Programme · presentations, e.g. short papers and snapshots). Notify speakers when (i) 15 mins remain and when (ii) 5 minutes remain (for long presentations, e.g. long papers, symposia

Proceedings of the Eighteenth

Annual Meeting of the

Mathematics, Science and Technology –

Crossing the boundaries Improving the quality of science, mathematics and technology education

through relevant research and a continued multi- and inter- disciplinary

approach to teaching.

18 – 21 January 2010

Hosted by the School of Science, Mathematics and Technology

Education

University of Kwazulu-Natal

Programme

Table of Contents 1 Message from LOC 3 2 General Information 4 3 Information for delegates in the Residence 5 4 Instruction to Chairpersons 6 5 Programme 7 6 Plenary Speeches 20

Message from the LOC

It is an absolute pleasure to welcome all delegates to the 18th Annual Meeting of the

Southern African Association for Research in Mathematics, Science and Technology

Education. Much work has gone into the preparation for this conference and it has not

been without problems. Funding was difficult to obtain, but our heartfelt gratitude to our

sponsors, who came on board with the explicit intention of ensuring that our research is

fruitful and is disseminated far and wide for the benefit of all educators.

Our paper reviewing process has been difficult, and at times very cumbersome. As is

usual getting feedback from some reviewers took some prodding but it suffices to say

that all long papers went to two reviewers and their suggestions were considered in

finally arriving at a decision. I must thank the discipline coordinators for doing this

unenviable task. Generally the reviewers for the long papers were drawn from the

SAARMSTE database of reviewers. The rest of the submissions had only their abstracts

reviewed. The reviewers for these abstracts generally came from within UKZN.

We wish all of you well and we hope that you go away after the 21st January with

pleasant memories of the Durban Conference. As an added bonus to all SAARMSTE

delegates, we invite all members to attend the Gala Dinner on Wednesday, 20th January

2010.

If you have any problems please contact the conference organisers.

General Information Registration The registration desk will be opened at 8 am on Monday 18th January 2010 and it will remain opened for the rest of the conference. Delegates can visit the registration for assistance related to the conference. You may contact Twané Palmer, any other assistants using a red T-shirt or one of the conference organisers. Lanyards All conference participants are requested to wear their name tags attached to the lanyards at all times. Entry into the conference venues and dining areas is reserved for conference delegates and invited guests. Social events All delegates are invited to the following social events: Monday, 18th January 2010 at 18:00: Welcome cocktail hosted by the UKZN Corporate Services Wednesday, 20th January 2010 at 19:00: Gala Dinner Photocopying facilities A photocopier will be available at the registration area. Please consult with the conference registration team for the use of the machine. There will be a cost attached to the photocopying. ATM Facility There is an ABSA bank ATM near the registration venue. Announcements Announcements will be made each day at the plenary sessions but delegates can also look at the notice board next to the registration desk.

Information for delegates in the residence We hope that your stay in the residence will be a comfortable one. Only

one set of linen will be provided per room. Please take note of the

following important points:

1. Lock your doors when leaving your room. If you did not carry a

padlock, please hire one from the registration desk. You will receive

a refund when you return it.

2. Please do not disturb other delegates at the residence.

3. Breakfast will be served in the dining area above the registration

hall. Ensure that you become familiar with the venue. Breakfast will

be strictly from 07:00 to 08:00.

4. No medical doctor is available on Campus but a Clinic is available

during working hours. If you have an emergency, please contact the

Campus Security, who will assist. You may contact them on the

number 031 2603493.

5. Please familiarise yourself with the Residence rules and regulations.

This will be made available when you are allocated a room.

SAARMSTE CONFERENCE Instructions to Chairpersons of sessions

HOW TO CHAIR A SESSION

1. Know the name, university/affiliation and presentation title of the talk (Check pronunciation of names beforehand if necessary)

2. Start ON TIME! 3. Be very strict with time management! With many parallel sessions running

simultaneously, even a small change in the one session will affect presenters in other venues with delegates arriving late.

4. Indicate the time remaining as listed below. This can be done by raising a hand to the presenters at indication (i) and by the chairperson standing up for indication (ii).

5. END ON TIME! TIMING NOTIFICATIONS Notify speakers when (i) 5 mins remain and when (ii) 1 minute remains (for short presentations, e.g. short papers and snapshots). Notify speakers when (i) 15 mins remain and when (ii) 5 minutes remain (for long presentations, e.g. long papers, symposia and round tables). AFTER THE PRESENTATION When the time is up, stop the speakers, and ask for questions from the audience. Oversee questions and ensure that presenters are not “picked on” or harassed unnecessarily – especially if the topic is a controversial one. Do not let one delegate “hog” the question time. Try and take questions from many different delegates (where time allows). The LOC takes this opportunity to thank you for chairing this session.

DAY 1- Monday 18 January 2010

8:30-9:00 VENUE: STUDENT UNION Registration and Tea

Venues 1 (G504) 2 (LT 2) 3 (LT 3) 4 (LT 4) 5 (LT 5) 6 (G502) 7 (G511) 9 :00-10:30 Workshops

Learning about the Real World Virtually: computer simulations from PhET Interactive Simulation Project N. Finkelstein

Working with Critical Realism in Science Education Research for a sustainable future R. O’Donoghue

Edukite Interactive Curriculum Software for Mathematics and Science - Grades 10, 11 and 12 V Aravind

10:30-11:30 11:30-13:00

OPENING CEREMONY

VENUE: STUDENT UNION Welcome from Faculty: Volker Wedekind – Acting-Dean of the Faculty of Education Welcome from City: Derek Naidoo – Deputy City Manager eThekwini Municipality

Welcome from SAARMSTE: Marc Schafer – President of SAARMSTE Vimolan Mudaly – Chair of SAARMSTE 2010 LOC

Plenary – Rob O’Donoghue Chair: Marc Schafer

13:00-14:00

Lunch

Venues 1(G504) Chair:

A Lelliott

2 (LT 2) Chair:

R Mudaly

3 (LT 3) Chair:

L van Laren

4 (LT 4) Chair:

B. Davidowitz

5 (LT 5) Chair:

N Govender

6 (G502) Chair:

G Stols

7 (G511)

14:00-14:30 Short Papers

‘Planet Earth and Beyond’ – lessons for teaching, learning and research in the General Education and Training phase A. Lelliott and M. Rollnick

Argumentation as a strategy of promotion of talking science in a physics classroom- M. Qhobela

Exploring the making and use of a Time Capsule for Psychosocial support in Mathematics classrooms of young learners – L. van Laren and I. Palmer

Strategies used by Grade 12 Mathematics learners in Transformation Geometry – J. Naidoo and S. Bansilal

Death of an Outcome – Heuristic Problem Solving in the New School Curriculum D.Clerk and D.Naidoo

Influence of the use of dynamic geometry software on students’ geometric development in terms of the Van Hiele levels – G. Stols


Modelling argumentation in whole-class discussion: A novice teacher’s adaptation of a teaching strategy A. Msimanga and A. D. Lelliott

Linking everyday practices and formal environmental knowledge- myth or reality? A case study of three Lesotho Secondary schools- L.Molapo

Integrating Mathematics and Science concepts: Some often forgotten considerations – L. J. Nyaumwe and J.C. Brown

Preference for Junior Engineers, Technicians and Scientists (JETS) Club membership among Female Pupils in Secondary and High Schools –C.Haambokoma

Purposeful and Targeted Use of Scientists to Support in-Service Teachers’ Understandings and Teaching of Scientific Inquiry and Nature of Science – K. White, N. Lederman and J.S. Lederman

The Verification of Euler’s Polyhedral Formula by In-service Teachers Using Zome Geometry- M. Mbekwa, R.Govender and C.Julie


Educators and students perceived difficult mathematics topics at the Further education and training (FET) band D. Mogari and U. I Ogbonnaya

Examining the C (condomise) in the ABC of HIV and AIDS prevention through the gendered lens of learners- R. Mudaly

Using dialogue in mathematics classes: Could it aid mathematical reasoning?- L. Webb and P. Webb

Argumentation teaching as a method to introduce IK into Science classrooms: Opportunities and Challenges – M.G. Hewson and M.B. Ogunniyi

Post-graduate Physical Science teachers’ Knowledge of and classroom practice of the nature of Science in KwaZulu Natal – K. K. Naidoo and N.Govender

How do school activities contribute to promote community environmental awareness? A. Uamusse, E. Cossa and A. Queba

15:30-16:00 Short papers

Fine-Grained Cognitive Diagnostic Assessment of Rational Number Misconceptions using the World-Wide Web

R Layton

Celebrating the benefits of participatory research while keeping an eye on the ethical challenges – C. Khupe and M. Keane

Mathematics teacher educators’ account of challenges in preparing pre-service multilingual teachers A. A. Essien

What can student-generated diagrams tell us about their understanding of chemical equations? B. Davidowitz G.Chittleborough and E. Murray

Exploring students’ interpretation of electric fields around molecules using a Haptic Virtual Model: An evolving study – K. Schonborn, G.Host and K.L. Palmerius

A genetic decomposition of the Riemann Sum by student teachers – D. Brijlall and S. Bansilal

16:00-16:30 Tea

Venues 1(G504)

Chair: M.Doidge

2 (LT 2) Chair:

D. Moyo

3 (LT 3) Chair:

M. du Toit

4 (LT 4) Chair:

M.Potgieter

5 (LT 5) Chair:

M.Sanders

6 (G502) Chair:

L Webb

7 (G511)


Factors that affect the implementation of a new biology curriculum in Malawi M. Mdolo and M.Doidge

Use of continuous assessment in Lesotho Secondary Schools M. Khaahloe and B. Makamane

Can Science Centres Support and Improve Chemistry Education in South Africa? M. du Toit

Small Frequent Tests: A Formative Assessment Enhancing Student Learning M. Lutz and H. Adendorff

Meeting the curriculum requirement of “learner-centredness” when teaching evolution: Giving learners a fair deal –M.Sanders

A genetic decomposition of the chain rule: work in progress- M. Jojo, D.Brijlall and A. Maharaj


