Mathematica Revisited

Mathematica Revisited

Redesigning a Math Education Museum Exhibit

Nancy Daniels, Karen Jia, Nina Weber, Jim VanidesStanford University Learning Design and TechnologyNovember 2001

Overview

ProposalMuseum ObservationsLearning TheoryExhibit Redesign and ExtensionsUser FeedbackNext StepsPersonal Reflections

Proposal

Initial Proposal: “The creation of the MathMania Discovery Center where interactive hands-on learning experiences are explored in a space that cannot be duplicated in a classroom setting or software environment.

Modified: “Redesign one of the Mathematica exhibits to improve its appeal and learning impact”

Mathematica Observation at the Exploratorium

WalkthroughDiscussion- selection of Multiplication Cube as focus of redesignVisitor Observation / Photographs / Field notesTeam discussion: Where’s the Learning?Problems with the existing designBrainstorm ideas for redesign

The Multiplication Cube

Mathematica Observations

-- Children usually need guidance by adult. No scaffolding available for just kids.

-- Kids can’t see over the controls

-- Lighting sequence doesn’t map to thecontrol input

--Position of the controls not optimalfor learning

Mathematica Observations

--Only one child was observed reading the surrounding notes. They provide some algebraic understanding but very hard to access.

-- Bulbs turn offtoo fast for young children to count

Learning Theory: How do we get them to know?

V=b•h•lIs that all?

From the Mathematica exhibit

What Should Learning Be?

interactive.

multi-sensory.

accessible.

built on prior experience.

conveyed through multiple representations.

beyond just knowing a formula.

The Challenge

Language of math can be confusing. 2 +3=5 but 1/2 + 1/3= 5/6

So much of math, relies on a solid foundation. If you miss one part, you may not get the next point.

There are many constraints in the classroom (materials, time, # of students, teaching philosophies of parents & of teachers)

“Assessment results are only estimates of what a person knows and can do.”--Knowing What Students Know, p2

My experience in the classroom

Some students just don’t get what we’re teaching

Some students really get the essence of what we’re teaching

Some students get a little bit of what we’re teaching

“ Learning entails the transformation of naïve understanding into more complete and accurate comprehension…” Knowing What Students Know, p4.

How can we tell a student knows?

A variety of assessments

Observations

Confusion

Boredom

Extending the activities

Misbehavior

“Not all children learn in the same way and follow the same paths to competence.”

--Knowing What Students Know, p. 4.

“Our knowledge can’t be a representation of that real world, but only as a key that unlocks possible paths for us.”

--Ernst von Glaserfeld, p.194.

“Constructivist theories assume that knowledge is actively constructed and reconstructed by the learner out of his or her experiences in the world.”

--Minds in Play, p.10.

Inspiration: Research and Theories

Ground the groundless. Make connections.

Build on what students know. Learning is recursive.

Vary the interactions by creating a multi-sensory experience: touch, hear, and see.

Think clearly about the goals before getting started. What do you want them to know?

Learning Theory Highlights

References

Knowing What Students Know: The Science and Design of Educational Assessment, Washington, DC: National Academy Press,2001.Kafai, Yasmin. Minds in Play: Computer Game Design as a Context for Children’s Learning, New Jersey: Lawrence Erlbaum Associates, Publishers, 1995.Klein, Elisa L. “Computer Graphics, Visual Imagery, and Spatial Thought” in New Directions for Child Development: Children and Computers, ed. E. Klein. Jossey-Bass Inc., 1985. Novak, Joseph D. and Gowin, D. Bob. Learning how to learn. Cambridge University Press, 1984.

