Moving towards understanding: Reasoning about graphs in primary mathematics education

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Moving towards understanding:

Reasoning about graphs in primary mathematics education

Carolien Duijzer

Supervisors and co-supervisors:

Marja van den Heuvel-Panhuizen Paul Leseman

Michiel Veldhuis Michiel Doorman

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Chapter 2 (Study 1)

Micro-

development Macro-

development Chapter 3

(Study 2) Chapter 4

(Study 2) Intervention

Reasoning about motion graphs, through physical experiences

(i.e., embodied learning environment)

Graphing motion

Chapter 5 (Study 3) Domain of linear

equations

Early algebra

Review Embodied learning

environments

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Part I

Theoretical framing

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Question 1

Embodied learning environment supporting students’ understanding of graphing motion, what is reported on:

1. Embodied configuration?

2. Presumed factors mediating learning?

3. Relationship between configuration and the factors mediating learning?

4. Efficacy of embodied learning environments.

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Embodied cognition theories differ in how they conceptualize the relationship between (lower- order) sensorimotor processes and (higher-order)

abstract cognitive processes:

simple to radical embodiment

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simple to radical embodiment?

MODERATE MODERATE

RADICAL SIMPLE

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Operationalizing embodied learning environments for graphing motion

A. Indirectly and Directly involving the physical body

• From observing and influencing other (human) movements to making movements oneself.

Levels of bodily involvement

B. Immediate and Non-immediate cognitive activities

• the immediacy of the embodiment of cognitive activities can differ between learning situations

Levels of immediacy

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• Observing and influencing others/objects’ motion in which bodily involvement takes place in the absence of direct environmental stimuli

• Whole and part bodily motion in which bodily involvement takes place in the absence of direct environmental stimuli

• Observing and influencing others/objects’ motion in which bodily involvement takes place in the presence of direct environmental stimuli

• Whole and part bodily motion in which bodily involvement takes place in the presence of direct environmental stimuli

Bodily involvement

Motor execution Motor mirroring

Immediacy Reactivated enactmentDirect enactment

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Immediate Non-immediate

Own motion Others/objects’ motion Own motion Others/objects’ motion

Looking at or observing others/objects’

motion Influencing

others/objects’

motion Whole bodily

motion Part bodily motion

Looking at or observing others/objects’

motion Influencing

others/objects’

motion Whole bodily

motion Part bodily motion

(n=26)^a (n=8)^b (n=4)^c (n=8)^d (n=3)^e (n=1)^f (n=6)^g (n=6)^h

Class I Class II Class III Class IV

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Mediating factors within embodied learning environments

• Real-world context Referring to experiences of students with or in the real world

• Multimodality Referring to intertwining modalities

• Linking motion to graph Referring to linking motion to the graphical representation

• Multiple representations Referring to multiple representations of a particular motion

• Semiotics Referring to the use of meaning-supported sign systems

• Student control Referring to students being in control of the learning activity

• Attention capturing Referring to aspects in the learning environment that captures students attention

• Cognitive conflict Referring to conflicting conceptions

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0 5 10 15 20 25 30 35 40

Immediate Own Motion (n=34)

Number of mediators in percentages

0 5 10 15 20 25 30 35 40

Immediate Others/Objects' Motion (n=12)

0 5 10 15 20 25 30 35 40

Non-immediate Own Motion (n=4)

0 5 10 15 20 25 30 35 40

Non-immediate Others/Objects' Motion (n=12)

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Question 1

Embodied learning environment supporting students’ understanding of graphing motion, what is reported on:

1. Embodied configuration?

2. Presumed factors mediating learning?

3. Relationship between configuration and the factors mediating learning?

4. Efficacy of embodied learning environments.

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Room for discussion I

How can we define the role of conceptual metaphors (or the building of conceptual metaphor) across embodied learning environments (differing on their embodied configuration)? Think for example about:

• Observing and influencing others/objects’ motion in which bodily involvement takes place in the absence of direct environmental stimuli

• Whole and part bodily motion in which bodily involvement takes place in the absence of direct environmental stimuli

• Observing and influencing others/objects’ motion in which bodily involvement takes place in the presence of direct environmental stimuli

• Whole and part bodily motion in which bodily involvement takes place in the presence of direct environmental stimuli

Bodily involvement

Motor execution Motor mirroring

Immediacy Reactivated enactmentDirect enactment

A. Learning environments providing a direct physical link with relevant source-domain (embodied) experiences versus learning environments that provide an indirect physical link with relevant source-domain (embodied) experiences.