Pre-service teachers’ conceptions of scientific literacy M. Good and N. Govender

Science Educators’ ideas of the ‘science’ in IKS and their willingness to include IKS in the School Science Curriculum.- M. Keane and D. Moyo

SCIENCE CENTRES: An instrument to change learners’ attitudes towards science and related careers. T.Z. Maqutu

Student performance at South African universities: the case for chemistry after 2008 –M.Potgieter and B. Davidowitz

The Impact of Learning by Teaching (LDL) Method. What are the Requirements for Successful LDL Strategy ?.- M. A. Mafa

Poetic reflections concerning issues in multilingual mathematics and science education in South Africa – L.Webb and L.Foster

18:00-19.00

VENUE: STUDENT UNION WELCOME FUNCTION

Michael Samuel – Acting Deputy Vice Chancellor Nadaraj Govender – Head of School of Science, Mathematics and Technology Education

19:00-19:30 BOOK LAUNCH 19:30-20:00 VENUE: STUDENT UNION

Astronomy

DAY 2- Tuesday 19 January 2010

8:30-9:55

VENUE: STUDENT UNION Plenary- Noah Finkelstein

Chair: Marissa Rollnick

Venues 1(G504) Chair:

D Brijlall

2 (LT 2) Chair:

H. Bentham

3 (LT 3) Chair:

S Jaffer

4 (LT 4) Chair:

T Mthethwa

5 (LT 5) Chair:

M Rollnick

6 (G502) Chair: C Julie

7 (G511) Chair:

M Moodley 10:05-10:50 Long papers

Beyond Constructivism: Enactivism as a Theoretical Lens in the Context of Figural Pattern Generalisation.-D.Samson and M. Schäfer

Teachers’ perceptions of the language of instruction and the contiguity argumentation theory E. Afonso and M. Ogunniyi

A description and analysis of the occurrence of shifts in the domains of mathematical operations produced by criteria regulating the elaboration of mathematics in five working class high schools in the Western Cape H.Basbozkurt

Main course: life preparation, Side order: mathematics A Mathematical Literacy case study- H. Venkat

Development of a Valid and Reliable Protocol for the Assessment of Early Childhood Students’ Conceptions of Nature of Science and Scientific Inquiry- J. S. Lederman

Reasoning and communicating mathematically focusing on word problems and construction of proof. B. Aineamani

Toward a discursive framework for learner errors in mathematics K. Brodie and M. Berger

10:50-11:20 Tea

Venues 1(G504) 2 (LT 2)

Chair: J Pool

3 (LT 3) Chair:

I Sapire

4 (LT 4) Chair:

U Ramnarain

5 (LT 5) Chair:

C.G. Benadé

6 (G502)

7 (G511)

11:30-12:00 Workshop/ Short papers

Workshop Exploring Complexity Thinking in Education C. Linder

Short paper Development of pedagogical content knowledge in pre-service Technology teacher training: Challenges and possible solutions J. Pool and G. Reitsma

Short paper Exploring the use of simultaneous interpreting in the training of mathematics teachers – H. Vorster and J. Zerwick

Short paper Black South African Learner Experiences of Transition from Natural Science to Physical Science from 2005 to 2008 –M. J. Peloagae and E. Gaigher

Short paper High Leverage Mathematics Teaching Practices Working with Students ‘Errors’ – J. Prince

Workshop Writing Computer Programs to learn Mathematics - E. Dubinsky

Workshop Coherence in

academic research writing H.Venkat, M. Setati and I. Christiansen

12:00-12:30 Short Paper Snapshots of South African Science classrooms: teacher self-reflections of their classroom practice – S. K.Singh,

Short paper Motivating mathematical exploration through the use of video clips: a collaborative research and development project between Switzerland and South Africa –H.Linneweber-Lammerskitten

and M. Schafer

Short paper Equity in the implementation of practical science investigations- U. Ramnarain

Short paper The Effect of Mathematics Game-Based Learning in Grade Seven S. Achary and R. Naidoo

12:30-13:00 Short Paper Planning a field trip to the Cradle of Humankind: A model of factors affecting the success of educational museum visits – M. Sanders

Short paper Investigating the take-up of Open Educational Resources for Maths Teacher Education: a case study in six Higher Education sites in South Africa.- I. Sapire

Short paper Curriculum Intended but not attained – Physical Science-Jennifer Case

Short paper The Transition from Secondary to Tertiary Mathematics – C.G. Benadé and S. Froneman

13:00-14:00 Lunch

14:00-15:30 VENUE: STUDENT UNION Plenary – Odora Hoppers

Chair: Sathiaseelan Pillay 15:35-16:05 Tea Venues 1(G504) 2 (LT 2) 3 (LT 3)

Chair: M Rollnick

4 (LT 4) Chair:

D Bessinger

5 (LT 5)

6 (G502) 7 (G511) Chair:

I Govender 16:15-16:45 Snapshots/Short papers

Snapshot Exploring the development of three Grade 9 Teachers’ understanding and practice of Education for Sustainable Development- H.Bentham

Snapshot Teaching and learning in large classes of students with wide-ability challenges: Lessons from the Botswana senior secondary school physics classes.- R. Rammiki and M. J. Motswiri

Short paper An investigation into the challenges of learning Linear Programming in two multilingual classrooms P. Mnyandu

Short paper A Step towards an effective and efficient mechanical Technology education model for South African schools.

D. Bessinger and J. Mammen

Snap shot Grade 11 teachers’ and learners’ understandings of scientific inquiry in relation to instructional practice. W. T. Dudu and E. Vhurumuku

Snap shot Making every dose count: Examining the Numeracy skills of Pharmacy students C. Klitsie and S. Burton

Short paper Using students’ experiences to investigate the effectiveness of a Learning management system to teach programming – I. Govender

16:45-17:15 Snapshots/Short papers

Snapshot Rwandan lower secondary school science teachers’ responses to the introduction of inquiry-based science teaching into the curriculum.- L. Mugabo and P. Hobden

Snapshot Creative Science: A transformative action research approach to utilise creativity and innovation to develop resources in an underprivileged classroom – F. Schabort

Short Paper An Indepth Exploration of Science Teachers’ Responses to the Changes in the Science Curriculum – PCK focused Case Studies M. Nakedi and M. Rollnick

Short paper Towards improving the academic performance in National Diploma Mechanical Engineering Thermodynamics students – C. Louw and J. Mammen

Snap shot South African learners’ and educators’ view of relevant contexts for Life Sciences learning. M. Kazeni

Snap Shot A student’s actions when he learns mathematics using MATHLAB J. Periasamy

Short Paper Developmental coherence: A stock-take of the enacted National Curriculum Statement for Mathematics (NCSM) at Further Education and Training (FET) level in South Africa – M. K. Mhlolo and H. Venkat

Venues 17:15-18:15 Chapter Meetings

1(G504)

2 (LT 2)

3 (LT 3)

4 (LT 4)

5 (LT 5)

6 (G502)

7 (G511)

17:15-18:15 VENUE: STUDENT UNION Book launch: Karin Brodie

DAY 3- Wednesday 20 January 2010

8:30-9:55 VENUE: STUDENT UNION

Plenary – Ed Dubinsky Chair: Deonarain Brijlall

Venues 1(G504)

Chair: A Roberts

2 (LT 2) Chair:

E Alfonso

3 (LT 3) Chair:

R Mackay

4 (LT 4) Chair:

A Lelliot

5 (LT 5) Chair:

B Mfolo-Mbokane

6 (G502) Chair:

K Brodie

7 (G511) Chair:

H Venkat 10:05-10:50 Long papers

The impact of language on the academic performance of Black students of Computer Science at Rhodes University. L. Dalvit,, S. Murray and A. Terzoli

Using a dialogical argumentation Model to enhance science teachers’ conceptions of selected phenomena-M. B. Ogunniyi

Assessment Feedback: Learners think out aloud – M. Naidoo

Bridging the Gaps between the Intended and the Implemented Science Curriculum – Mismatches in Teachers’ Beliefs, Knowledge and Actions- M. Nakedi and

M.Rollnick

Role of Mathematics Teacher Educators in Promoting Local Languages as the Language of Learning and Teaching in Schools N. Chitera

Addressing concerns with the NSC: An analysis of first-year student performance in mathematics and physics N. Wolmarans, R. Smit, B. Collier-Reedand H. Leather

An investigation into the relationship between mathematical background and performance in first-year pre-calculus mathematics V. Govender

10:50-11:20 Tea

Venues 11:30-13:00 Short Papers

1(G504) Chair:

S H Gervaius

2 (LT 2) Chair:

R de Villiers

3 (LT 3) Chair:

M Stears

4 (LT 4) Chair:

N Lederman

5 (LT 5) Chair:

M Botha

6 (G502) Chair:

B W -Thompson

7 (G511)

11:30-12:00 Exploring the learning of fractions at grade seven- J. Molebale, D. Brijlall and A. Maharaj

Professional development networks: Towards capacity and knowledge building K.M. Ngcoza, P. Irwin and S. Southwood

Are we teaching learners to reason? – Developing a reasoning test to find out. P. Hobden

Year Three: Linking Progressive Development of Teachers’ Understandings of Nature of Science and Scientific Inquiry with Progressive Development of Instructional Ability-N. Lederman, J. Lederman and K.White

Improving The Conceptual Understanding of Three Dimensional Geometry of Primary School Learners Using Technology.- P. Yegambaram and R. Naidoo

Investigating factors influencing access and retention of girls in primary schools: a case study – R. M. T. P. Munguambe and E. F. R. Cossa

12:00-12:30 An investigation of the factors that affect Grade 11 learners’ performance in Mathematics as perceived by learners and teachers in selected secondary schools in the Omusati education region of Namibia. S.H. Gervaius and C.D. Kasanda

Tracing the School, DoE, industry interface in respect of Skills, Knowledge, Attitudes and Values (SKAVs) development in context of Biotechnology – A. Singh - Pillay and B. Alant

Evaluating the level of scientific literacy within the framework of Curriculum 2005: a case study – M. Stears and K. Erickson-Pearson

In-service physical science teachers’ concepts maps on global climate change: An investigation of content knowledge M. Mokeleche, T. Lelliott and M. Rollnick

