The document discusses a study that explored using an exploratory learning environment and speech-enhanced interaction to help primary school children develop conceptual understanding of fractions. It found that encouraging students to verbally reflect on fraction tasks while interacting with the exploratory environment helped facilitate conceptual knowledge development. The study also found that prompting students to explain their thinking and having them verbalize concepts using proper terminology supported learning, even for students not talking aloud.

1. Promoting primary children’s verbal
reflections on fraction tasks in an
exploratory learning environment.
Manolis Mavrikis, Eirini Geraniou, Alice Hansen
London Knowledge Lab,
Institute of Education, London
BSRLM – London – Feb 2014 1

2. Pedagogical rationale of iTalk2Learn
1. The development of conceptual and procedural
knowledge leads to robust learning
2. Students develop conceptual knowledge by
interacting with Exploratory Learning Environments
3. Speech-enhanced interaction may facilitate
conceptual knowledge through reflection
BSRLM – London – Feb 2014 2

3. Pedagogical rationale
1. The development of conceptual and procedural
knowledge leads to robust learning
2. Students develop conceptual knowledge by
interacting with Exploratory Learning Environments
3. Speech-enhanced interaction may facilitate
conceptual knowledge
BSRLM – London – Feb 2014 3

4. “With increases in one type of knowledge leading to gains in the other type of
knowledge, which trigger new increases in the first” (Rittle-Johnson, et al., 2001)
• Understanding about
underlying principles
and structures of a
domain
• Understanding
connections
BSRLM – London – Feb 2014 4
• Knowledge about and
application of
procedures
• An action sequence
• Knowing how to
apply a rule in order
to solve a problem
Procedural
knowledge
Conceptual
knowledge

5. BSRLM – London – Feb 2014 5
Task design
• Understanding about
underlying principles
and structures of a
domain
• Understanding
connections
• Knowledge about and
application of
procedures
• An action sequence
• Knowing how to
apply a rule in order
to solve a problem
Procedural
knowledge
Conceptual
knowledge
• Holistic approach
• Student-directed
activity
• Constructivist
• Student role is
active/major
• Atomistic approach
• Student activity is
directed
• Behaviourist
• Student role is
passive/minor
Structured
tasks
Exploratory
tasks

6. Pedagogical rationale
1. The development of conceptual and procedural
knowledge leads to robust learning
2. Support students develop conceptual knowledge by
interacting with an Exploratory Learning Environment
- Multiple traditional and dynamic representations
- Tools for comparison, addition, partition, equivalence
- Exploratory tasks
3. Speech-enhanced interaction may facilitate
conceptual knowledge
BSRLM – London – Feb 2014 6

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11. Dave was using rectangles to add
fractions. He made the fraction
below, but forgot how he did it.
Show a sum he might have made,
using two fractions with the same
BSRLM – London – Feb 2014 11
denominator.
Make three fractions equivalent
to ½
Which fraction is the odd one
out?

12. Pedagogical rationale
1. Supporting the development of conceptual and
procedural knowledge leads to robust learning
2. Support students develop conceptual knowledge by
interacting with an Exploratory Learning Environment
3. Speech-enhanced interaction may facilitate
conceptual knowledge
BSRLM – London – Feb 2014 12

13. Thinking and reflecting aloud
• Importance of language as both a
psychological and cultural tool that mediates
learning
• Young children often talk aloud while engaged
in demanding activity (Flavell et al., 1997; Berk
& Landau, 1993; Englert et al., 1991)
• Inner speech develops gradually (7-8 year
olds)
BSRLM – London – Feb 2014 13

14. Importance of reflection
• Bell and Woo (1998) analyse how some words
influence the conceptual structures developed
• Kafai & Harel (1991b);Ackermann (1991; Hoyles (1985)
and others identify the importance of reflection
• Freudenthal (1981) suggests that reflecting is the
answer to stimulating retention and that the skill of
reflection must be taught at an early age.
• Goodchild (2001) ‘blind’ vs ‘reflective’ activity
• Schoenfeld (1987 ) and others: metacognitive
instruction that uses self-directed speech improves
students’ mathematical reasoning
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15. Methodology
• Design-based research
– Series of small design experiments
• To simulate a functional system an ‘operator’
(Wizard-of-Oz) sends messages remotely to
students
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16. Examples
1. “What have you learnt about equivalent fractions?”
Video: “doubling and halving.mp4” @ 0.50sec
2. “Can you use the terms numerator and denominator?”
Video: “Use nominator, denominator.mp4” @ 1.16sec
3. “I can’t really explain”
Video: “I can't really explain,mp4” @
BSRLM – London – Feb 2014 16

17. Prompts during interaction
Pragmatic
Remember that you can talk aloud.
Can you explain a little more?
Tell me what you are thinking.
Can you explain what are you
doing?
What help would you like?/What
would you like to know?
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Elaboration
What’s another way to say that?
Can you explain that again using
[specific terminology]?
Reflection-in-action (Schon, 1987)
What do you notice about […]
Why do you think it has done that?
What relationships have you noticed?
Why did that surprise you?
Did you think that those were equivalent?
(Why / why not?)
Reflection-on-action
What did you learn from this task?
Are there any conclusions you can make?
What are you thinking about…
(now that you have done this activity)?
What was surprising?
Describe what you did to another student

18. Preliminary Findings
• Encouraging students to interact with the
FractionsLab can provide concrete ways to
construct and negotiate meanings about fraction
representations.
• Challenging students to verbalise their thoughts
and reflect can act as a springboard for
conceptual understanding
• Even the ones who are not talking aloud may be
encouraged to engage in inner-speech and get
familiar with terminology
BSRLM – London – Feb 2014 18

19. How does talking to the computer help
you to think?
• It helped me think because when you’re talking to a
computer you’ve got a different mindset, especially
when you’re looking at the screen.
• It helps me to learn because when you’re working
out you have someone talking to you about what
can help you but when you’re talking to it and
asking questions it’s just like a teacher but its
basically like 1-1 because you don’t have anyone
else butting in ... It’s just you and the computer and
it helps me learn more because I can understand it
more instead of everybody else around me talking.
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20. How does talking to the computer help
you to think?
• When me and Lucas were on it we thought of one of
[the fractions] and then we kept on [shading all the
parts of the whole] next to each other and then the
computer said ... adjacent ... So you can split it all up
and it will still be the same but you can move [the
shaded parts of the whole] around and I thought
that was really helpful.
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21. Discussion
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• Challenges
– Balance what is technically possible with what
seems empirically to work
– Encourage realistic interaction and reflection
– Context matters

22. Examples
1. “What have you learnt about equivalent fractions?”
Video: “doubling and halving.mp4” @ 0.50sec
2. “Can you use the terms numerator and denominator?”
Video: “Use nominator, denominator.mp4” @ 1.16sec
3. “I can’t really explain”
Video: “I can't really explain,mp4” @
BSRLM – London – Feb 2014 22

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24. Student difficulties
• E.g., for part-whole students need to
understand:
– the parts into which the whole is partitioned must
be of equal size
– the parts, taken together, must be equal to the
whole
– the more parts the whole is divided into, the
smaller the parts become
– the relationship between the parts and the whole
is conserved, regardless of the size, shape or
orientation of the equivalent parts
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