©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: stemproj@msu.edu

STEM 2012 – Math-in-Action: Volume-Capacity