Institut for Naturfagenes Didaktik

APEC CONFERENCE FEB. 2017, U. OF TSUKUBA, TOKYO

Researching Lesson Study with
The Anthropological Theory of the Didactic
Carl Winsløw, winslow@ind.ku.dk
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1. LS as a ”research topic”
à Need for scientific models of LS
2. Model of ”what LS is about”
à mathematical and didactic praxeologies
3. Model of ”what LS is”
à institutions and infrastructure
4. Some results of ATD research on LS
5. The Anthropological Theory of the Didactic (ATD)à Outlook and invitation
6. Conclusion
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1 – Lesson Study as Research topic

149 research publications
in mathematics education
journals and books with“Lesson Study” in title

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1 – Lesson Study as Research topic

Some overall research directions pursued:
Examplesof work
Descriptive
research:what islesson
study?

Intervention
research:
Results ofspecific LS,
LS asmethodDias 4OBJECT:
授業研究 in JapanOBJECT:
Lesson study (…) abroad

Lewis (1998): A lesson is Yoshida (2012): Mathematics
like a swiftly flowing
lesson study in the United
river
States: Current status andIsoda, Stephens, Ohara, ideas…
Miyakawa (2007):
Japanese Lesson Studyin Mathematics

授業研究 itself

Hart, Alston, Murata (2011):
Lesson Study Research andPractice in Mathematics
Education

Rasmussen (2016): Lesson
study in prospective
mathematics teacher
education: didactic andparadidactic technology in thepost-lesson reflection.

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1 – Lesson Study as Research topic

Descriptive research considers LS as an object ofstudy with research questions like- what does LS consist of (description of « interior »)- what are the essential parts (« boundary »)- what are the essential conditions for LS (« external »)- what is LS about (what are its objects and aims)
Intervention research uses LS as a method to study
more specific questions about mathematics teaching
and learning, teacher education etc.
In both cases, researchers need explicit models of LS
and its « surroundings » in order to make theirquestions, methods and results precise, intelligible andopen to scientific debate (including critique).
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2 – Model of what LS is about

Scientific Models”complex reality”

Model: ”simple” structure,described within a theory(i.e. a scientific register)All models are wrong but some are useful(George E.P. Box)
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2 – Model of what LS is about

What is LS about?
FIRST NAIVE ANSWERS
Object of LS (what it is ”about”)- Teaching, by teachers (more specifically, a lesson)- Learning by students
Objective of LS : to promote- Learning, by teachers, related to the previous two objects.
LS thus is about very specific forms of human practice andknowledge:- Mathematical Practice and Knowledge (by students, teachers)- Didactic Practice and Knowedge (by teachers)
Here, ”didactic” refers to the act of teaching or, more broadly,”inducing others into a certain practice / knowledge”.
”Didactic practice” is fundamental to all human practices!
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2 – Model of what LS is about

THEORETICAL BLOCK:‘know that and why’
Techniques can be
described and justified
PRACTICAL-BLOCK:‘know how’Tasks, solved bytechniques

TheoryTechnology(techne + logos)
TechniqueType of tasks

Example of a mathematical praxeology:- Type of task: find all x satisfying

PRAXEOLOGYThe notion of praxeology (praxis + logos) (Chevallard, 1999)(for given , , )- Technique: computation based on- Technology: explanation of formula (how, when, why,…))- Theory:
DidMatValgebraic definitions, rules, etc. to justifytechnology

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2 – Model of what LS is about

Mathematical and Didactic Praxeologies
Mathematical Praxeology (MP):this is what the students should learn; we can model it with theprecision we need for a given research project
Didactic Praxeology (DP):
1. the didactic practice (tasks and techniques) can be observed:- tasks of teaching (related to some MP)- techniques (what the teacher does to some those tasks)
2. the corresponding didactic technology and theory is notobservable in the classroom – the teacher (normally) neither
explains nor justifies his practice there.
NOTICE:- A DP is always intimately related to a MP (for math. teaching)- technology and theory of DP ~ core of teacher knowledge- The tendency of didactic technology and theory to be”personal”,”non-shared”, fragmented etc.
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2 – Model of what LS is about