B. Conceptual metaphors that are explicit and active versus conceptual metaphors that are implicit and internal

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Room for discussion II

In the review we did not distinguish between observing the movement of a human model, and observing the movement of an object, even though some studies have suggested this is a relevant aspect to consider.

A. Mirroring hypothesis

B. Action repertoire (human versus object)

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Part II-A

The potential of an embodied learning environment – consisting of a six-lesson teaching sequence – to support students’ HOT, as their

reasoning about graphing motion

15|11 Math seminar – Nov 2020

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16|11

Understanding graphs describing dynamic situations (e.g., distance changing over time) is a core goal of mathematics education

Developing a “graph sense”  Higher-order thinking

“looking at the entire graph (or parts of it) and gaining meaning about the relationship between the two variables and, in particular, their pattern of co-variation”

(Leinhardt et al., 1990)

Some empirical evidence that students are able to reason with, and construct (graphical) representations of dynamic data:

• Inventing graphical representations (of motion) (DiSessa et al., 1991)

• Reasoning about graphical representations generated by motion sensors (Nemirovsky & Tierney, 1998)

• Reasoning about discrete and continuous change (deBeer et al., 2015)

Introduction

(Duijzer et al., 2018) Math seminar – Nov 2020

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Math seminar – Nov 2020 17|11

Teaching sequence

• Goal: Understanding of graphs describing motion (i.e., time-distance graphs)

• 6 lessons

• Embodied learning environment

Participants:

• 70 fifth-grade students (about 10/11 years)

The current study

Students move in front of a motion sensor, graphing their movement

(Duijzer et al., 2018)

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• Activities in which students create time-

distance graphs describing their own movements

• Coupling action, perception and reasoning

m

Alignment with embodied cognition theory

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Motion sensor technology

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‘This is an example of a graph, can you try to

walk this graph?’

Example: Bas (7 years old)

Math seminar – Nov 2020

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‘This is an example of a graph, can you try to

walk this graph?’

Example: Timon (11 years old)

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Question 2

How does students’ reasoning about graphing motion develop over a six-lesson teaching sequence within an embodied learning

environment?

Micro-development over the teaching sequence

Focus:

A. Changes over time in students’ level of reasoning when interpreting and constructing graphs.

B. Relation between students’ perceptual motor experiences and their reasoning about the graphs.

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Teaching sequence

Task 1

Task 2 Task 2

Task 3 Task 3

Task 4 Task 4

Task 5 Task 5

Task 6 Lesson-specific

graph interpretation

tasks

LESSON 1

Motion: o

reflecting and representing

LESSON 2

From discrete too

continuous representations

of change

LESSON 3

Continuouso

graphs of

‘distance to’ (1)

LESSON 4

Continuouso

graphs of

‘distance to’ (2)

LESSON 5

Scaling on theo

graphs’ axes

LESSON 6

Multiple o

movements and their graphical representation

From home to school

DISCRETE

GRAPHS CONTINUOUS

GRAPHS INFORMAL

GRAPHS

(individual) (individual) (individual)

(collective) (collective)

Task 6

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1

3

2 3 3 3

Graph construction Graph interpretation Lesson-specific testitems (micro)

Example: Example:

Assessing the students’ ability to (1) represent, (2) interpret and (3) reason about graphical representations

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Lesson-specific testitems (micro)

Assessing the students’ ability to (1) represent, (2) interpret and (3) reason about graphical representations

1 2 3 4 5 6

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Example 1: reasoning on the tasks

[YES] “The ferris wheel makes a

turn, and the line is slanted”

[NO] “The boat goes straight over the water, and does not go

up sidewards.”

[YES]“The hot air balloon goes

up sidewards, just as the line”

(Duijzer et al., 2019)

Lesson 2

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[YES] “The distance becomes bigger,

just as with the hot air balloon, and that is also represented in

the graphs.”

