strengthening early mathematics education5 math is clearly seen as a critical skill subject – both...

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Douglas H. Clements, Julie Sarama, University of Denver

Arthur J. Baroody, University of Illinois, Urbana-Champaign & DU

Strengthening Early Mathematics Education

© 2007, 2008, 2009, 2010, 2011, 2012, D. H. Clements, Ph.D., & Julie Sarama, Ph.D. All rights reserved. Do not use or duplicate without permission.

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© D. H. Clements. Do not use or duplicate without permission.© D. H. Clements. Do not use or duplicate without permission.

Math’s Predictive Power

Large-scale research, predicting school success(Duncan et al., 2007)

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Math Predicts

• Early math best predictor of elementary school success

• Math best predictor of high school graduation and college entry (Duncan et al., in press)

• Persistent math problems most reliable predictor

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Math is clearly seen as a critical skill subject – both in theory but also in a practical sense

!Both children and parents rate it as the most important school subject.

!Both see it as the subject most needed for a good job.

!Both see it as the subject they will use the most.

Source: PROMISE research, Phase 2, June 2010

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!Most kids (8 of 10) like school and like math (7 of 10) !Only 55% think they are excellent or very good at it !All three measure slip in 7-8th grade

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. . . And parents may be feeding into this: !Parents’ ratings of their children in math also decrease as they age.

Source: PROMISE research, Phase 2, June 2010

An Issue of Confidence

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!7Source: PROMISE research, Phase 2, June 2010

Some parents believe that certain children will simply “never get math

29%think math ability

is hereditary

39%say it’s okay for a child to say “I’m not good in

math”

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Opportunities needed

Our schools need to find ways to ensure the

brightest kids have opportunities to take

higher-level courses in areas like math and

science !

91%


© D. H. Clements. Do not use or duplicate without permission.

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Professional Development&

Best PracticesBoth involve:

• Promoting a conceptual understanding of the mathematics, especially the big ideas (e.g., for PD, comprehending the mathematics taught in pre-K-grade 8);

• Engaging learners in actual mathematical inquiry (e.g., problem solving, reasoning, justifying);

• Integrating/connecting instruction with other domains (e.g., for PD, integrating math content instruction with mathematical psychology and best instructional practices);

• ensuring instruction is engaging or motivating.

Professional Development and Best Practices

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Zork Odd and Even


Example:Zork Odd and Even

1. Provides a basis for understanding

A. zero is an even number and

B. the concept of even and odd numbers.

by relating these concepts to the big idea of equal partitioning (fair sharing):

an even number of items can be shared fairly by exactly two people.

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Equal partitioning also underlies variety of other concepts

• division—either a partitive meaning (An amount shared fairly among a certain number of people yields a share of what size for each person?) or quotitive division (An amount divided into equal shares of a particular size yields how many shares?);

• fractions—both a quotient meaning (e.g., 3/4 can be viewed as: Sharing three candy bars among four people, what is the size of each person’s share?) and a and part-whole meaning (e.g., 3/4 can be viewed as: What part of the whole candy bar is three of four equal size pieces?);

• measurement—a continuous quantity such as length or area can be subdivided into equal size parts or shares (units), which can then be counted;

• mean—to find what a typical share size (score) would be if everyone had the same size share, combine all shares (scores), and then divvy them up the total fairly among the number of people (number of scores).

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2. Zork Odd and Even could be used by PD trainers or teachers of children to involve learners in solving a genuine problem by simply posing such questions as: Is the game fair? Is there a winning strategy?

3 & 4. Addressing those questions could create a real need to collect and analyze data

or

addressing such questions as If we have 10 fingers and group by tens, what do you suppose they group by on planet Zork? or How would a Zorkian represent 4, 7, 8, 15, 16 with a written number? provides a basis for exploring and understanding grouping and place-value concepts and skills

—in an entertaining manner that promotes a positive disposition.

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Learning Trajectory for Counting

• Art’s Alison

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• Precounter Says number words, not sequence: "one, two, four…”.

• Chanter Says in sequence but may run together

• Reciter Verbal counting to 5, then 10

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• Corresponder Counts correctly using 1-1 correspondence, at least up to 5 objects in a line

• Counter (Small Numbers) Counts 1-5 objects in a line meaningfully (i.e., employ the cardinal rule)

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• Producer (Small Numbers) Counts out a collection up to 5

• Counter (10) Counts collections up to 10

• Counter and Producer (10+) and keeps track of unorganized collections

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• Counter from N Can count from any number; +1 and -1

• Counter On Using Patterns

• Counter on Keeping Track (includes hidden object tasks)

• Counter of Quantitative Units Counts unusual units, including “wholes” of broken objects

• Counter Forward and Back Counts embedded sequences, decades, etc.