Foundation Phase student teachers’ self-reported conceptions about science teaching and learning

M. Botha and G. Onwu

Identities of intermediate phase teachers in the context of changing curriculum – B. Wilson-Thompson

12:30-13:00 Influence of a motion sensor on the understanding of graphs – J. Kriek

The use of animals in education and research: do animals have rights?- R. de Villiers

Implementation and Evaluation of a Service-Learning Component in a Second Year Organic Chemistry Course – S. Abel, J. Sewry, D. Coleman and A. Hlengwa

Learning to teach the Chloralkali Industry to Grade 12- P. Pitjeng and M. Rollnick

A study of learners’ conceptual development in mathematics in a Grade 8 class using concept mapping – U. Moodley

Culture and religion: Do they affect the teaching of human reproduction in primary schools – M. Doidge and A. Lelliott

13:00-14:00 Lunch

Venues 14:00-14:45 Long Papers

1(G504) Chair:

M Naidoo

2 (LT 2) Chair:

H Bazboskurt

3 (LT 3) Chair:

J Lederman

4 (LT 4) Chair:

A Maharaj

5 (LT 5) Chair:

K Benald

6 (G502) Chair:

M Berger

7 (G511) Chair:

H Venkat

14:00-14:45 Teachers’ evaluations of learners’ acquisition of criteria for the reproduction of mathematics – R. MacKay

An investigation into orientations towards privileged texts in Grade 8 mathematics classrooms –S.Jaffer

Using an Argumentation-Based Instructional Model to enhance Teachers’ Ability to Co-construct Scientific Concepts C. S. Siseho and M. B. Ogunniyi

Terminological primacy in high school learners’ geometric conceptualization – H. U. Atebe and M. Schafer

The contexts Albanian students prefer to use in Mathematics and relationship to contemporary matters in Albania – S. Kacerja , C. Julie and S. Hadjerrouit .

Conversations with the mathematics curriculum: Testing and teacher development – K. Brodie, Y. Shalem, I. Sapire, and L. Manson

Teachers’ views on the role of context in Mathematical Literacy –T. Mthethwa

14:45-15:15 Tea Venues

1(G504) Chair:

D Samson

2 (LT 2) Chair:

J A T Tholo

3 (LT 3) Chair:

S Murray

4 (LT 4) Chair:

N Chitera

5 (LT 5) Chair:

K Brodie

6 (G502) Chair:

H Leather

1(G511) Chair:

R Mackay 15:15-16:00 Long Papers

Researching the constitution of mathematics in pedagogic contexts: from grounds to criteria to objects and operations – Z. Davis

Effective teaching through Creativity and Assessment in large and under resourced Primary Science and Technology classes in Kasungu- A. Nchessie

Language and Mathematics classrooms – catalyst or impediment to learning? – A. Roberts

Students’ difficulties in interpreting and translating from graphs: A study on Visualisation B.Mofolo-Mbokane

An APOS analysis of students’ constructions of the concepts of monotonicity and boundedness of Sequences D. Brijlall and A. Maharaj

Teacher trainers’ views of student learning and sense making of experiences with activities offered to support learning – Y. Mudavanhu

Stats Talk: Learning to interpret Statistics in context E. Lampen

16:00-18:00 19:00-24:00

AGM : STUDENT UNION

GALA DINNER: GELOFTE HOER SKOOL

DAY 4- Thursday 21st January 2010

8:30-9:55

VENUE: STUDENT UNION Plenary- Cedric Linder Chair: Nadaraj Govender

Venues

1(G504)

2 (LT 2)

3 (LT 3)

4 (LT 4) Chair:

S Davis

5 (LT 5) Chair:

H Ayouyo

6 (G502) Chair:

M Young

7 (G511) Chair:

A Nchessie 10:05-10:50 Round Tables/ Long Papers

Round Table What is a Socially Responsible Science Education? Perspectives from Students in Project SUSTAIN A.T. Sinnes, W.C. Kyle, Jr., and B.P. Alant

Round table Mathematics teacher change and identity in evidence-informed practice. M. Chauraya

Long paper Learner mathematical thinking on Grade 12 introductory differentiation: An analysis of errors and misconceptions K. Luneta and P. Makonye

Long Paper A Study of the Self-efficacy Beliefs of Maths Learners and the Impact on Maths Learning – M. Moodley and S. Hobden

Long paper Training Needs Assessment for Mathematics and Science teachers: Agenda for DMSE-INSET Training program M. J. Motswiri, K. A. Ramatlapana and R. Rammiki

Long Paper An approach for the implementation of technology education in schools in the North West Province – J.A.T. Tholo, R.J. Monobe and M.W. Lumadi

10:50-11:20

Tea Meeting: LOC 2010 and LOC 2011

VENUE: G511

Venues 11:30-13:00 Short Papers/ Round Tables

1(G504) Chair:

J Mammen

2 (LT 2) Chair:

O Koopman

3 (LT 3) Chair:

S Bansilal

4 (LT 4) Chair:

M Kazima

5 (LT 5) Chair:

T Penlington

6 (G502) 7 (G511)

11:30-12:00 Short Paper Exploring Grade 10 Physical Science Learners’ Conceptions of Nature of Science A. Moodley and P. Hobden

Short Paper Analysis of the FET Physical Science External examinations in South Africa O. Koopman

Short Paper Mathematical Literacy teachers’ engagement with contexts related to personal finance S. Bansilal , T.Mkhwanazi and P.Mahlabela

Short Paper Using local language as a resource in teaching and learning mathematics M. Kazima, F. Pwele and M. Kasakula

Short Paper An exploration of grade 9 learners’ fractional conceptions in two secondary schools: A Case Study-T.Makhathini and D.Brijlall

Round Table 11:30-12:15 The integration of Indigenous Knowledge of Zulu medicinal plants in pre-service chemistry education at the University of KwaZulu-Natal (Edgewood Campus) D. Moodley and N. Govender Round Table 12:15-13:00 Evolutionary nature of explanatory frameworks and their impact on school science explanation of natural phenomena T. Mamiala

Round Table 11:30-12:15 School-Based Professional Development: An Action Research Initiative by Science Teachers T. Mokuku and M. Nthathakane Round Table 12:15 – 13:00 An Investigation into alignment with respect to the theme ecology at Lesotho Junior Secondary Science S. Majara

12:00-12:30 Short Paper Importance of student performance in undergraduate science courses: factors to blame and lessons to learn -. J. Mammen

Short Paper Performance analysis of Science Education undergraduates: (A Case Study of Biology Education Students) C.E. Ochonogor

Short Paper Information and Communication Technology (ICT) in the Teaching and Learning of Functions U. I. Ogbonnaya

Short paper Proficiency in the multiplicative conceptual field: Using Rasch measurement to identify levels of competence C. Long , T. Dunne, and T. Craig

Short Paper Can our Mathematics teachers measure up? T.Penlington

12:30-13:00 Short Paper Students’ beliefs about statistics: Lessons from a business administration statistics class D Kasoka

Short Paper A case study of first year chemistry students’ metacognitive judgements about their performance in a Stoichiometry test K Mathabathe

Short Paper Exploring the teaching of fractions in grade 7 J. Molebale, D.Brijlall and A.Maharaj

13:00-14:00

Lunch

14:00-15:30

1(G504) Chair:

S Hobden

2 (LT 2) Chair:

N Govender

3 (LT 3) Chair:

J Mammen

4 (LT 4) Chair:

V Paideya

5 (LT 5) Chair:

W Rauscher

6 (G502) Chair:

D.Huillet

7 (G511)


Looking back at school Mathematics: Insights from pre-service teachers who struggled to learn Mathematics S.Hobden

A Learning theory approach: Motivating a theoretical framework-N.Maharajh, D. Brijlall and N.Govender

Towards the development of an instructional framework in line with Van Hiele Phases to teach geometry in senior secondary school – J.K. Alex and J.Mammen

Engaging first year Engineering students in Chemistry Supplemental Instruction sessions V.Paideya and R.Sookrajh

School -Based Teacher Training and Education- an Alternate Certification Approach: A Mafikeng Campus Experience: An Impact Study – F. Kwayisi and T.E.B. Assan

Why are learners’ and teachers’ experiencing difficulties with the concept of zero? Z. Jooste

Symposium 14:00 -15:30 The usefulness of taxonomies for alignment between the intended and examined curriculum: the case of the 2008 science and mathematics grade 12 examinations- M.S. Rollnick, H. Venkatakrishnan, A. Lelliott, M. Berger, D. Taylor, E. Lampen, P. Nalube, M. Graven, A. Msimanga, G. Moletsane, M. Doidge, L. Bowie, L. Nyaumwe, M. Nakedi, F. Mundalamo and M. Mokeleche


Triadic dialogue: An analysis of interactions in multilingual mathematics primary classrooms P.Sepeng

When learners study what do they say they do? N.P. Ngwira

Beyond numeracy: Values in Mathematical Literacy classrooms S Rughubar - Reddy

An instructional model for promoting critical thinking in a crowded Physical Science curriculum: The ladder approach A.Stott and P.Hobden

The knowledge-generating activities drawn upon by technology education students when they design and make artefacts W.Rauscher

Introducing the concept of limits of functions through WINPLOT – D.Huillet

15:00-15:30 Snapshot Whats in the lunch box: Stigma or exploitation? D Kroone

15:30-16:00 Tea

16:00-17:00 Closing ceremony

Final Comments – Marc Schafer – Final closing comments Vote of thanks – Vimolan Mudaly

Handing over of the banner to the 2011 LOC

PLENARY SPEECHES

The APOS Theory of Learning Mathematics: Pedagogical Applications and Results

Ed Dubinsky Department of Curriculum and Instruction, Florida International University