What is LS about? Praxeological models
Object of LS (what it is ”about”)- Didactic Practice (planned, observed, discussed)- Mathematical Praxeologies (of students – in LS,planned/anticipated, observed and discussed)
Objective of LS : to develop teachers’ didactic
knowledge (technology, theory)- Shared didactic technology and theory needed,and developed, in both planning and reflection
meetings
An example of analysing a lesson study activity(mainly lesson plan, lesson and reflection) can befound in Miyakawa & Winsløw, JMTE 2013.
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3 – Model of what LS is

LS is often described in terms of ”cycles” or otherindications of a process that unfolds in time

But to analyse our observations of a specific LS, thismodel is far too simple. We are interested in thespecific MP and DP at stake. So what is LS itself,
relatively to these?
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3 – Model of what LS is

All human activity suppose what I call a praxeologicalinfrastructure… [which] comprises in particular somesmall and large facilities [dispositifs] which are works,
and which allow the development of superstructuralactivities – the execution of some technique being
supported by an infrastructure…
(Chevallard 2009, summer school at Clermont-Ferrand)

Mathematical infrastructure (for MP)
Works as ”facility” for mathematical practice
Ex1 ℂ used to compute rotation by a
Ex2 Computer algebra system
Didactic infrastructure (for DP)
Artefacts and ressource systems as facility for didactic
practice
Exs: textbook, smartboard, CAS, lesson plan,…
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3 – Model of what LS is

Paradidactic practice

Teachers’ work outsideclassrooom

Didactic practice

Teachers’ work inclassroom
Paradidactic infrastructure: in addition to some ofthe didactic infrastructure, special ”facilities” for thework of t, such as office space, teacher guide,
personal notes etc. Paradidactic praxeologies
develop personal knowledge of t.
How about institutionalization?
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3 – Model of what LS is

Elements of paradidactic infrastructure inJapan:• Konaikenshyuu (study groups of teachers withineach school), with grants, activity reports etc.
• Lesson study• Mathematical circles (teacher seminar accrossschools)• Open lessons (municipal, regional national) withinvited and registered participants• Conferences with invited and registeredparticipants• Teacher journals and books• School math dictionnary• Videos with ”super-lessons”• TV-shows with famous teachers(”fun math” with kids)
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3 – Model of what LS is

The case of lesson study (Miyakawa & W, 2013)
Pre-didactic
praxeology

Lesson plan
Teachers
prepare lesson
related to MP

PARADIDACTICINFRASTRUCTURE
OBSERVATIONLESSON

Post-didactic
praxeology[discussion]
Teachers
analyseobserved DP,MPDP
developed in classMPdeveloped in classtime
Paradidactic infrastructure of a school institution: the total ofDias 15
conditions
for teachers work outside ”class” (MW,2013)

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4 – Some results of ATD research on LS

Miyakawa & Winsløw (2013). For a specific LS, coherent analysisof “what teachers learn” (DP theory block), the lesson itself (DPpractice), and “what students learn” (MP).
Rasmussen (PhD-thesis, 2015). Analysis of postdidactic
praxeology in LS embedded in multidisciplinary teacher
education, showing the potential for developing didactic
knowledge “to the benefit of prospective teachers, educators andresearchers alike”.
Østergaard (PhD-thesis, 2016). Analysis of the common “gap”
between DP practice and knowledge blocks, and between MP andDP, as developed in teacher education … interventions with LS
has strong potential to bridge those gaps
Carlsen (PhD-thesis, forthcoming). During practicum LS, teacher
students develop (new) didactic technology about CAS-use inlower secondary algebra.
Bahn (Phd-thesis, forthcoming). Experimenting, with success, a
concrete paradidactic infrastructure for implementing LS in DK.
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4 – Some results of ATD research on LS

