Analysis of some primary lesson segments using variation - PowerPoint Presentation

Slide1

Analysis of some primary lesson segments using variation 

Anne Watson

University of

Witswatersrand

March 2017Slide2

Thanks to:Thabit Al-Murani, Cecilia Kilhamn, Debbie MorganSlide3

Exchange visits between English and Shanghai teachers of mathematics

Political intention: change methods to 'mastery' to improve learning for all

Background: major

differences in training,

workload, culture and funding

Major

difference observed by

teachers

is the grain size of the focus of

lessons on

critical aspects

of a mathematical

idea

Input about variation to structure

what is available to be learntSlide4

Role of variation Design tool for teachersAnalytical tool for researchersSlide5

Mastery

the

idea that all pupils should learn core mathematical concepts together, with appropriate extra support where

necessary

no implications for particular methodsSlide6

Variationtheory or lens or principle?

learning by

discerning

variation against

a background of

invariance

teachers

have

a detailed conceptual focus

does pupils

'

lived

object of learning (LOL

) match the

intended object of learning (IOL

)?

enacted object of learning (EOL) – how variation is used to connect LOL to IOLSlide7

What is varied?dimensions of possible variation (

DoV

)

range

of permissible change (

RoC

)

DoV

and

RoC

provide the

EOL (?)

interplay of variation and invariance:

Vary the feature that matters against an invariant background

Vary features that do not matter so that the important feature can be recognised in different contexts

Underlying dependency relationship does not vary

Slide8

Enacted object of learning (EOL)dimensions of possible variation

ranges of permissible

change

w

hat is compared to what and how that comes about

population and structure of the example space

density of population

connectivity

generality

generativitySlide9

The studylessons on NCETM websitevariation theory as an analytical toolSlide10

Background: Chinese meaning of variation (Sun 2011)

One problem, multiple methods of solution: representations, actions, diagrams, notations

Transformations/variations of the problem: same structure, different formats

One solution method, multiple problems: the class of problems that can be solved this way

Examples and non-examples (contrast)

Procedural and conceptual (!) (

Gu

, Huang

Marton

, 2004)Slide11

Contested termsProcedural and conceptualWhat has to vary

Critical aspects (

Marton

and

Tsui

)

Points:

key

difficult (misconception?)

critical Slide12

Our research focusto identify the enacted objects of learning when primary teachers in England make use of variation to achieve mastery

we look for what is varied and how is it varied, and evidence of LOL (very incomplete)Slide13

Our methodthe NCETM videos were watched by the researchers,

together and separately

researchers produced chronological reports

on the

lesson

reports

and commentaries were compared and collated to generate a shared view of the EOL in segments of the lesson, and how this

was enacted through variationSlide14

Our perspectivesAl-Murani: how, when and who introduces which

DoV

and

RoC

and exchange between teacher and students (

Gothenburg and Oxford influence)

Kilhamn: identified the EOL from analysis of the

DoV

and

RoC

, separation, contrast, fusion etc.

(

Gothenburg influence)

Morgan: knowledge of teachers, their intentions and how VT had been introduced through NCETM and Shanghai

exchange, lesson observation of

the actual lessons and prepared the videos for website use (

Shanghai and policy influence)

Watson: how conceptual content was managed through use of variation; what dependency relations are expected to be, or available to be, learntSlide15

Excerpt 1Year 1

IOL: to move pupils from understanding subtraction as 'take away' to understanding it as 'difference‘

children mainly know additive facts about numbers to 10 and also know that these can be expressed in a range of ways, including part-part- whole diagrams

2

nd

lesson on ‘difference’

(lesson 1 was about “more

than, less

than”)Slide16

Before this clip …..

‘Take away’

Invariants so far (INV):

cars

in a car

park as context

u

se of counters

to represent cars

o

nly numbers 5,3,2 have been used

part-part

whole diagram use for additive

relationship

DoV

/

RoC

from pupils:

transformations of the part-part-whole relationship

5 - 3 = 2; 5 - 2 = 3; 2 + 3 = 5; 3 + 2 = 5; 5 = 2 + 3; 5 = 3 + 2Slide17

What teacher presents

Variation

analysis

"Now represent: there are five red cars and there are three blue cars.”