[email protected] APOS Theory is a constructivist theory of how students might learn mathematical concepts and is based on the ideas of J. Piaget. The fundamental principle is that in order to learn a mathematical concept, the subject must apply certain cognitive mechanisms to construct mental structures to use to learn the concept. Thus the research component of work consists in determining specific mental mechanisms and mental structures for specific concepts. The pedagogical component consists in helping students make mental constructions that the theoretical analysis proposes. This uses the technology of students writing and using computer programs. After students have engaged in this activity, it turns out that using pedagogical strategies such as cooperative learning, small group problem solving and student-directed discussions is substantially more effective than traditional educational approaches. The paper describes APOS Theory in general and gives examples of theoretical analyses (genetic decompositions) of various mathematical topics. This approach can be used in a very wide spectrum of mathematics, from infinite decimals and fractions at the secondary level to abstract algebra and countably infinite sets at the post-secondary level. Some examples of the pedagogy which follows the computer activities and samples of results are given. Introduction I am very happy to be here in Durban for this conference and to give this Plenary Address. If I may begin with a personal note, I would like to observe that I received my PhD. in 1962 and am now close to the end of my professional career. This career began with faculty positions at University College of Sierra Leone (1962-63) and University of Ghana (1963-64) and it is fitting that this visit followed perhaps with some ongoing interaction with graduate students and young research faculty allows me to return near the end of my career to the continent on which its beginning took place. In this talk, I would like to begin with a discussion of the APOS Theory of learning mathematics and how this theory can be applied in studying how certain mathematical concepts might be learned. Then I will describe how APOS-based theoretical analyses can be applied in the classroom using certain pedagogical strategies including having students write computer programs. Finally, I will mention some of the results we have obtained from this approach. The work I will describe is a cooperative effort of a loosely-organized group of researchers in mathematics education which we like to call a Research in Undergraduate Mathematics Education Community (RUMEC). As will be seen, we have not restricted ourselves to the undergraduate level, because as Piaget observed, the fundamental principles of the development of mathematical knowledge and understanding are not changed throughout the range from infants to mathematical researchers. In any case, I would like to thank the many researchers with whom I have collaborated and interacted and apologize to them for not mentioning the names of any of the very large number of individuals involved. APOS Theory

For the first 25 years of my career, I was a relatively traditional teacher of college mathematics trying to balance my teaching work with my efforts in research in Functional Analysis. I think I was a good lecturer and I very much liked doing it, but I was saddened by the inescapable fact that my students were really not learning mathematics at the level I hoped for. I tried all sorts of faddish pedagogical methods but nothing seemed to help much. Finally, I decided that if I really wanted to help my student learn, I needed to study the mental processes of learning mathematics as intensively as I was studying Functional Analysis. Having just completed by best research in the latter area, I thought I could divide my time in half --- 3 days a week studying learning and 3 days a week studying nuclear Frechet spaces. I was wrong. My interest in the latter quickly dried up and I found myself devoting full time to research and curriculum development in mathematics education. But I was still a theoretician at heart and so I began to think about theories of learning. I spent two years ploughing through the literature in mathematics education research, focusing on secondary and post-secondary levels and the role of theory. What are the roles of theory in mathematics education research? Following some ideas of Alan Schoenfeld, I came up with the following list.

• Tool for doing research • Enhance our understanding of: How mathematics can be learned How pedagogy can help • Guide for analyzing data • Communication of ideas and results

I feel very strongly that in thinking about theories of learning, we should keep these items very much in mind. In spite of what the author(s) might say, if something is referred to as a theory, it should relate to at least some of the items on this list. I did not find much that satisfied this requirement nor did I find many papers that presented much of a challenge to understand. In my experience with mathematics, if I decided to study a 20-page paper and try to understand its contents fully, I would have to plan on spending at least a month. For most papers in mathematics education research of a similar length, a full day sufficed. One of the few exceptions was the work of Piaget. Although he knew very little mathematics and made many mathematical errors (confusing relations and functions, claiming that the integers formed a field, etc.) his views and theories of learning mathematics resonated with my own experiences as a mathematician and his work was profound, requiring as much effort as I had to put into understanding works in mathematics. So I focused on the work of Piaget and, in trying to understand his theoretical ideas and relate them to my (at that time) 27 years of teaching, I began use his ideas and my experiences to develop APOS Theory. A sketch of APOS Theory In the subsequent 25 years, APOS Theory has proven highly versatile, having been applied to a vast set of mathematical topics. Following is a (probably not complete) list. I believe that for each item on this list, there is at least one published study that uses APOS Theory.

Infinite Repeating Decimals Integers and Rational Numbers Sets Permutations and Symmetries Mathematical Induction Propositional Calculus Predicate Calculus

Variables Functions Composition of Functions Graphs of Functions Limits Derivatives Definite Integrals Indefinite Integrals Chain Rule Leibniz Formula Sequences and Series Differential Equations Binary Operations Groups Subgroups Cosets Normality Quotient Groups

The theory begins with an assumption about the nature of mathematical knowledge and an hypothesis on how it can be acquired. NATURE OF MATHEMATICAL KNOWLEDGE

An individual’s mathematical knowledge is her or his tendency to respond to perceived mathematical problem situations and their solutions by reflecting on them in a social context and constructing or reconstructing mathematical actions, processes and objects and organizing these in schemas to use in dealing with the situations.

This paper focuses on the constructions HYPOTHESIS ON LEARNING MATHEMATICS

An individual does not learn mathematical concepts directly. Rather, he or she applies mental structures to make sense of the concept. If the individual possesses such structures appropriate for a given mathematical concept, then it is easy, almost automatic to learn the concept. On the other hand, if appropriate mental structures are not present, then learning the concept is almost impossible. Thus, the goal of teaching is to help students build appropriate mental constructions. In APOS Theory, the constructions are actions, processes, objects and schemas.

This hypothesis relates to what I believe is a universal experience with mathematics. “Everyone” learns how to count, almost without effort. As one goes through school, one increases one’s mathematical knowledge, at first with little effort, but then there comes a point where understanding is no longer automatic and one has to struggle. Finally, the time comes for everyone, even the greatest mathematicians in history when ideas are encountered that cannot be understood, problems that cannot be solved. I think that the above conjecture offers an explanation for this phenomenon. We all have a greater or lesser ability to construct the mental structures necessary to understand mathematical concepts. We can do this spontaneously at first, without aid or conscious effort, just by reacting to our environment. That is the mathematics we learn almost automatically. The reason it gets harder is that we no longer have appropriate structures and must make an effort, even get help, to build them. Eventually we don’t have the structures and we are unable to build them. That is where it stops. In some sense, one can define mathematical talent by the ability to construct these structures spontaneously and without conscious effort or external assistance.

Let us turn now to the construction of the specific mental structures posited by APOS Theory. Mental Constructions for Mathematical Knowledge

The mental constructions in APOS Theory are: Actions, Processes, Objects and Schemas. Following is a diagram illustrating how the first three of these are interrelated.

A mental or physical transformation of mental or physical objects is considered to be an action when it is a reaction to stimuli which the subject perceives as external. When an individual reflects on a repeated action, it can become perceived as a part of the individual and he or she can establish control over it. A mental structure, called a process is then constructed which does exactly the same transformation as the action except that it takes place in the mind of the individual rather than the external world. The mental mechanism by which an action understanding is converted to a process understanding is called interiorisation. When an individual reflects on a need to apply actions or processes to a particular process, becomes aware of the process as a totality, realizes that transformations (whether they be actions or processes) can act on it, and is able to actually construct such transformations, then he or she is thinking of this process as an object. The mental mechanism by which a process understanding is converted to an object understanding is called encapsulation. A schema for a specific mathematics concept is a collection of actions, processes, objects and other schemas that has coherence relative to the concept in the sense that the individual has some means, either explicit or implicit, either conscious or not, of deciding whether a given situation comes under the aegis of the schema. Example: The Concept of Function With an action understanding of function, the individual requires an explicit mathematical expression before he or she considers that a function is present. The only transformations that can be performed consist of substituting numerical values for the variable in the expression or, perhaps, substituting another expression for that variable and calculating or simplifying. If two expressions

are used to define the function, then the subject considers, as did Euler at one time, that there are two functions. After repeating such actions and reflecting on them, an individual may construct a mental structure that allows her or him to think about exactly the same actions, but in her or his mind without necessarily performing the calculations explicitly. This structure is a process conception of function. It allows the individual to think about such ideas as reversing the transformation, which leads to inverse functions and combining two functions in such a manner that leads to the composition of functions (see below for more detail on compositions). The individual may confront situations in which he or she desires to apply various actions and/or processes to a function. These might include thinking about an operation that takes two functions and produces a new function such as arithmetic operations or compositions. Or the operation might be the derivative or the integral. A differential equation is also an operation (the left hand side) applied to an unknown function to obtain a given function and the task is to reverse this operation to find the unknown function. Such actions cannot be performed if the function, known or unknown, is considered as a process because a process is dynamic. It cannot be tied down and submitted to a transformation. In order to operate on a function, the process understanding must be encapsulated and converted to an object. Finally, the schema for a function might consist of all of the functions understood as actions, functions understood as processes and all of the transformations a subject can apply to functions considered as objects. The coherence might lie in the understanding that to have a function, there must be a domain set, a range set, and a means of transforming elements of the domain set to elements of the range set. Composition of Functions: Genetic Decompositions Consider that a certain gas is kept at a fixed temperature and the following function C gives the compressibility c of the gas in terms of its volume v:

3

32

32

3

( 2) ( 6)1 if 0 654

( ) 0.048 1.73 7.7 if 6 903000

4 1 if 901005 800000

v v v

vc C v v v v

v vvv

⎧− −⎪ + ≤ <⎪

⎪⎪= = − − ≤ <⎨⎪⎪⎪ − ≤⎪ −−⎩

Consider also that the following function V describes the variation in volume v as the pressure is varied through time t

2

3

1.26 1 if 0 10( ) 4 ( 7) 2 if 10 t

3

t tv V t

tπ

⎧ + ≤ <⎪= = ⎨

− + ≤⎪⎩

The problem then is to construct a function CV that will describe the variation of the compressibility

in time. Mathematically, CV is the composition of C and V and CV(t) = C(V(t)) With just an action conception, if all you can do is substitute values and expressions for v and t, if you see five functions corresponding to the five expressions in the definition of the function, then there is no way you can understand how to do this problem. Indeed, you will need to use these substitutions, but along with process and even object conceptions of the two functions involved in the problem. To see how such an understanding can be constructed, let us begin with what is called a genetic decomposition of a mathematical concept such as composition of functions. This is a description, often displayed in a diagram, of the actions, processes, objects and schemas and their relationships, that might be involved in such a construction. Consider the following diagram:

Object Process F (x F(x) ) De-encapsulation Process Object Coordination x F(G(x)) F oG Encapsulation

Object Process G (x G(x))

De-encapsulation The overview of this diagram is that one begins with two functions, F and G and transforms them into a single function, F o G. The transformation begins by de-encapsulating F back to the process x F(x) from which it came and then doing the same to the function G to obtain a second process x G(x). The two processes are coordinated to obtain the process x F(G(x)), which is then encapsulated to the object F o G. We will return to this problem below when we discuss how we use this theoretical analysis in our pedagogical strategies.