Learning Trajectories: 3 Parts

1.Goal

2.Developmental Progression

3. Instructional Activities

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Geometry and Measurement

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• Which are rectangles?

• Slightly > 50%

• 4s were more likely toaccept the squares

• All accepted “long” quads w/ pair of parallel sides#3, 6, 10, and 14

• Properties less frequently

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1

2

3

4 5

68

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10

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14

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Rectangles

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Low SES 4-6Mid SES 6-12

Compare to Elem.—Rectangles

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What Children See

• Ring the triangles


A Trajectory for

Composing Geometric

Shapes

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Pre-Composer

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Picture Maker

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Shape Composer

What’s important? —Think, pair, share


Take Away

• All 3 components of LT’s

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Brief #3: Interventions

• What is a research-based intervention?

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TRIAD Scale-up Model

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• 167 classrooms in 3 states

• Schools publicly, randomly assigned to:

• TRIAD

• TRIAD-with Follow Through

• Control

Design


Control

TRIAD

Rasch scores

p < .0001

ES = .72

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Building Blocks in the News

• http://www.thebostonchannel.com/news/15776035/detail.html

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Ctrl-Non Ctrl-AA TRIAD-Non TRIAD-AA

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Follow Through


Control

TRIAD

TRIAD Follow Through

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ControlTRIAD

TRIAD Follow ThroughTo do better than this…

We need this…

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Relationship to Other Domains: Example—Language and Literacy

Does that area suffer?

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Assessments• Letter Recognition

• Renfrew Bus Story Assessment


Results

• Letter recognition: ns

• Bus Story, ns on:

• Sentence length

• Listening

• Story duration

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Results• Sig. higher for TRIAD on:

• Information .29

-1.10

-1.00

-0.90

-0.80

-0.70

-0.60

-0.50 Control Building Blocks

Information (Rasch) - OL Scores by Treatment Group - Beginning of K (Fall 2007), TRIAD II

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• Information .29

• Complexity .16

0.95 1.00 1.05 1.10 1.15 1.20 1.25

Control Building Blocks

Complexity (Raw) - OL Scores by Treatment Group - Beginning of K (Fall 2007), TRIAD II

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• Information .29

• Complexity .16

• Independence .36

36.00 36.50 37.00 37.50 38.00 38.50 39.00 39.50 40.00


Independence (Raw) - OL Scores by Treatment Group - Beginning of K (Fall 2007), TRIAD II



• Information .29

• Complexity .16

• Independence .36

• Inferential Questions .16

3.40

3.50

3.60

3.70

3.80

3.90

4.00


Inferential Reasoning: Q2 Practical (Raw) - OL Scores by Treatment Group - Beginning of K (Fall 2007), TRIAD II

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Is Early Math “Research-based”?

• You can not consider yourself child-centered without substantial mathematics

• The whole child

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Math and Play

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Math, Literacy, and Play

• Curricula focus lead to stronger emphasis in subject-matter

• Children in content-focused classrooms more likely to engage at high-quality level during free play

• Those focusing on both math and literacy more engaged at high level than neither or only one!


© D. H. Clements. Do not use or duplicate without permission.© D. H. Clements. Do not use or duplicate without permission.© D. H. Clements. Do not use or duplicate without permission.© D. H. Clements. Do not use or duplicate without permission.

NRC Recommendation #1

“A coordinated national early childhood mathematics initiative should be put in place to improve mathematicsteaching and learning forall children” (p. 3).

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Professional Development

• All teachers, children 0-8, need

• Pre-service: Minimum of 2 courses…

• Combining 3 components of LTs, and affect, as ‘Zork’ modeled

• Inservice: 50 to 75 hours

• Grounded in practice

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Professional Development• Performance assessment of teachers seeking

certification, including

• mathematical competencies (show two ways to add),

• interpreting children’s behaviors (e.g., diagnosing errors), and

• pedagogy (design and implement a research-based activity that addresses a key goal of the CCSS).

• States’ standards for research-based professional development

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Math Specialists

• Specialist just teaching math

• Team member specializing

• At least: Specializing coach


strengthening early mathematics education5 math is clearly seen as a critical skill subject – both...

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