References

Papert, Seymour. The Children’s Machine: Rethinking School in the Age of the Computer. New York: Basic Books, 1993.Von Glaserfeld, Ernst. The Construction of Knowledge: Contributions to Conceptual Semantics.Seeing FractionsThe Visual Arts and Early Childhood Learning, ed. C.M. Thompson. The National Art Education Association. 1995.www.exploratorium.edu

Exhibit Redesign: The Multiplication Cube

Exhibit LayoutControl Panel UIMuseum and Classroom Extensions

New Exhibit LayoutReorient Exhibit to facilitate visibility

Move control panel to corner

Lighting sequence one dimension at a time

Eliminate graphic panels

Add overhead algebraic display

Current Control Panel

Multiplication Machine

It will multiply the numbers you choose from these three dimensions of numbers.

1) press only one button in each direction2) press the start button to multiply

x x =

1

2

3

4

5

6

7

8

2345678

23

45

67

8

STARTSTART




1 x 4 1x =

1

2

3

4

5

6

7

8

2345678

23

45

67

8

STARTSTART




3 x 4 1x =

1

2

3

4

5

6

7

8

2345678

23

45

67

8

STARTSTART




STARTSTART 3 x 4 5x =

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 3 x 4 5x =

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 1 = 1

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 2 = 2

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 3 = 3

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 4 = 4

1

2

3

4

5

6

7

8

2345678

23

45

67

8




Watch!Watch! 4 = 4

1

2

3

4

5

6

7

8

2345678

23

45

67

8

1 x




Watch!Watch! 4 = 8

1

2

3

4

5

6

7

8

2345678

23

45

67

8

2 x




Watch!Watch! 4 = 12

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x




Watch!Watch! 4 = 12

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x 1x




Watch!Watch! 4 = 24

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x 2x




Watch!Watch! 4 = 36

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x 3x




Watch!Watch! 4 = 48

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x 4x




Watch!Watch! 4 = 60

1

2

3

4

5

6

7

8

2345678

23

45

67

8

3 x 5x

Museum and Classroom Extensions

Multiplication Activities (touchscreen)Multiplayer Multi-CubeCube City (classroom)Online Fraction Game

Cube Extension Activities on Touch Screen Panels

Real world application contentMultiplication in contextVisible problem solving process and scaffolding Downloadable simulationsExamples:

Multiplication in Life

Name: Rio Kenny Brian Carmen Sally Carlos Jenny # of P: 1 2 3 4 5 6 7

You’re throwing a big birthday party, and have invited seven friends. All of them are taking some additional people with them except for Rio, who will be coming by herself. The number below the name is the total number of people coming that each friend represents. How many people will be coming?

Multiplication in Life

Name: Rio Kenny Brian Carmen Sally Carlos Jenny # of P: 1 2 3 4 5 6 7

1 + 7 = 8 2 + 6 = 8

Hint Hide Hint

Multiplication in Work

Claire’s family lives in the residential area where the school bus makes the last stop. The area is 10 miles from the school. It’s 7:00 in the morning, Claire wants to feed her dog before she leaves, but her mom is rushing her off: “Class begins at 7:30, you’ll be late!” The bus drives at ½ mile per minute. Will Claire really be late?

Multiplication in Work

Step 1: How far can the bus drive in 30 minutes => 0.5mile/min * 30min = 15 miles Step 2: How far does Claire live from school => 10 miles

Step 3: The amount of distance left over => 15miles – 10miles = 5 miles Step 4: The amount of time left over => 5miles/.5mile/min = 10 minutes

Hint Hide Hint

Multiplication in Community

You’re helping the city planning the green area between two administrative buildings. There are three types of trees you can plant. The big one takes 10 square feet per tree, the medium one takes 5 square feet per tree, and the small one takes 3 square feet per tree. The area to be covered is 100 square feet, and each size of tree has to be planted at least once. How can you plant the trees so that two kinds of trees get to planted only once?

