promoting primary children’s verbal reflections on fraction tasks in an exploratory learning...

$: Promoting primary children’s verbal reflections on fraction tasks in an exploratory learning environment$
BSRLM – London – Feb 2014 1

Promoting primary children’s verbal reflections on fraction tasks in an exploratory learning

environment.

Manolis Mavrikis, Eirini Geraniou, Alice HansenLondon Knowledge Lab,

Institute of Education, London


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


Pedagogical rationale


2. Students develop conceptual knowledge by interacting with Exploratory Learning Environments

3. Speech-enhanced interaction may facilitate conceptual knowledge


• 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

“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)

Procedural knowledge

Conceptual knowledge


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




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



http://link.lkl.ac.uk/FractionsLab


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

denominator.

Make three fractions equivalent to ½

Which fraction is the odd one out?



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



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)


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


Methodology

• Design-based research– Series of small design experiments

• To simulate a functional system an ‘operator’ (Wizard-of-Oz) sends messages remotely to students


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” @ 1.16sec3. “I can’t really explain”Video: “I can't really explain,mp4” @


Prompts during interaction

PragmaticRemember 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?

ElaborationWhat’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


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


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.


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.


Discussion

• Challenges – Balance what is technically possible with what

seems empirically to work– Encourage realistic interaction and reflection– Context matters


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” @ 1.16sec3. “I can’t really explain”Video: “I can't really explain,mp4” @

BSRLM – London – Feb 2014 24Robust Learning in Elementary Mathematics , 1st iTalk2Learn Review

Slide 24 of 18

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 1

6124

>

BSRLM – London – Feb 2014 25Robust Learning in Elementary Mathematics , 1st iTalk2Learn Review

Slide 25 of 18

promoting primary children’s verbal reflections on fraction tasks in an exploratory learning...

Presentations & Public Speaking

procedural knowledge

type of knowledge

reflection bsrlm london

fractions6 bsrlm london

londonbsrlm london

procedural skills

students difficulties

robust learningstudents