Integers Here we give two examples which span a wide spectrum from very elementary to very advanced mathematics. First we consider the very elementary notion of integer. In moving from the natural numbers which are based on ideas of forming collections, adding objects and taking objects away, we change the metaphors. Addition, for example, becomes a trip along a line which has discrete stops. A movement of 3 stops in one direction is the integer 3 and a movement of 3 stops in the opposite direction is the integer −3. This leads to a process conception of integer. At the same time, an integer is a location along this line of stops. Moreover, actions such as arithmetic operations can be performed on integers which strengthens an object interpretation. Students’ understanding of integers is enhanced by reflecting on the interplay between process and object conceptions. Consider the formula, a b b a+ = + On the left, a is a location (object) and b is a movement (process) from that location to a new location. On the right side, however, b is the start location (object) and a is the movement (process) from that location to a new location. Infinity The mathematical concept of infinity is as advanced as integers are elementary. Some millennia ago, Aristotle spoke of potential infinity by which he referred to an action that was repeated indefinitely, such as laying out the natural numbers, one by one. Seeing this repeated, unending process as a completed totality with all of the objects it produces present in one’s mind, all at once, was not, in Aristotle’s view and the view of many mathematicians over the next 3000 years (including H. Poincare), a possibility for human beings. In other words, what is called actual infinity does not exist. This position, although controversial, was the reigning view amongst western mathematicians, at least until the time of G. Cantor who developed the idea of actual infinity in mathematics. APOS Theory provides theoretical support for the existence in people’s minds of both potential and actual infinity. We are able to establish, in what seems to be all of the examples arising in this controversy, a plausible explanation of potential infinity as a process whose encapsulation is an object which meets all of the requirements of actual infinity. We have used this proposed resolution of a controversy in the Philosophy of Mathematics to explain student errors in certain mathematical situations (e.g., the union of a countable collection of increasing sets) and to help students understand the relation between an infinite repeating decimal and the rational number to which it corresponds (e.g., 0.999 . . . and 1). We are currently working on a description of how the mind comes to understand uncountable sets such as the set of all subsets of the natural numbers. Abstract Algebra For many students of mathematics, a course in Abstract Algebra is a major stumbling block and often spells the end of a career in mathematics. One possible reason is that unlike many other undergraduate mathematics courses, the topics in Abstract Algebra do not lend themselves very much to procedural knowledge. The student simply has to confront the deep abstractions in this subject and make appropriate mental constructions that can be used to understand topics such as groups, subgroups, cosets and quotient groups. Following are some examples of these constructions for cosets.

An action conception consists of forming a coset in a familiar situation where there are formulas such “multiples of 4” in the group of natural numbers Z with addition as the operation or in the group Z20, where the set is {0,1,2,…,19} and addition mod 20 is the operation. More explicitly we have, in the latter case,

Group (Z20,+ 20) = ({0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}, + 20)

Subgroup H = {0, 4, 8, 12, 16}

Coset 3 + H = {3, 7, 11, 15, 19}

A more complicated context, in which it is not so convenient to write everything out and do all calculations explicitly is provided by the permutation groups Sn of permutations of n objects with composition as the operation. Even in the case in which the value of n is fairly small, such as 4, it is necessary to have a process conception in order to think about the following situation,

Group S4 = The set of permutations of [1, 2, 3, 4] with composition as the operation. We will use the cyclic notation in which (124) is the permutation that sends 1 to 2, 2

to 4, 4 to 1 and leaves 3 unchanged. Also, the permutation (12)(34) sends 1 to 2, 2 to 1, 3 to 4 and 4 to 3. The operation of composition will be indicated by concatenation.

Subgroup H = {(1), (12)(34), (13)(24), (14)(23)},

Left Coset (123)H = The set of all products (123)x where x is an element of H

It is this last notion that strongly requires a process conception. The individual must be able to think about iterating through each element x of H and every time forming the product (123)x all without actually making any of these calculations. More generally, a process conception leads to the ability to think of cosets in groups of permutations of n objects where n is large, and in a generic group

Finally, there is the mathematical situation in which one has a group G together with a subgroup H and wishes to form a set consisting of all of the left cosets xH of H by elements x in G. In order to make sense of this, the subject must move from just thinking about cosets in terms of the process used to form them and also think of cosets as objects to which actions, such as forming a set of cosets can be applied, G mod H = {xH : x in S4} Even further, one wants to equip this set with an operation, called coset product and this brings an even stronger need to think about cosets as objects. When this is done, the set G mod H together with the coset product is called the quotient group of G mod H, and the mental constructions we have just described seem to be necessary and perhaps sufficient in order to understand this difficult mathematical concept. Paradigm The theory we have been discussing is one of three components that fit together in what we call a

paradigm for research and curriculum development. Following is a graph that displays these components and their interconnections.

Work on a particular mathematical concept begins with a theoretical analysis such as we have just been describing. Using our own knowledge of the mathematics and of APOS Theory together with the historical development of the concept we form a preliminary genetic decomposition. It is preliminary in the sense that it is not yet based on research. Then, using certain pedagogical strategies which we will describe below, we design instruction for the concept. The instruction focuses on helping the students construct the mental structures called for by the theoretical analysis. Then by completing certain tasks, participating in class discussions facilitated by the instructor and even listening to explanations by the instructor or perhaps other students, the students construct their understanding of the mathematical concept. The designed instruction is implemented and both quantitative and qualitative research is conducted. In this research the following three broad questions are asked. Did the students appear to make the mental constructions called for by the genetic decomposition? Did the students appear to learn the mathematics in question? Does there appear to be a relationship that could be causal between making the mental constructions and learning the mathematics? Thus, this research is guided by the theoretical analysis which is the meaning of one of the directions of the arrow between the theory and data boxes in the diagram. The meaning of the other direction is that, based on this research, the genetic decomposition and the instruction that is based on it are reconsidered and revised, if necessary. The cycle in then repeated using the revised instruction and this continues, perhaps with other students, until the genetic decomposition stabilizes and the level of learning is satisfactory. We have discussed the Theoretical Analysis component of our paradigm and now we will consider the other two components Pedagogical Applications In our pedagogy, we de-emphasize lecturing in favor of strategies such as cooperative learning and small group problem solving. This helps, but it is not enough to foster the mental constructions of interiorizing an action to a process or encapsulating a process to an object. Constructing these mental mechanisms is the key to our pedagogy and our application of APOS Theory to teaching and learning. Our pedagogical strategy is embodied in what we call the ACE Teaching Cycle. It has three

components: Activities, Classroom, Exercises. The entire course is broken into repetitions of this cycle of components, with each repetition taking about a week. First we describe these components in general terms and after that we consider examples. Activities The Activities component takes place in a computer lab where students work in teams on computer tasks designed to foster construction of the requisite mental structures. Students write computer programs that implement procedures that model action understandings of a concept. Expressing these procedures as computer programs with inputs and output represented by parameters and running these programs for various values of the parameters leads to the student reconstructing an action which he or she has repeatedly performed in what is her or his external world (that to which access is obtained through the five senses) but as a mental transformation that lives in her or his mind. This tends to foster the interiorisation of the actions to become mental processes. More important is the issue of fostering encapsulation which involves seeing a process as a completed totality and thinking of performing an action on this process so that it becomes a mental object. Studies have shown that this can be very difficult for students. We have found that if a student writes a program P that represents a mental process and then writes another program which accepts P as an input, manipulates it in some way and perhaps produces a new program P′ which represents another process, then the student tends to encapsulate the processes represented by P and P′ into objects. An additional role of the computer lab is to provide an experiential base for the abstract mathematical concepts that the students are in the process of learning. Classroom In the classroom, the students work cooperatively on mathematics tasks designed to help them use the mental structures that they were expected to build in the computer lab to construct their understanding of the mathematics the teacher wants them to learn. In some cases, students will work on a task as a group and in other cases they will work individually, compare notes and negotiate a group solution to the problem. Often, they will report to the class their results. Throughout this process, the emphasis is on discussion, reflection explanations by the teacher where appropriate and, of course, success in completing the task and understanding the underlying mathematics. Exercises The exercises are fairly traditional. There role is to reinforce the learning that has taken place in the other two components, provide opportunities for students to make applications of the mathematics they have learned, and prepare them for the study of the mathematics that is about to be taken up. Examples for the study of functions and composition We begin with the compressibility and volume functions C and V defined above in our discussion of composition of functions. The computer language that we use is called ISETL. We refer to it as mathematical programming language because it supports many mathematical expressions in the same or very similar notation and syntax as that used in mathematics. Here is a program that expresses the function C

C := func(v);

if v < 0 or v = 100 then return; end; if v <= 6 then return 1 + ((v-2)**3 * (v-6))/54; elseif v <= 90 then return (v**3/(3.0e3)) - 0.048*v**2 + 1.73*v -7.7; else return 4*v**(3/2)/(5*sqrt(v**3 - 7.0e5)) - 1/(v-100); end;

end; And here is a program for the function V

V := func(t); if 0 <= t and t < 10 then return 1.26*t**2 + 1; else return (4/3)*3.14159*(t-7)**3 + 2; end;

end; It is not very difficult given just a few notes on translation (e.g., * indicates multiplication, ** indicates exponentiation, <= indicates less than or equal, / indicates divide) and a very minimum of programming experience (which students can pick up in a short time) one can see that this computer code expresses exactly the same procedure as does the split domain definition of these two functions. The students write these programs in the computer lab, evaluate them for different values of the variable and graph them (which requires a computer instruction of one line). In the classroom, we ask them to explain verbally what their computer program does when the computer is given instructions such as: C(5.38); C(42); C(1000); C(-7.4); V(0); V(4.73); V(10); V(12.654); As the students explain how their program makes decisions and calculates, the external action is transformed into an internal process in their minds. Once the students can understand a function as a process, they are ready to learn about coordinating the processes of two functions to obtain a function which is their composition. Again, we have them construct in their minds a coordination for the functions C and V by writing and running a computer program such as the following which is almost a direct translation of the definition of composition..