Multiplication in Community

Step 1: If the large tree and medium tree get planted only once, then the remaining area is 100 – 10 – 5 = 85 Step 2: The maximum number of small trees that can be planted in the remaining area 85/3 = 28 1/3 => 28 Step 3: If the medium tree and small tree get planted only once, then the remaining area is 100 – 5 – 3 = 92 Step 4: The maximum number of large trees that can be planted in the remaining area 92/10 = 9 1/5 => 9 Step 5: If the large tree and small tree get planted only once, then the remaining area is 100 – 10 – 3 = 87 Step 6: The maximum number of large trees that can be planted in the remaining area 87/5 = 17 2/5 => 17

Hide HintHint

Multiplayer Multiplication Cube - Objectives

Increase Mathematical Discourse and collaboration at the exhibitIncrease the cognitive challenge by providing an alternative game that works backwards from a target numberEncourage mathematical inquiry


Player 1

x x =

1

2

3

4

5

6

7

8

It will multiply the numbers you and your team-mates choose from these three dimensions of numbers.

Choose which game you would like to play: “Free Explore” or “Find-a-Number”

Press one button from your dimension, while your team-mates pick their own numbers from their dimensions.

Free Explore

Find-a-Number


Player 1

x x =

1

2

3

4

5

6

7

8

Work with your team to find three numbers, that when multiplied together, equal the Challenge Number below:

Find-a-Total

60

4 = 4

1

2

3

4

5

6

7

8

1 x 1x


Player 1Work with your team to find three numbers, that when multiplied together, equal the Challenge Number below:

Find-a-Total

60

4 = 4

12

34

56

78

1 x 1x



Find-a-Total

60

4 = 12

12

34

56

78

3 x 1x



Find-a-Total

60

4 = 12

12345678

3 x 1x



Find-a-Total

60

4 = 60

12345678

3 x 5x



Play Again!

60

Cube City

Teacher Introduction/DemoHands-On ActivityClassroom Sharing and LearningSharing Beyond the ClassroomStandards Alignment

Image to be added here

Online Fraction Game - Objectives

Help users see that fractions are related to multiplicationBuild a stronger conceptual understanding of the meaning of a fraction through multiple representations (grid, numeric, bar)Model the simplification of fractions

Fraction Fun!There are two ways to play. Choose one of the buttons below:

press buttons to build a fractionpress buttons to build a fraction

choose numbers tobuild a fraction


64

Fraction Fun!

0

Move the hand up or down to select the numerator…

Start Over…Start Over…

64

Fraction Fun!

1



64

Fraction Fun!

12



64

3

16

=

Fraction Fun!There are two ways to play. Choose one of the buttons below:

press buttons to build a fractionpress buttons to build a fraction



64

Fraction Fun!

0

64

Press as many buttons as you would like, and see howthe pattern represents a fraction…


# of lights“on”

out of atotal of…

Fraction Fun!

1

64



# of lights“on”

out of atotal of…

Fraction Fun!

2

64



# of lights“on”

out of atotal of…

1

32

=

Fraction Fun!

3

64



# of lights“on”

out of atotal of…

Fraction Fun!

4

64



# of lights“on”

out of atotal of…

1

16

=

User Feedback

.. "Some younger kids might see the interface design as a hexagon with an upside down peace symbol, not a representation of a cube" => Consider using a 3D display!"Too much to watch... break it down into easier to see portions that go slower"It wasn't immediately obvious that the exhibit was about volume. "It's a lousy way to teach someone how to multiply." => Change the name of the exhibit from Multiplication Cube to Volume Visualizer, or something that explicitly states that this is about VOLUME

Next Steps

Incorporate additional user feedback (students & teachers)Debrief with ExploratoriumSome of us will continue into Masters’ Project

Personal Reflections

Karen “A good learning design should start with the learner and end with the learner. ”

Nancy “Having Mathematica in San Francisco now is a wonderful opportunity. I originally studied it as a graphic designer, but now I could study it much deeper, as a learning designer. And working with a team offered multiple perspectives”

Jim “Teamwork is great when it works like this… we were all synchronized, speaking the same language of learning design…”

Nina “Constant, relentless reflection and research were essential to help us better address our learning problem and enhance our learning design.”

Mathematica Revisited

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