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Linking 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

Contact Information

- clarkeb_at_uoregon.edu
- Phone
- (541) 342-8471
- Special thanks to David Chard, Scott Baker,

Russell Gersten, and Bethel School District

Warm 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?

A 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)

Numbers 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).

The Number 7

- Could be used to describe
- Time
- Temperature
- Length
- Count/Quantity
- Position
- Versatility makes number fundamental to how we

interact with the world

Mathematical 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)

Proficiency 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)

Despite 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)

Achievement 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.

Trajectories 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)

Developmental 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)

Discussion 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)?

Discussion 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?

The Ghost in the MachineNumber Sense

- Number sense is difficult to define but easy to

recognize - (Case, 1998)
- Nonetheless he defined it!

Case (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.

Cases 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

Number 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)

Math 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

Math 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.

Initial 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.

Purpose

- To analyze findings from experimental research

that was conducted in school settings to improve

mathematics achievement for students with

learning disabilities.

Identifying High Quality Instructional Research

Method

- 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).

RESULTSFeedback 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.

Feedback 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).

Feedback 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.

Goal 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?

Peer 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)

Curriculum 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

Key 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

Overview 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.

Big 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

Big 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

Five 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.

Big 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)

Counting

- 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

Counting (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

Counting (cont.)

- 5 Principles of Counting
- One to one correspondence
- Stable order principle
- Cardinal principle (critical)
- Item indifference
- Order indifference
- (Gelman Gallistel, 1978)

Cardinality

- 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)

From counting to addition

- Addition makes counting abstract
- Addition is counting sets
- 2 apples and 3 apples

The 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)

Development 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

Addition 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

Addition (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

Big Idea Effective Instruction

- Five key components of effective instruction are
- Big Ideas
- Conspicous strategies
- Review/Reteaching
- Scaffolding
- Integration

Mathematical 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.

Mathematics 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)

Conspicuous 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.

Time

Instructional 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.

Integration

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)

Review/Reteaching

Review must be

- sufficient to enable a student to perform the

task - without hesitation,

- distributed over time,

- cumulative with less complex information

integrated - into more complex tasks, and

- varied to illustrate the wide application of a

students - understanding of the information.

Discussion Point Effective Instructional

Principles

- How are effective instructional principles likely

to vary by student skill level - Big Ideas
- Conspicuous Strategies
- Scaffolding
- Integration
- Review/Reteach

Big 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.

Sequencing 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

Sequence of Instruction

1.

2.

4.

3.

1.

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?

1.

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?

1.

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?

1.

2.

4.

3.

3.

1.

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.

Scaffold the Instruction

Time

Lesson 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

Selection of examples

Selection of Examples

Which problems would be appropriate for

introducing students to addition?

What misrules might students make?

Selection 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.

Introduction to the Concept of Addition

Addition of Semi-concrete Representational Models

5 3 ?

The Min Strategy

5 3 ?

8

Missing Addend Addition

4 ? 6

Mental Math

5 1

1 2 3 4 5 6 7 8 9 10

Mental Math

5 1

1 2 3 4 5 6 7 8 9 10

Mental Math

5 2

1 2 3 4 5 6 7 8 9 10

Mental Math

5 2

1 2 3 4 5 6 7 8 9 10

Number Families

4 3

7

Fact Memorization

4 3

1 8

5 2

6 0

Lesson 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

Steps 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

Invented 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

Invented Strategies (cont.)

- Invented Strategies are flexible rather than

rigid - 7000 -25 traditional algorithm requires complex

steps

Exercise

- Work with a partner to come up with three ways

students could solve - 46 38 ?

Invented 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

Invented 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

Invented 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

Discussion 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.

Big 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.

Sequencing 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

Sequence of Instruction

1.

2.

4.

3.

1.

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?

1.

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

1.

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?

1.

2.

4.

3.

3.

1.

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.

Introduction to the Concept of Functions

Input 2

Output 6

Functions with increasingly complex operations

y x12

Functions to Ordered Pairs Ordered Pairs

to Functions

Math 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

Problem 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?

Reasoning/Problem Solving

Commutative Property of Addition

Knowledge Forms

Equality

Number families

Complex Strategies

Divergent

Rule Relationships

Knowledge Forms

Basic Concepts

Convergent (Conventions)

Facts/Associations

Solving 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.

Three Part Plan for Problem Solving Instruction

- Get students mentally ready to solve
- the problem. (Preteach)
- Be sure expectations are clear.

Before

Generic Problem Solving Strategy

Check for accuracy

Find/calculate the solution

Plan a strategy to solve the problem

Analyze the problem

Read the problem

Something Else?

Number Family Strategy

(No Transcript)

See 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?

Three types of information you can provide

students during problem solving

- Conventions (facts, symbols, reminders of rules)
- Alternative methods
- Clarification of student work

- Get students mentally ready to solve
- the problem. (Preteach)
- Be sure expectations are clear.

Before

Promoting 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?

Sample 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?

Possible Solution Strategies

Moving 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)

What 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

Designing 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)

Curriculum content

- Scope and sequence based on 4 integrate strands
- Numbers and operation
- Geometry
- Measurement
- Vocabulary

Curriculum 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

Structure

- 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

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