Variation Theory as a research tool for identifying learning in the design of tasks

Boris KoichuTechnion – Israel Institute of Technology

bkoichu@technion.ac.il

ICMI Study 22 Task Design

Oxford, July 22-26 2013

Theories of learning

2http://cmapspublic3.ihmc.us/rid=1LGVGJY66-CCD5CZ-12G3/Learning%20Theory.cmap

Why Variation Theory?

For many reasons, but in particular, because it postulates that:

"there is no learning without something being learned“(Runesson, 2005, p. 70)

or

“learning always has an object” (ibid, p. 70)

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Types of objects of learning

Intended object of learning (IOL): the capabilities the teacher wants the learners to develop.

Enacted object of learning (EOL): what is possible to learn in a particular situation.

Lived object of learning (LOL): what is actually learned from the point of view of a learner.

4)Marton & Booth, 1997; Runesson, 2005; Ling & Marton, 2012(

Identification of objects of learning(in connection to task design)

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For intended objects of learning: Prospective analysis of tasks, stated (by the designers) goals, their intentions and anticipations. Possible sources: reflections of task designers, observation of design processes

For enacted object of learning: Identification, based on empirical evidence, which experiences the task affords and to which extent those are beneficial for co-constructing the intended object of learning.Possible sources: observations of how learners deal with the task, written artifacts

For lived object of learning: identification, also based on empirical evidence, what actual learning outcomes of dealing with the task for the learner are. Possible sources: observations, written artifacts, exams, interviews, reflections.

)Marton & Booth, 1997; Runesson, 2005; Ling & Marton, 2012(

Methodological issues

• Which empirical evidence is needed for revealing enacted and lived objects of learning in the design of particular tasks?

• How empirically-based conclusions about the enacted and lived objects of learning can be reliably drawn?

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Three examples

Example 1: EOL≈IOL; LOL is uncertain

Example 2: EOL≠IOL; EOL≠LOL

Example 3: EOL≈IOL; EOL≈IOL

EOL: enacted object of learning

IOL: intended object of learning

LOL: lived object of learning

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 The intended object of learning: mathematics teachers' awareness of structural similarities and differences among the basic mathematical concepts of analytical geometry.

Object of didactical manipulations (variations): three versions of Locus of Points sorting task (LP task) in the context of analytical geometry.

Data: videotape-recorded small-group and whole group discussions, videotaped sorting actions, sorting sheets.

Data analysis: identification of the affordances of each version of LP task by constructing its enacted variation space (sorting criteria; the order of their appearance; actual experiences with the task, i.e., the use of prior knowledge or “solving” the items).8

Koichu, Zaslavsky & Dolev (2013), ICMI Study 22

LP task: examples of items

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1. Locus of points such that the ratio of

distances from the points to point (8,0)

to the distances from the points to

straight line x=12.5 equals 0.8.

16. Locus of points such that the sum

of the distances from the points to

points (-8;0) and (8;0) is 20.

12. Locus of points such that the ratio

between the distances from the points

to point (3;4) to the distances from the

points to point (-3;4) equals 1.

Ellipse in disguise

Ellipse

Straight line in

disguise

Three iterations of LP sorting task

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Version 1 Version 2 Version 3

Enacted object of learning (experiences)

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Conclusions-1

The data afforded us to make conclusions about the enacted object of learning.

It was difficult to reveal the lived object of learning, that is, to unequivocally claim what mathematics the participants had learned through dealing with the task.

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Example 2: EOL≠IOL; EOL≠LOL

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Intended object of learning: university students’ capability to appreciate some of the problems and their solutions as interesting, appealing or beautiful.

Object of variations: A particular algebraic problem.

Data: videotape-recorded small-group and whole group discussions.

Data analysis: identification of the experiences of the learners with the tasks (ways of solutions), categorization of their responses to the question “which problem did you like the most and why”?

Koichu, Katz & Berman (2007)

In the letters shop, one can buy letters. The cost of the letters needed to write the word ONE is $6. The cost of TWO is $9 and the cost of ELEVEN is $15.

a) What is the cost of TWELVE?

b) What is the cost of THIRTEEN?

c) What is the cost of TEN?

Letter Shop Task

Koichu, Katz & Berman (2007)

Solutions and experiences

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9

15

O N E

T W O

E L E V E N

a) T+W+E+L+V+E=(E+L+E+V+E+N)+(T+W+O)-(O+N+E)=15+9-6=18.

b) T+H+I+R+T+E+E+N=? c) T+E+N=???

Conclusions-2

Enacted object of learning (ways of solutions, the students’ opinions about the problems)

TWELVE: solving the system for T+W+E+L+V+E or a shortcut.

THIRTEEN: attempts to solve the system for T+H+I+R+T+E+E+N, realization that it is impossible as some of the letters (I, H) are not mentioned among the givens

TEN: continuous unsuccessful attempts to solve the system of equations for T+E+N.

As a rule, no problem is evaluated as beautiful or interesting.

Lived object of learning: the use of algebraic manipulations (“nothing new”, in words of one of the students).

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Intended object of learning: the middle-school students’ capability to appreciate some of the problems and their solutions as interesting, appealing or beautiful.

Object of didactical manipulations (variations): A particular geometry problem and the order in which the problem and its variation are given.

Data: videotape-recorded protocols of individual interviews.

Data analysis: identification of the actual experiences of the learners with the problems (ways of solution, time and the extent of autonomy), categorization of their responses to the question “which problem did you like the most and why”?

Katz (2011), Koichu, Katz & Berman (in prep.)

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Bisector problem Height problem

Does a triangle with a particular property exist?

yes

no

no

yes

Katz (2011), Koichu, Katz & Berman (in prep.)

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The second problem is the most beautiful!

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31

310

5

10

15

20

25

30

35

אחר שתי השאלות השאלה השניה השאלה הראשונה השאלה המועדפת

םדי

מיתל

הר

ספמ

Katz (2011)

First pr. Second pr. Both. None

Conclusions-3

Enacted object of learning (ways of solutions, students’ opinions about the problems)

Bisector Problem: in most cases – the pattern “yes-no” was observed and the problem was evaluated as interesting or beautiful.

Height Problem: in most cases – the pattern “no-yes” was observed and the problem was evaluated as interesting or beautiful.

In most cases, the second problem was appreciated as more interesting or beautiful among the two.

Explanation (works in the majority of the cases): the first problem created an appropriate background for surprise when solving the second problem.

Lived object of learning: most of the students learned that it is legitimate to talk about problems in terms “more beautiful – less beautiful” and that a problem with counterintuitive solution can be beautiful. 20

Back to the methodological issues

• Which empirical evidence is needed for revealing enacted and lived objects of learning in the design of tasks?

• How empirically-based conclusions about the enacted and lived objects of learning can be reliably drawn?

The presented examples employ different data sources and, of course, are hardly generalizable.

So far, I keep thinking anew about the above questions in every new task-based situation I deal with in my studies.

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Shall I use VT in my future studies?

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Apparently yes.

Yes!

Because elements of VT pop up for me now in many situations, sometimes spontaneously or ad hoc.

In addition, VT can be a very practical thing, which can be used in PD of teachers as task designers.

But I’d take into account the lessons learned so far:

1. One should be extra careful when formulating what an (intended) object of learning is.

2. One should worry about having both observational and reflective data in order to be in position to say something about a lived object of learning.

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Shall I use VT in my future studies (cont.)?