Early Math: We Need Better Math - NCCA · is important to engage them in ‘math talk’ –...

Douglas H. ClementsUniversity of Denver

Early Math:Surprisingly Important

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

We Need Better Math

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

What’s Surprising?

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Surprise #1: Math’s Predictive PowerLarge-scale research, predicting school success(Duncan et al., 2007)

4Early_Math_Research_(c)_Clements.key - November 25, 2014

Foundations for Success

“Most children acquire considerable knowledge of…mathematics before … kindergarten.

…related to their mathematics learning for years thereafter - even high school.”

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Young children can learn amazingly broad, complex, and sophisticated mathematics.

Surprise #2: Children’s Math Potential

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Preschool Potential

• Cory isputting 4 trianglestogetherto makesquares

Cory makes a new shape:A unit of units

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Another boy sees the square structure,butbuildswrongsquare


Finishing, Cory shows adult, who asks:“How many triangles did you use?”

Cory counts: “24”

“24 what?”

“Triangles.”

“How many squaresdo you have?”

Puts 4 fingers ontriangles in eachnew unit and countseach square: “6!”

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© Douglas H. Clements. Do not copy or use without permission.

• Reintroduction: 64 -28

• ”Now, I can’t take 8 from 4, so…”

-Kye, a 3rd grade boy, interrupted:

• Yes, you can! 8 from 4 is negative four 64 -28 -4

Third Grade Subtraction

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• “…and 20 from 60 is forty: 64 -28 -4 40

• …and negative four and 40 is 36” 64 -28 -4 40 36

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“In the curriculum, a view of all children as having the capacity to engage with deep and challenging mathematical ideas and processes from birth should be presented.”

— Primary Maths Reports…


“Mathematics is viewed not only as useful and as a way of thinking, seeing and organising the world, but also as aesthetic and worthy of pursuit in its own right (Zevenbergen, Dole, & Wright, 2004).

All children are viewed as having an ability to solve mathematical problems, make sense of the world using mathematics, and communicate their mathematical thinking.”

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A Surprise to Educators

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Surprise #3: Surprise to Educators

• What young children can learn is a surprise to most early/primary educators.

• Therefore, they do not challenge children or use formative assessment effectively (especially “ends”).


Surprise #4: Most Children Need a Math Intervention

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Not Just the Poor

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Equity

• Especially low-income children who hear about 1,500 number words a year

• Compare to middle-income: 93,000

• Middle hear 60 times as many words!

‘mathematics for all’ —

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Surprise #2 Revisited: Children’s Math Potential

Children invent mathematics…


How much is 9 take away 4?

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Why Not Just the Algorithm?

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Why Develop, Discuss, Use?

Promotesconcepts,

skills,problem solving,

creativity

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Difficult subtraction: 1000 - 359



Surprise #5: We Know a Lot

• About how children think about and learn math

• Learning trajectories

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Learning Trajectories: 3 Parts

1.Goal

2.Developmental Progression

3. Instructional Activities

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Scientific Approach to

Learning Trajectoriesweaves the 3

parts together

Mathematics Goal

InstructionalTasksDevelopmental

Progression


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Small Numbers and Counting

• Finger plays:

• One, two, buckle…

• When I was one…When I was one, I was so small, (hold up 1 finger)

I could not speak a word at all. (shake head)

When I was two, I learned to talk. (hold up 2 fingers)

I learned to sing, I learned to walk. (point to mouth and feet)

When I was three, I grew and grew. (hold up 3 fingers)

Now I am four and so are you! (hold up 4 fingers)

• Later: Five Little Monkeys, etc.


Number Jump

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https://phonics.opencourtonline.com/inttools/one_page/launchdemo.html

Joelleb, password recycle

Games

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• Count the dots and move that number of jumps

• Connecting different concepts of number

Road Race:Counting in Two “Worlds”

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• Count the sides of a shape and move that number of jumps

• Connecting new concepts of number

Road Race Shape Counting -

Another Variation

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• Choose the “better” of two numbers

• Comparing but also reasoning: Which is better in this case?

Space RaceNumber Choice

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Encourage counting on from

numeralAdd numerals

Addition “choice” game

Two-digit addition

Arithmetic Sequence



Surprise #5: We Know a Lot

About how to scale up teaching and learning with learning trajectories

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Control

TRIAD

Rasch scores

p < .0001

ES = 0.72

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“Of the assessment approaches available, formative assessment offers most promise for generating a rich picture of young children’s

mathematical learning (e.g., NCCA, 2009b; Carr & Lee, 2012).

Strong conceptual frameworks are important for supporting teachers’ formative assessments

(Carr & Lee, 2012; Ginsburg, 2009a; Sarama & Clements, 2009). ”

Learning Trajectories for Formative Assessment



Surprise #5a: Math + Play

Building mathematics knowledge does not require sacrificing play.

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Average percentage of minutes in which mathematical activity occurred:

Mathematical Activity in PreK Play

42%

Important: Only an intuitive foundation!

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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!

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Play with Ideas

• Regular play with blocks, puzzles, socio-dramatic play (with self-regulation), and

• Enhancement of math in that play, and

• Intentional, planned, math (LTs)...and

• Play with mathematics


Surprise #5b: Language and Literacy Do Not Suffer

• No difference on letter naming or 3 expressive language measures.

• Sig. higher for TRIAD on:

• Information

• Complexity

• Independence

• Inferential Questions

3.40

3.50

3.60

3.70

3.80

3.90

4.00

Control Building Blocks

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

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“In addition to introducing young children to mathematical vocabulary, it is important to engage them in ‘math

talk’ – conversations about their mathematical thinking and reasoning .”

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Building Blocks of Math

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Control

TRIAD

TRIAD Follow Through


ControlTRIAD

TRIAD Follow ThroughTo do better than this…

We need this…#5c: We Throw Gains Away

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Fall 06-07

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26

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Computer General Curriculum Small Group Whole Group

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Surprise #5d: Sustainability

0

13

26

39

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Fall 06-07 Spring 06-07 Spring 09

Computer General Curriculum Small Group Whole Group

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“Mathematics education should address the range of mathematical ideas that all children need to engage with. It should

not be limited to number.”


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


Substitution Composer

•Finds different ways to fill a frame, emphasizing substitution relationships.

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Super Shape Series

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Teachers’ Representations of

Learning Trajectories


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Teachers of Older Students• Herb Gross’ sites:

www.adjectivenounmath.comwww.lovemath.org

• http://www.wirelessgeneration.com/posters

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Early Math Surprises

1. Early math has surprising predictor power.

2. Young children have the potential to learn powerful math.

3. #2 is a surprise to most educators.

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Early Math Surprises

4. Most children need an intervention.

5. We know a lot. LT + Interventions

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Justin

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References• Sarama, J., & Clements, D. H. (2009). Early childhood

mathematics education research: Learning trajectories for young children. NY: Routledge.

• Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. NY: Routledge.

• Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333, 968-970.

• Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127-166.

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• Clements, D. H., & Sarama, J. (2007). Building Blocks Curriculum, Grade PreK. SRA/McGraw-Hill.

• Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443-494.

• Clements, D. H., Sarama, J., & Liu, X. (2008). Development of a measure of early mathematics achievement using the Rasch model: The Research-based Early Maths Assessment. Educational Psychology, 28(4), 457-482.

• Clements, D. H., Sarama, J., & Wolfe, C. B. (2011). TEAM—Tools for early assessment in mathematics. McGraw-Hill.


Early Math: We Need Better Math - NCCA · is important to engage them in ‘math talk’ –...

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