CV := func(t); return C(V(t));

end;

They then write and run code such as the following. (Here the first line is the code and the second is the result returned by the computer.

CV(0); CV(4.73); CV(10); CV(12.654); 1.093; 10.19; 1.02; 0.799;

This solves the compressibility problem introduced earlier. It is solved in the minds of the students

and can be expressed in mathematical terminology in the external world. The examples just shown focus on process conceptions of functions. The operation of composition can also have a process conception because it takes two functions and composes them to construct a new function. However, in order to think of composition in this way, which is how mathematicians think of it, it is necessary to understand the input functions as objects. Following is a computer program which fosters both understanding functions as objects and understanding composition as a process.

comp := func(f,g); return func(x);

return f(g(x)); end;

end; After writing this program, and keeping C and V as defined (of course any other functions could be used as well) the student can write the following single line with the result below it.

(C .comp V)(4.73); 10.19;

All of this leads to the development, in the student’s mind, of C and V (or any other functions that are used) as objects and composition as a process. Some of the computer activities involve reversing the order of composition and solving composition equations (given any two of the three functions in a composition, find the third). These all become classroom tasks. Examples from Calculus With a strong understanding of functions, students are ready to develop an understanding of the deep concepts of calculus. We begin with a description of our treatment of the derivative and we will use the split-domain function g as an example. Following is the ISETL representation of g, code to calculate its value at 0.25 and the one-line code that will plot it.

g := func(x); if x < 1 then return x**2 else return x/2 + 0.5; end;

end;

g(0.5); 0.25; plot(g,-1.25,4.25);

This plot instruction will cut the domain of this function down to the interval [-1.25,4.25]. The students use an approximation of the derivative to work through some of the basic ideas involving differentiation. This approximation is a function ad that accepts an ISETL representation of a function and returns an ISETL representation of a function which is an approximation to its derivative.

ad := func(f); return func(x);

return (f(x+0.00001) - f(x))/0.00001; end; end;

The student can easily replace 0.00001 in this definition by a smaller number to get a better approximation. The following code is very important. In the classroom, the students have the task to explain in words such as “ad takes the function g and constructs a new function, ad(g) whose value at 0.5 is 1”.

ad(g)(0.5);

1; The following code graphs two functions: g and the approximation to its derivative ad(g) on the same interval [-1.25,4.25].

plot([g,ad(g)], -1.25, 4.25); In the classroom, the students discuss the relationship between the two functions g and ad(g). They also work on tasks such as the following. Sketch a graph of a function h which satisfies the following conditions: h is continuous h(0) = 2, h′ (−2) = h′ (3) = 0 and 0limx→ h′ (x) = ∞

h′ (x) > 0 when − 4 < x < −2, and when − 2 < x < 3,

h′ (x) < 0 when x < −4, and when x > 3, h′′ (x) < 0 when x < −4, when −4 < x < −2, and when 0 < x < 5, h′′ (x) > 0 when − 2 < x < 0, and when x > 5, lim ( ) and lim ( ) 2x xh x h x→−∞ →∞= ∞ = −

As a final example from calculus, consider the chain rule. Given what the students have developed up to this point, they are ready for a discussion that leads to them seeing the chain rule as a coordination of the derivative and composition. Therefore, they can construct the following one-line of code that implements the chain rule and run this code to calculate the value of the derivative of the composed function.

C V′ := ad(C .comp V);

C V′ (4.73); With this computer structure, the students can work traditional problems involving the chain rule both on the computer and with pencil and paper. An important exercise is to have the students use the following code to graph the four functions, C, V, CV, CV ′ on the same graph and figure out which curve is the graph of which function. plot([C,V,CV, CV’],0,10); Turning now to the integral, we begin with a computer construction for Riemann sums. Most students need some help to write the following code. Note that the ISETL symbol %+S, where S is a set of numbers stands for S∑ , the sum of the elements in S.

RiemLeft:= func(f,a,b,n); x := [a+((b-a)/n)*(i-1) : i in [1..n+1]]; return %+[f(x(i))*(x(i+1)-x(i)) : i in [1..n]];

end;

An interesting exercise is to have the students take a particular function, such as the function g defined above, graph g, run the following code, and then draw on the graph a representation of what this code is calculating, even if they have not seen this before. Most students will draw the standard picture that divides the area under the curve into rectangles. They can then calculate the sum of the areas of these rectangles and see how the Riemann sums approximate the area under the curve.

RiemLeft(g, 0, 4, 10); The power of ISETL can be used to construct a computer representation of the Fundamental Theorem of Calculus as follows. First we have the program that will take a function f defined on the interval [a,b] and construct an anti-derivative defined on the same interval.

Int := func(f,a,b); return func(x);

if (x < a) or (b < a) then return; end; return RiemLeft(f,a,x,1000);

end; end;

Next, the students run this code on (an approximation to) the derivative of the function g, evaluated at 2.

Int(ad(g),0,4)(2);

They also approximate the derivative of the anti-derivative of g

ad(Int(g,0,4))(2); And finally,

g(2); They will see (assuming that g(0) = 0) that all three calculations give the same result. More generally, they can graph the three functions with the following code.

plot([Int(ad(g),0,4),ad(Int(g,0,4)),g], 0, 4);

A lively discussion will ensue when they try to understand why they only see one curve on this graph of three functions. If the same exercises are done with a function whose value at 0 is not 0, then classroom discussion of why one of the three functions is translated can lead students to discover for themselves the idea of the “constant of integration” and its role in the definite and indefinite integral. For a more detailed discussion of this paradigm that we are using, see Asiala et al. (1996). Results In this section we give just two of the many results which have been obtained using this approach. These are for calculus. A more comprehensive report is given in Weller et al. (2003). For the first example, we report on a study of 41 students at a major university in the US, 17 of whom took our calculus course (EXP) and 24 of whom took a traditional, lecture-based calculus course (TRAD). Using a combination of a written instrument and intensive interviews, we rated student understanding and performance on four aspects of the derivative. We used the following rating system.

3–Student appeared to understand completely. 2–Student appeared to have the main ideas. 1–Student displayed little or no understanding.

The aspects of calculus and the percentage of students receiving each rating are given in the following table. Aspect of Calculus EXP TRAD Understands that the value of f ′ (x) is the slope of the tangent at (x, f(x)). 3 rating 2 rating 1 rating

71% 18 12

46% 13 42

Can deal with the derivative of a function based only on graphical information without explicit use of expression.

3 rating 2 rating 1 rating

71% 29 0

50% 0 50

Understands that, as the value of the derivative approaches infinity, the slope of the graph grows without bound, resulting in an asymptote. 3 rating 2 rating 1 rating

82% 0 18

63% 4 33

Understands how to use the derivative to determine intervals of monotonicity. 3 rating 2 rating 1 rating

100% 0 0

71% 0 29

These results were obtained within a few months after the completion of the course. We also did a long range study over a 4-year period of 4636 students, 205 who took our course and 4431 who took traditional calculus. We asked about the average of the students’ final grade in the first (of 3) semesters of calculus, the average of their final grade in all mathematics courses, the average number of the 3-course calculus sequence they completed and the average number of other mathematics courses the student subsequently took. The grades were rated on a scale of 0-4 (4 corresponding to an A and 0 for failing the course) and weighted by the number of hours of credit given for the course to obtain a grade-point average. In all cases, we report the difference between the TRAD and EXP students and the p-value of significance for each numerical result. Following is a table of the results. In all cases, the differences favored the EXP students. Question Difference p-value Average final grade in first calculus course 0.42 0.0001 Average final grade for all math courses 0 NA Overall grade point average 0.09 0.05 Average number of the 3 calculus courses taken 0.24 0.0001 Average number of non calculus, math courses taken 0.16 0.025 The following summarizes what these results suggest. • The EXP students were more likely to finish the calculus sequence than were the TRAD students. • The EXP students earned higher grades in calculus courses than the TRAD students. • EXP students were more likely to take non-calculus mathematics courses than were the TRAD students. • The EXP students did about as well in mathematics courses beyond the calculus programs as did the TRAD students. Conclusions We feel that the following conclusions are supported by the results we have obtained.

• APOS Theory is a useful tool for analyzing how students do or do not come to understand a mathematical concept. • Pedagogy based on APOS Theory is difficult to implement. • Pedagogy based on APOS Theory seems to lead to encouraging results both in comparison with students taking traditional courses and in absolute terms. • Pedagogy based on APOS Theory seems to act more like a pump than a filter as regards student interest in mathematics. References Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K.(1996 ). A Framework for Research and Curriculum Development in Undergraduate Mathematics Education. Research in Collegiate Mathematics Education, Issues in Mathematics Education, 2, 1-32. Weller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M., & Merkovsky, R. (2003). Student performance and attitudes in courses based on APOS Theory and the ACE Teaching Cycle. In A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Research in Collegiate Mathematics Education, Issues in Mathematics Education, 5, 97-131.

Perceptivism and Representation in a Discourse Model of Science Learning Cedric Linder

Division of Physics Education, Department of Physics and Astronomy, Uppsala University, Sweden, and Physics Department, University of the Western Cape, South Africa.

Abstract

Relating characterizations of learning to meaningful and effective classroom practices is a seemingly endless challenge to many educators. In this spirit I propose that there is an essential common thread running through contemporary views of learning whose essence can be captured by a synthesis of models of perceptivism and representation, and aspects of the phenomenography variation theory. The outcome is used to suggest implications that arise for a notion of the crafting of teaching practice, which is defined as a constituted practice that is skillful, knowledgeable, reflective and conceptually based.

