.
102Three Part Plan for Problem Solving Instruction
- Get students mentally ready to solve
- the problem. (Preteach)
- Be sure expectations are clear.
Before
103Generic Problem Solving Strategy
Check for accuracy
Find/calculate the solution
Plan a strategy to solve the problem
Analyze the problem
Read the problem
104Something Else?
105Number Family Strategy
106(No Transcript)
107See examples on pages in Problem Solving Chapter
(Stein et al.)
Matt had some money. Then he lost 14 dollars. Now
he has 2 dollars. How many dollars did he have
before he lost those dollars?
108Three types of information you can provide
students during problem solving
- Conventions (facts, symbols, reminders of rules)
- Alternative methods
- Clarification of student work
109- Get students mentally ready to solve
- the problem. (Preteach)
- Be sure expectations are clear.
Before
110Promoting Mathematical Discourse
Why do you think your solution is correct and
makes sense?
How did you solve the problem?
Why did you solve it that way?
111Sample Problem
Miss Spider is hosting a tea party for her 3
insect friends. If she wants each friend to have
two cookies with their tea, how many cookies will
she need to make?
112Possible Solution Strategies
113Moving Back to Instruction
- Children enter school with a base of math
knowledge and the ability to interact with number
and quantity. Very context dependent. (6) - Instruction in math is based on the interactions
between student, teacher, and content. Students
must link informal knowledge with formal often
abstract knowledge.(9)
114What to do
- Three year grant to develop and refine
Kindergarten math curriculum - Y1 Intervention development and refinement
Measurement refinement and validation - Y2 Intervention efficacy Implementation
analysis and hypothesis development - Y3 Hypothesis testing re differential
effectiveness of intervention
115Designing Interventions
- What to do?
- Few teachers have instructional strategies in
mathematics (Ma, 1999) - Lack of evidence regarding effects of mathematics
reforms (Heibert Wearne, 1993) - Few experimental studies examining specific
instructional practices (Gersten, Chard, Baker,
2000)
116Curriculum content
- Scope and sequence based on 4 integrate strands
- Numbers and operation
- Geometry
- Measurement
- Vocabulary
117Curriculum Content (cont.)
- Key goals
- Building conceptual understanding to abstract
reasoning via mathematical models - Building math related vocabulary
- Procedural fluency/automaticity
- Building competence in problem solving
118Structure
- Lessons sequenced in sets of 5
- Designed for whole class delivery in 20 minutes
- Culminates with group problem solving activity
which integrates math discourse with strands
taught during the previous 4 lessons