Example 1: reasoning on the tasks

[YES] “He goes straigt up, so

the distance becomes larger,

which is also shown in the

graph.”

[NO] “The ferris wheel makes a

turn, so the distance has to become bigger

and smaller, small bumps [in

the graph]”

Lesson 3

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R0 Unrelated reasoning

“Because you cannot make turns”

R1 Iconic reasoning

“In the graph it goes up and here the airplane also goes up”

R2 Single variable reasoning

“The boat moves forwards and the graph as well, if the graph goes up it means you go forward.”

R3 Dual variable reasoning

“Yes, because he travelled a ‘certain amount of distance’ within a certain amount of time”

Coding students’ levels of reasoning

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RQ 1: Development over time

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0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Task 1

Lesson 1 Task 3

Lesson 2 Task 3

Lesson 3 Task 5

Lesson 4 Task 5 Lesson 5

Percentage of students

Graph interpretation

Students’ level of reasoning (N^{Lesson 1}=67, N^{Lesson 2}=63, N^{Lesson 3}=67, N^{Lesson 4}=69, N^{Lesson 5}=65, N^{Lesson 6}=64)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Task 2

Lesson 1 Task 2

Lesson 2 Task 4

Lesson 3 Task 4

Lesson 4 Task 6

Lesson 5 Task 6 Lesson 6

Percentage of students

Graph construction

43 63

R0R1 R2R3

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Perceptual motor experiences and students’

reasoning about graphs

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Celine: “He is making them bigger.”

Celine: “They have to be closer together”

Teacher: “How could we make the graphs more similar?”

Celine: “A little faster…a little faster and a slightly shorter distance?”

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Question 2

How does students’ reasoning about graphing motion develop over a six-lesson teaching sequence within an embodied learning

environment?

• Reasoning became more sophisticated

(i.e., iconic reasoning  multiple variable reasoning)

• Students’ reasoning about the graphical representation of the motion was stimulated and strengthened by:

- Perceptual-motor experiences (provided by the motion sensor)

- Working together in groups and looking at each others’ movements

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Part II-B

The potential of an embodied learning environment – consisting of a six-lesson teaching sequence – to support students’ HOT, as their

reasoning about graphing motion

31|11 Math seminar – Nov 2020

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Question 3

To what extent does embodied support in a six-lesson teaching

sequence on graphing motion affect the development of students’

graphical reasoning?

Macro-development over the school year

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Condition 1

Indirect embodied support (on paper – digital blackboard)

Condition 2

Direct embodied support (physical – digital blackboard)

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Embodied condition Non-embodied condition Conditions: Differences & similarities

1. Embodied tasks (lesson 1) 2. Worksheet model:

3. Embodied motion sensor tasks (lessons 2-6)

1. Disembodied tasks (lesson 1) 2. Worksheet model:

3. Paper and pencil (worksheet) tasks/Digibord tasks (lessons 2-6)

Experimental set-up:

- Two parallel conditions: same content and tasks - Different tools

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Embodiment setting:

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Taking into account students’ level of reasoning

Four categories:

R0 No reasoning

R1 Iconic reasoning

R2 Single-variable reasoning R3 Multiple-variable reasoning

36|10

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Geen redeneren

[R0] Iconisch

redeneren [R1] Redeneren met 1

variabele [R2] Redeneren met meerdere

variabelen [R3]

It was a guess [F] Because this is the highest point

[B, C] Between point B and C, there the line goes straight up, and he travels 8 kilometer

[B,C] The car travels 8 kilometer in 5

minutes and that is the longest distance in the shortest time period

4 8 12 16 20 24

Distance (in kilometers)

0 10 20 30

Time of travelling (in minutes)

B A

C

D E

F

Between which points does the car travels

fastest?

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Geen redeneren

[R0] Iconisch

redeneren [R1] Redeneren met 1

variabele [R2] Redeneren met meerdere

variabelen [R3]

20 40 60 80 100 120

10:00u

Time

Distance(in kilometers)

11:00u 12:00u 13:00u

3a. Draw a graph that fits the description above.

3b. How do you know?

Score: correct (1), incorrect (0)

A train ride.