Introduction

Ever since formal schooling began, different characterizations of learning have strongly influenced the form and content of classroom experience for both learners and teachers. Some well-known examples are behaviourism, cognitivism, constructivism, and more recently, how social and cultural aspects can be used to characterize learning. Perhaps, in this genre of new formulation, the influences of social identity and the benefits of experiencing patterns of variation, are gaining increasing attention. Although quite different in epistemic terms, it is possible to see themes of a common thread running through the grounding of these characterizations; learning as a coming to see new things and/or a coming to see things in a new ways. From a contemporary viewpoint this common thread can be unravelled to argue that disciplinary knowledge sharing (with its associated particular ways of knowing, applying, doing and extending knowledge) is inseparable from its discursive representations (for example, see Roth, & Welzel, 2001, Lemke 1990, 1995, 1998, 2003, Swales 1990, Säljö 1999, Wickman & Östman 2002, Northedge 2002, 2003, Florence & Yore

2004). It is possible to see complimentary aspects in the current ongoing exploration of experiencing of patterns of variation (for examples, see Fraser & Linder 2009, Pang & Marton 2009, Pang et al. 2006, Marton & Tsui 2006, Runesson 2005). A reasonable claim could be made that this kind of meta-analysis could generate a respectable theoretical platform for a pragmatic learning in practice orientation (for example, see Säljö 2000). In the broad terms of the science, mathematics and technology disciplines, however, an accompanying constituted teaching practice that is skillful, knowledgeable, reflective and conceptually based*, still lacks much distinctive, enabling clarification. In particular, insight into what kind of conceptual framing could fruitfully be used by teachers to craft their teaching practice around vis-à-vis the how, what and why of the representation drawn on, and to draw on, for optimizing classroom knowledge sharing for particular contexts of learning∗∗, is still lacking.

From this standpoint my aim is to make a case for a conceptual framing that would reflectively draw on a synthesis of Brauner's (1988) modeling of perceptivism, aspects of the phenomenography variation theory, and Airey & Linder's (2009) modeling of attaining discursive fluency in a critical constellations of modes of representation. Perceptivism An insightful introduction to perceptivism can be found in Charles Brauner’s 1988 brief for the Canadian British Columbia Royal Commission on Education:

Perceptivism is built around the idea that the educated person has a greater command of more worthwhile ways of perceiving things than someone who has not had the opportunity to get an education. Hence the purpose of education is to equip the learner with the capacity to perceive things in the different ways that the most worthwhile approaches to understanding provide (Brauner, 1988, p. 2)

An striking example of how a lack of this kind of perceptivism can block knowledge sharing, and thus praxis change and further productive conceptual development, can be found in the work of Hungarian medical practitioner, Ignaz Semmelweis (1818-1865). From observations taken and made sense of in a quite different perceptivism orientation to the dominant medical one of the time Semmelweis proposed that child birth disease (puerperal fever) was being propagated by agents too small to be seen that were living on the hands of medical students to women in labour and insisted that medical students should wash their hands, clothes and equipment in a chlorine based lime solution before attending deliveries (Burns 2007). This policy led to a profoundly significant decline in childbirth mortality in the hospitals Semmelweis worked with; the rate declined from around 12% to almost 0% (op. cite). However, the medical fraternity at the time did not have access to the perceptivism that underpinned Semmelweis’ theorizing about what today we commonly know as germs. They, thus, strongly rejected his ideas. This so profoundly affected him that he ended up having a mental breakdown and he later died in a Viennese mental hospital when, ironically, after being severely beaten by hospital personal, he died of septicaemia. It was not until Louis Pasteur’s work many years later that Semmelweis’ ideas became accepted. In his 1988 document Brauner went on to present a model of nine modes of perception as follows

1. Standard perception - The world of ordinary experience, that is, seeing trees and horses and houses and knowing what they are.

2. Mythic perception: The extraordinary world of mythology - seeing Santa Claus or seeing

*Drawing on the work of John Dewey and Donald Schön, I have characterized this kind of framing for the reflective design of teaching practice as crafting of teaching practice. ∗∗See enacted object of learning discussed later.

someone “healed” by faith. 3. Operational perception: The technical world of how things work – taking apart an alarm

clock and seeing how it works. 4. Theoretic perception: The non-conventional view of experience offered by the sciences, for

instance seeing germs through a microscope. 5. Thematic perception: The special view of things generated by purely intellectual themes –

seeing justice as the projection of the plutonic form. 6. Thesistic perception: The unique view of things produced in the social sciences. For

example, seeing expensive cars as instances of conspicuous consumption. 7. Relational perception: The relationships revealed by formal systems, such as, grasping the

logic of having two negatives cancel each other out in a sentence. 8. Primary perception: The fresh perspective offered by literature. For example, recognizing

Captain Ahab in Melville’s Moby Dick as the embodiment of Puritanism’s obsession with overcoming the forces of nature.

9. Primal perception: the fresh pre-conceptual experience conveyed by the visual arts. For example, seeing the green and yellow colour swirls of an Emily Carr painting in the rain forests of British Columbia. (Brauner, 1988, p. 3):

Brauner used this nine-mode model of perceptivism to make the case that it is only when learners have their perception possibilities broadened across all modes of perception that they will truly gain access to coming to understand things in an appropriate and holistic manner; providing learners with what is needed in order to provide the possibility of the attainment of a much more “complete education”. What Brauner does not expand upon is how these modes of preceptivism can be productively attained and extended. This is where the theoretical and experimental work in the area of what has become widely known as the Variation Theory of Learning (for example, cf. Marton and Booth 1997, Bowden & Marton 1998, Marton & Pang 1999, Marton & Trigwell 2000, Marton and Morris 2002, Marton and Tsui 2004, Runesson 2005), and the discursive fluency modeling (Airey & Linder 2009) that I introduced earlier come into the formulation of my synthesis.. Linking Enhancing Perceptivism to Experiencing Designed Patterns of Variation and the Discursive Fluency Model The learning advantage: experiencing designed patterns of variation An integral epistemological viewpoint that underpins the variation perspective on learning is the notion of an “object of learning”; that in order for learning to take place there must be a “something” to be learnt, and this “something” is what is referred to at the “object of learning”. Then, any classroom environment carries a continuously interacting mixture of objects of learning – the lived object (from the point of view of the learner what is learnt), the intended object (what teachers would like to have learnt), and the enacted object (the learning that takes place as part of in class interactions; between teacher and learners, and learners and learners). It is how the enacted objects of learning are constituted that is central to the synthesis that I am proposing (cf, Runesson 1999):

Some ways of experiencing are more powerful than others (in relation to certain aims). So, the way something is experienced is fundamental to learning. (Runesson 2005 p. 70)

In order to master an object of learning, the enacted object of learning – in other words the

knowledge building and sharing that underpins learning – involves one being able to discern its critical features and also to focus on them simultaneously. Variation theory proposes that in order to discern such critical features it is a great advantage if one experiences patterns of variation that distinctly help make those features noticeable. If such a pattern of variation carries a design that has some features being kept invariant while others are varied, then the learning advantage becomes distinctive (for example, see Marton & Tsui 2004, Carstensen & Bernhard 2004, 2007, 2009, Runesson 2005, Fraser & Linder 2009). My synthesis formulation now has an essential thread; a “more complete” education involves developing perceptivism across the nine modes proposed by Brauner (1988). At the same time, perceptivism in and across these modes has a significantly enhanced possibility of becoming an essential part of learning if the learning experience is embedded in the basic fundamentals of variation theory. For my synthesis I have chosen to characterize this kind learning environment using an enacted object of learning model where the form and content of the associated interactive knowledge sharing and building takes on critical significance in relation to gaining access to what is needed to constitute appropriate and holistic learning, particularly in science, mathematics and technology contexts. Interactive knowledge sharing and building: the discursive fluency model

Discourse, as a synonym for dialogue, conversation, communication and speech, has led to the notion of “disciplinary discourse” being taken to be synonymous with the specialized language that is used in a particular discipline; a somewhat limiting view. As an alternative, Hall, for example, describes discourse in terms of

… ways of referring to or constructing knowledge about a particular topic of practice: a cluster (or formation) of ideas, images and practices, which provide ways of talking about, forms of knowledge and conduct associated with, a particular topic, social activity or institutional site in society. (Hall 1997 p. 6)

One can also include Kress et al.’s (2001) notion of affordances as being able to provide a powerful “generative metaphor” (Schön 1979) for the modeling of the relationship between disciplinary ways of knowledge sharing and knowing, and the representations that make up the “text” of this disciplinary discourse (for example, see Figure 1.):

Several issues open out from this starting-point: if there are a number of distinct modes in operation at the same time (in our description and analysis we focus on speech, image, gesture, action with models, writing, etc.), then the first question is: “Do they offer differing possibilities for representing?” For ourselves we put that question in these terms: “What are the affordances of each mode used in the science classroom; what are the potentials and limitations for representing of each mode?”; and, “Are the modes specialized to function in particular ways. Is speech say, best for this, and image best for that?” (Kress et al. 2001 p. 1)

Figure 1. The modal relationship between disciplinary ways of knowledge sharing and knowing, and the representations making up disciplinary discourse (Airey and Linder 2009 p.29).

Figure 1. illustrates an unpacking of “disciplinary discourse” vis-à-vis Kress et al.’s suggestion that there are “a number of distinct modes in operation at the same time”. The attributes of this Figure, are, I would argue, easily taken to be learning resources in Schön's (1983) sense of being a set of inter-related generative metaphors; when things that are

seen as similar are initially very different from one another, falling into what is usually considered different domains of experience, then seeing-as takes a form I call ”generative metaphor”. (p. 183-184).

