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©STEM @ MSU 2012 – Math-in-Action Conference –
Grand Valley State University
Strengthening Tomorrow’s Education in Measurement (STEM) Project
Elementary Curricula’s Treatment of Volume and Capacity
Dan Clark, Danny Johns, Nic Gilbertson
STEM 2012 – Math-in-Action: Volume-Capacity
Session Overview
Background of STEM Project
Volume-Capacity tasks
A look at some textbook explanations
Connections to the Common Core
Questions & Discussion
STEM Project Overview (briefly) STEM: Strengthening Tomorrow’s Education in Measurement
Purpose: Produce a fine-grained analysis of the character and limitations of spatial measurement content in elementary mathematics curricula.
Close-up analysis of the spatial measurement of three elementary curricula
Choices: Everyday Math, Scott-Foresman/Addison Wesley, Saxon
What knowledge element is expressed; how expressed (textual form)
Knowledge categories: Conceptual, Procedural, Conventional
Textual Categories: Statements, Questions, Demos, Worked Examples, Problems
Result: Confirmation of Stephan & Clements (2003): Heavy procedural focus
Conceptual knowledge is weak (overall) and spotty (by element)
Heavily procedural for length (K-3, curricula X grade greater than 75% procedural)
Stronger for area (K-4 has 88% procedural overall )
Currently working on volume and capacity analysis
STEM 2012 – Math-in-Action: Volume-Capacity
Image from http://www.micapp.org retrieved 2-9-11
We work with…
Pre-service teachers at MSU and
33 Professional Development Facilitators
in 23 Regions across the state
STEM 2012 – Math-in-Action: Volume-Capacity
STEM 2012 – Math-in-Action: Volume-Capacity
The task…
With the tools available at your table, determine the volume/capacity of as many objects as you can.
As you are working, two things we would like you to think about (to discuss in 8-10 minutes)
1) What does a student have to know about measurement to be able to correctly find the volume/capacity of each object?
2) What were the benefits and limitations of the tool you had to use?
2-cm cubes, 1-cm cubes, rulers, graduated cylinders
STEM 2012 – Math-in-Action: Volume-Capacity
The task…
Were there any objects that were particularly problematic to measure in your groups?
Were you measuring volume, capacity, or both?
STEM 2012 – Math-in-Action: Volume-Capacity
The task…
Were you measuring volume, capacity, or both?
Sponge Colander Balloon
Three Different Distributions of Definitions
• SFAW: Defines volume and capacity most in second and third grade, then less so in fourth and fifth grade for a total of 33 times
• EM: Defines volume and capacity increasingly until a high in fifth grade for a total of 33 times, including four joint definitions
• Saxon: Only 6 total definitions of volume or capacity in grades K-5, including one joint definition
Typical Capacity Definitions
• “The amount something will hold” –Saxon 3
• “The amount a container can hold” –EM 2,3,5
• “The amount a container will hold” –SFAW 2,3,4
• “The amount of liquid that an object can hold” –SFAW 3,4,5
• “A measure of the amount of liquid or other substance a container can hold” –EM 4
Typical Volume Definitions
• “The amount of space a shape occupies” –Saxon 5
• “The number of unit cubes and fractions of unit cubes needed to fill the space taken up by the [3-dimensional] object” –EM 4
• “A measure of how much space an object takes up” –EM 4
• “The amount of space taken up by the figure” –SFAW 4
• “The number of cubic units needed to fill a solid figure” –SFAW 4
So...
• This is all pretty cut and dried...
• Well...
• Except for...
Blended Definitions
• “Capacity is a measure of an amount of liquid” –SFAW 5
• “Volume can also refer to how much goes inside a container” –SFAW 3
• “Capacity is a measure of the amount of space something occupies or contains” –EM 2
• “The volume of a container that holds liquids is often called its capacity” –EM 3
Joint Definitions
• “The amount of liquid a container will hold is called its ‘capacity’ or ‘volume’” –Saxon 3
• “Both volume and capacity are measures of the amount of space something occupies” –EM 3
• “Volume (or capacity) is the measure of the amount of space inside a 3-dimensional geometric figure” –EM 5
Those are the curricula of today
• How might the curricula of tomorrow differ?
Volume Occurrences in the CCSSM by Grade
• At what K-12 grade level(s) does the CCSSM concentrate most on volume?
Volume Occurrences in the CCSSM by Grade
0
2
4
6
8
10
12
14
16
18
20
K 1 2 3 4 5 6 7 8 High School
Volume in the CCSSM: Grades K-2
• No mention of volume
Grade K
• Decompose 2-D, 3-D
• "Background of Measurement"?
Grade 1 • Cubes
• Less guidance for 3-D
Grade 2
Volume in the CCSSM: Grades 3-5
• Fill space?
• Add, Subtract, Multiply, Divide problems using volumes in same unit
• No mention of cubic units
Grade 3
• No mention of Volume
• Volume units (L, mL) used
Grade 4 • Volume -> 3-D space
• Equivalence relations (formulas)
• V = l x w x h = B x h
Grade 5
Volume in the CCSSM: Grades 6-8
• Fractional side lengths
Grade 6
• "Real World" vs. "Mathematical”
Grade 7 • "Completing work
on Volume"?
• 3 new formulas
• Memorization?
Grade 8
Volume in the CCSSM: High School
• Proofs!
High School
• Cavalieri's Principle
High School • Density
• Physics
High School
Volume in the CCSSM: Summary
• Follows progression
• Shapes -> Measurement (Length/Area) -> Units of Volume -> Calculating volume -> More shapes -> Formulas -> Proofs
STORYLINE
• What volume is vs. How it is found
• Volume -> 3D space implicit until 5th grade
• Capacity never mentioned Concerns
Discussion Questions
• What’s your role as a teacher if you notice that a textbook definition is misguided or insufficient?
• How do you see this as being problematic with students?
Take Home Points
• Look in curricula materials across grades to see how they define topics
• Think together as a school how you want to achieve internal consistency
• The CCSSM provides a rough guide
– Some things are misplaced
– Capacity is never mentioned
Math in Action Conference
• We are presenting this at NCTM
• We welcome your feedback on how to make the presentation better
We want to thank the National Science
Foundation for funding this work
For more information :http://www.msu.edu/~stemproj
If you have any questions please e-mail us at: [email protected]
STEM 2012 – Math-in-Action: Volume-Capacity