A train travels twice as fast between 10:00 and 11:00 o’clock than between 11:00 and

12:00 o’clock. The train stands still from 12:00 to 13:00 o’clock.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 2 3 4 5 6

Proportion use per occasion

Virtual Measurement Occasion R0R1 R2R3

Indirect embodied support

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 2 3 4 5 6

Virtual Measurement Occasion

R0R1 R2R3

Direct embodied support

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 2 3 4 5 6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 2 3 4 5 6

Indirect embodied support Direct embodied support

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Example: graph construction

Student 1

Pre-measure

A train ride.

A train travels twice as fast between 10:00 and 11:00 o’clock than between 11:00 and 12:00 o’clock. The train stands still from 12:00 to 13:00 o’clock.

3a. Draw a graph that fits the description above.

3b. How do you know?

Score: correct (1), incorrect (0)

20 40 60 80 100 120

10:00u

Time

Distance(in kilometers)

11:00u 12:00u 13:00u

Student 1

Post-measure

R1

R3

“Guessed” “Standing still = straight line

between 10:00 and 11:00 it must take him less time so a

steep slope

and between 11:00 and 12:00 it must take him longer so in between a slanted and straight

line”

42|10 (Duijzer et al., 2020)

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Question 3

To what extent does embodied support in a six-lesson teaching

sequence on graphing motion affect the development of students’

graphical reasoning?

• Strong effect of the intervention (both direct and indirect embodied support)

• Condition was found to be a predictor of the intervention effect. Thus student’s’ receiving direct embodied support during the teaching sequence displayed higher levels of reasoning after the intervention than students that received indirect embodied support

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Room for discussion III

How does the embodied cognitive mechanism (metaphor?), that mediates bodily

experiences directly,

bring forth new abstract forms of meaning (HOT)?

HOT

EMBODIMENT LOT

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Room for discussion IV

What are opportunities and challenges to measure embodied learning. Which

tools could be used; how do we obtain a deeper understanding of how perception- action processes in embodied learning environments activate, change, combine, and blend elementary embodied cognitions to ground abstract mathematical concepts? (

What are opportunities and challenges when these types of activities (embodied

learning tasks) are implemented in the primary school (mathematics) classrooms?

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REASONING ABOUT GRAPHS IN

PRIMARY MATHEMATICS EDUCATIONREASONING ABOUT GRAPHS IN PRIMARY MATHEMATICS EDUCATION

REASONING ABOUT GRAPHS IN PRIMARY MATHEMATICS EDUCATION CAROLIEN DUIJZER

M O VING M O VING

TOWARDS

UNDERSTANDING

REASONING ABOUT GRAPHS IN PRIMARY MATHEMATICS EDUCATION CAROLIEN DUIJZER

MOVING TOWARDS UNDERSTANDING: REASONING ABOUT GRAPHS IN PRIMARY MATHEMATICS EDUCATION CAROLIEN DUIJZER

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Publications related to this thesis (further reading)

• Duijzer, A.C.G., Van den Heuvel-Panhuizen, M., Veldhuis, M., & Doorman, M.

(2019). Supporting primary school students’ reasoning about motion graphs through physical experiences. ZDM, 51(6), 899-913. doi:10.1007/s11858-019- 01072-6

• Duijzer, A.C.G., Van den Heuvel-Panhuizen, M., Veldhuis, M., Doorman, M., &

Leseman, P. (2019). Embodied learning environments for graphing change: A

systematic literature review. Educational Psychology Review. doi:10.1007/s10648- 019-09471-7

• Duijzer, A.C.G. (2018). Het fizier gericht op…grafische representaties van verandering. Euclides, 93(5), 22-23.

• Duijzer, A.C.G., Van den Heuvel-Panhuizen, M., Veldhuis, M. & Doorman, L.M. (20- 01-2017). Greep op grafieken. In Marc Van Zanten (Eds.), Rekenen-wiskunde in de 21e eeuw – Ideeën en achtergronden voor primair onderwijs - Jubileumbundel ter gelegenheid van het 35-jarig bestaan van Panama (pp. 53-58). Panama -

NVORWO - Universiteit Utrecht - SLO.

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