From teacher:

DoV

: image;

RoC

: 'remove' or 'compare' numbers

Pupils told to use a row of 5 and a row of 3:

●●●●●

●●●

From pupils:

DoV

: layout

Colours of the counters seen as irrelevant

DoV

: different coloured counters were used and the irrelevance of this was discussedSlide18

What teacher presents

Variation

analysis

"Looking at this picture, what is the difference between the number of red cars and the number of blue cars. Tell the person next to you."

From the pupils:

DoV

: the ways the difference is worked out and expressed;

RoC

: counting on, using number facts, comparing numbers, one to one matching

T draws a part-part-whole diagram to represent the situation with the cars, and overlays that onto a picture of counters:

All chant "we can use part-part-whole to tell us about difference"

From the teacher:

DoV

: the ways in which the part-part-whole diagram can be drawn;

RoC

: either schematically or drawing around, and labelling, the

material

representation.Slide19

What teacher presents

 Variation analysis

T gives a new task: “There are 7 children and there are 4 dinner tokens. Represent that.”

T says: “What [one pupil] said is that 3 and 4 equals 7. And we could use that addition fact to help us find the missing part. To help us find the difference”

DoV

: what a part-part-whole diagram can be used for;

RoC

: 'take away‘

and

‘difference’

DoV

: numbers change;

RoC

small

integers

DoV

: actions associated with ‘compare’

DoV

: meaning of 'difference';

RoC

: compare quantities of two similar objects; do a one-to-one matching of different objectsSlide20

Invariantscounters to represent cars and numbers

linear layout of counters

part-part-whole diagram represents the additive relationship

use of p-p-w to find the differenceSlide21

IOL, EOL, DoV, LOL

IOL: to focus on part-part-whole so they 'understand it in a way that they can apply it to a different structure’; dependency relation between three quantities through addition and subtraction

EOL: same numbers, same diagram, same representations, similar context

EOL : a

'compare' situation can be represented

using two lines of

counters, in a linear layout.

DoV

: ways

they

find numerical difference

EOL: name missing part in diagram as 'difference'

EOL: represent

and work out the difference in a new

situation (7,4,3 and matching): LOL

is not the same for all students as some of

them do not set up the diagram for this new situationSlide22

Reflections on our observationscoherence of IOL

:

a

whole lesson is devoted to developing a new meaning for

subtraction

because they know number facts, the

lesson is not

experienced as being about

counting

but about additive structure

various

DoV

are opened up and then

pinned down to become invariant so that eventually

the relationship between

'difference'

and p-p-w becomes

the only change around, and is then emphasised through a chanted

phrase

EOL can be discerned from looking at differences within invariant features; LOL is how pupils discern difference - this is not possible to know for sure but can be deduced from what they say and do, and how this relates to the EOL

 Slide23

Current thinking

Perceptual (seeing); enactive (doing); transformational (seeing differently); representational (translation); procedural (order of actions); structural (relations between objects); conceptual (patterns among objects

)

Contrast (similarity); separation; generalisation; fusion (does order matter?) (

Marton

et al.)

Population; connectivity;

generalisability

;

generativity

of PES (Sinclair et al.)

What is the same/ what is different? (delineating/defining) (with the grain – expanding the population of PES and non-examples)

What varies/what stays the same?(relational/functional) (across the grain – connectivity in the PES)Slide24

Value of methodTool to make conjectures about difficulties:

it took us several cycles to recognise a variation in meaning of ‘compare’

the word 'whole' in the phrase 'part-part-whole' no longer has meaning when the objects are different

representing the missing number using p-p-w instead of imagination (?) Slide25

Excerpt 2Year 6Lesson about reading from graphsSlide26

anne.watson@education.ox.ac.ukPMTheta.com