An example (Miyakawa & Winsløw, 2013)
School festival, June 2009
• Primary school attached to Joetsu U.
• Held every year at this school (Friday+Saturday)• 70 open lessons in all subjects, in two days
• 700 guests in 2009 (from all of Japan) – teachers and…
• All aspects of the school life on display, but the primary
aim is to share and develop knowledge for teaching

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The lesson

4 – Some results of ATD research on LS• Grade 2 (students about 7 years old), 40 students• Lesson no. 16 in a series of 17 lessons, duration 65 mins
• Series title: Sukkiri as we draw!
(sukkiri » experience of clarity, « aha »)• Japanese « open approach » (Nohda, 2004)• Student assignment : find solution and make a drawing
which, to a group of students, give « clarity » (of solutionand method, and on who are convinced of what and why)

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4 – Some results of ATD research on LS

A mathematical task, wide choice of techniquesTask• [t0 : to formulate t] [didactic technique]• t (the task) : There were 16 persons on a bus. Later,more people got on, so there were 34 in total. How many
had got on the bus ?
• T (type of task) : Given a total N and a part A, find rest B(N = A + B; given N and A, find B).
Techniques• t0 : identify numbers (16, 34); choose the right operation(+/-); execute (results: 16+34 or 34-16 or …)• t1 : represent situation symbolically and solve(34 = 16 + rest à rest = 34 – 16).
• t2 : represent situation by a drawing (abstraction fromconcrete situation) showing situation and solution « inone » (as opposed to arithmetic/algebraic approaches).
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4 – Some results of ATD research on LS

Reflection meeting : the meaning of « sukkiri »Main questions• Why this acitivity « sukkiri by drawing » ?
• What is the meaning of « sukkiri » ?
Comment of a participant• « sukkiri » it’s to have a drawing that helps them to get
the answer, to set up a calculation, or to see what is theobject of the interrogation sign – not to have drawing
which depicts the problem exactly
The paradidactic practice• The « new » notion of sukkiri (a priori not didactictechnology) facilitates an shared development of didactic
technology and theory, focusing on the production anduse (by students) of more or less non-standard
mathematical ostensives
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4 – Some results of ATD research on LS

Reflection meeting : techniques for
managing the students’ presentations
Participants• Evoke pupil mathematical techniques observed in
the lesson (e.g., 34 + 16) and possible
explanations (e.g., the wording of the problem
implies addition since « the whole » ~ sum) ;• Propose alternative or motified didactic techniques
for an DO in which the task is to teach « drawing
techniques » for subtraction (t2).
The paradictic practice• Question the planned and observed DO by taking
into accound the MO realised in the class, as wellas the didactic technology of the lesson plan

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4 – Some results of ATD research on LS

Reflection: theory level
Principal question• To what extent are the choises and arguments of theplanned DP justified?
Justifications mentioned• National programme (curriculum);• Philosophy of this school (« prepare the pupils to live inhuman society ») ;
à Comment of the guide (prof. Nunokawa): theimportance of interaction in society, even if you have astrong personality (« sukkiri »)Paradidactic practice• The discussion allows for reflection on more general
didactic principles, firmly based on the common
observation of a concrete DP• Development of possible new didactic technology(”sukkiri”)
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5 – The anthropological theory of the didactic

ATD welcomes you with:- An elaborate system of theoretical
tools to model and design
mathematical and didacticalknowledge and practice- A research community with active
researchers in many countries(Japan, Korea, France, Spain,Germany, Denmark, Sweden,Brazil, Canada, USA, …),international congresses etc.
- A research programme with a tightnetwork of questions, methods,results and so on- A definite potential for LS research
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6 – Conclusion

Conclusion: the profession of teaching
Etzioni, quoted by Chevallard (2009):

Crucial: existence of paradidactic infrastructure!
… to enable a collective development of knowledge(DP technology and theory), as in LS.
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Thanksありがとう