However, particularly from an enhancing teaching and learning perspective, each representation form (mode) has its own particular, and sometimes unique, set of seeing-as aspects. Thus, a variation of representation does not necessarily make up a set of generative metaphors that one would need to come to appropriately understand, conceptualize and utilize before one could be considered to have become discursively fluent; becoming discursively fluent is

a process through which handling a mode of disciplinary discourse with respect to a given disciplinary way of knowing in a given context becomes unproblematic, almost second-nature. Thus, in our characterization, if a person is said to be discursively fluent in a particular mode, then they come to understand the ways in which the discipline generally uses that mode when representing a particular way of knowing. (Airey & Linder 2009, p. 33)

From a crafting of teaching practice perspective, how should the notion of becoming discursively fluent become part of the conceptual framing that underpins such crafting of practice? Drawing on the work of diSessa & Sherin (2000) and Ainsworth (1999, 2006) I would argue that there are distinctive mode-related facets of knowledge that teachers need to have to draw on in crafting of their practice.

Facets of knowledge

To illustrate what is meant by “facets of knowledge”, the kind of diversity that may be involved, and

its relevance for teaching and learning in science, I present examples, which I have drawn from physics and chemistry. The physics example consists of a set of representations that Van Heuvelen and Etkina (2006) have shown to be highly relevant for the attainment of what Airey and I have characterized as discursive fluency, in introductory kinematics (see Figure 2). The figure presents a set of common

representations in such a way that the figure facilitates an appreciation of the different facets of knowledge that each of these representations may be able to offer.

Figure 2. A basic set of representation modes for introductory kinematics (Van Heuvelen & Zou 2001 p. 185)

The next example is taken from electrical circuits and deals with Kirchhoff's voltage law (also known as Kirchhoff's loop rule). In a written-language-mode it could be represented as follows, “Since electric fields are conservative the total work performed in moving a test charge around a closed path is zero”. In an oral language-mode it could be represented as follows, “The track-directed sum of the electrical potential differences around any closed circuit must be zero”. And in the image-mode, in a very simplified form for this example, it could be represented as in Figure 3. (for, say, 4 circuit components).

Figure 3. An image-mode representation of Kirchoff’s voltage law (also known as the loop law). A closed loop means starting at any node in the circuit, trace a path through the circuit that returns you to your origin starting-point node. In writing KVL equations, the following convention is used: for a loss in potential across a particular element (i.e., + to -) give the voltage for that particular element a minus sign and vice-versa.

Then, a mathematics-mode representation would be as follows:

These are just four of the possible ways that a student could experience facets of knowledge relating to Kirchoff’s voltage law. Other representation-mode experiences could feasibly involve circuit diagrams, demonstrations, hands-on activities (with batteries, wires, resistors, bulbs, voltmeters and ammeters), a table of voltages and currents for a given circuit, and graphical illustrations.

My last example illustrates an array of some of the more commonly used representation in chemistry, in this case to present facets of knowledge about a reaction sequence.

The reaction formula is one type of representation that gives one general facets of knowledge about reagents and products:

HO— + CH3CH2Br -> CH3CH2OH + Br— The reaction mechanism is another representation that gives access to understanding the ordering of different bonds being broken or formed.

Starting node

OH C

CH3

BrH

H

CH3

BrHO

H H

C

CH3

HOH

H

+ Br

Transition state A mathematical representation of this reaction is the rate expression, which gives you the rate of formation of the product:

Rate = k[CH3CH2Br][OH—]

In their own ways, each of these modes of representation provide access to facets of knowledge, which collectively start to provide the basis for a more holistic understanding. Modes that were not included would probably have other facets that would, for a new learner, remain in the background or simply not be present. Using this modelling John Airey and I have argued that it is only through an appropriate combination of a particular array of these modes that a holistic learning experience can be made possible (in a way, analogous to viewing a physical object from different angles). We did this in an idealized manner, which I will now repeat using Figures 5. through 9 (Airey and Linder 2009, p. 31-33). In Figure 5. an illustrative hypothetical disciplinary way of knowing is formulated to illustrate access to facets of knowledge in relation to the notion of becoming discursively fluent. These facets of knowledge are metaphorically shown as the sides of a hexagon. Figure 6. illustrates how a mode such as mathematics may open access to three of these facets, and Figure 7. how an experimental-mode may enable access to two further facets. Figure 8. illustrates a missing access to a facet of knowing, which is denoted by a question mark to illustratively reflect a lack of teacher craft-knowledge about the constellation of modes needed to enable access to a holistically complete representation of the given disciplinary concepts. In this spirit, Figure 9. illustrates how the addition of an image-mode (a diagram) may fail to represent this missing facet, but could provide a link between the mathematical and experimental modes.

Figure 5. Disciplinary ways of knowing have multiple aspects or as we term them facets. Here we have an idealized representation of a disciplinary way of knowing using a hexagon. Each side of the hexagon represents one facet of the disciplinary way of knowing.

Figure 6. A representation using the mathematical mode of disciplinary discourse allows access to three facets of the disciplinary way of knowing.

Figure 7. Experimental work allows access to two further facets of the disciplinary way of knowing.

Figure 8. To complete constitution of the disciplinary way of knowing access to the sixth facet is still needed. Here we have chosen to label the mode which gives access to this final facet with a question mark, highlighting what we believe is the present situation in much science teaching: there is little teacher craft-knowledge about the particular constellation of modes that may be required to provide a holistically complete possibility for the learning of a given disciplinary way of knowing.

Figure 9. In this final figure, the visual mode is added in the form of a diagram. In this particular case, the addition of the diagram provides a link between the mathematical and the experimental modes, but complete holistic constitution of the disciplinary way of knowing is still impossible. Modes and appresentation

An interesting relationship between modes of disciplinary discourse and disciplinary ways of knowing can be characterized in terms of the phenomenological concept of appresentation, which, in turn can be linked to variation theory. Marton and Booth’s (1997) description below can be seen illustrate this linkage:

…. in addition to what is “presented” to us—that is what we see, hear, smell—we experience other things as well. If we look at a tabletop from above, for instance, we hardly experience it as a two-dimensional surface floating in the air, in spite of the fact that what we see is, strictly speaking, a two-dimensional surface separated in some mysterious way from the ground. But in looking down on a tabletop we experience the legs that support it as well, because the experience is not of a two-dimensional surface, but of a table… That which is not seen, is not even visible is appresented … We wish to apply the concept of appresentation to experiences of abstract entities as well as concrete ones. If we think of the gravitational constant, g, for instance, then the highly abstract formulation made by Newton of how bodies affect one another at a distance is appresented, given that we have acquired sufficient education in and experience of classical physics (pp. 99-100).

In the modeling illustrated in Figures 5. through 9. the idea of appresentation can be depicted as the ability to spontaneously infer the presence of further abstract and concrete facets of a disciplinary way of knowing, over and above those made available through the mode a learner has been presented with. Put in another way, while a single mode of disciplinary discourse can open up the possibility to experience a particular number of facets of a disciplinary way of knowing, in order to holistically experience this way of knowing, there are most likely to also be other facets of the way of knowing that need to be appresent. From here Airey and I argue that learners of a particular discipline may be unable to fully experience a disciplinary way of knowing unless two important criteria are met (Airey & Linder 2009)..

Firstly, at some stage they must have experienced each of the various facets of the way of knowing. This we argue entails multimodal representation, and we hypothesize that a degree of discursive

fluency in a mode may be necessary before some of the facets of knowing that are made available by the mode can be appropriately related to the whole.

Secondly, they need to be able to experience these facets simultaneously (cf. variation theory); that is, when one facet or group of facets is presented to them through a particular mode of disciplinary discourse, the other necessary facets need to be appresent. We suggest this second criterion can only be met after a person has familiarized themselves with each of the relevant modes of disciplinary discourse sufficiently well that experiencing the various facets simultaneously becomes “second nature”, or as we characterize it, when they have become discursively fluent in a critical constellation of modes. An important corollary is that in order to appropriately come to know and understand an intended object of learning, the object of learning must be experienced within an essential configuration of modes of representation. Concluding thoughts In this article I set out to illustrate a case I wanted to make for a conceptual framing that would inform the crafting of teaching practice that could significantly contribute to the optimization of classroom knowledge sharing. For the illustration I have proposed a synthesis of Brauner's (1988) modeling of perceptivism, aspects of the phenomenography variation theory, and Airey & Linder's (2009) modeling of attaining discursive fluency in a critical constellations of modes of representation. At the same time a parallel illustration has emerged; that it is unlikely that a limited number of representation modes can ever generate an appropriate and holistic learning experience as an enacted object of learning. Therefore, a teacher’s crafting of practice must, arguably, need to include an appreciation of what modes of representation and perceptivism are necessary to open up access to all the needed facets of knowing. At the same time this conceptual framing needs to simultaneously bring to the fore that the use of a given set of modes of representation in classroom knowledge sharing can never be sufficient in itself to open up learning opportunities. A linking to perceptivism and variation theory is necessary to create an enhanced possibility for learners to become discursively fluent across modes of representation and perceptivism. In other words, in science, mathematics and technology classroom knowledge sharing there is there are critical constellations of learning experiences that are needed, and I suggest that my proposed synthesis formulation can offer a fruitful platform to build that teaching and learning knowledge on. Acknowledgements Funding from the three research boards at the University of Kalmar for John Airey’s collaboration and from the Swedish Research Council is gratefully acknowledged. The classroom representation reflections that Inger Edfors and Susanne Wikman shared with me are also gratefully acknowledged.

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The Role of New Technologies in Science and Mathematics Education: Understanding When and Why High Tech Education Tools Work

Noah D. Finkelstein, 1 Wendy Adams, and Katherine Perkins 1 Department of Physics, University of Colorado at Boulder

1 [email protected] We examine the development, role and promise of technology in education, focusing on work in the PhET Interactive Simulations project (PhET). The PhET team has developed a suite of more than 80 free, downloadable simulations that span introductory physics, as well as simulations on more advanced physics, chemistry, earth science, biology, and mathematics. These research-based simulations are designed to promote student understanding and interest in science and to provide complementary tools to the canonical materials (real equipment, textbooks, etc.) used in educational environments. We present the research design and sample studies that document the utility of these simulations. In comparisons of the use of PhET simulations to the use of more traditional educational resources in lecture, laboratory, recitation and informal settings of introductory college physics, our studies demonstrate that simulations can be as productive, or more productive, for developing student conceptual understanding and appreciation of science as real equipment, reading resources, or chalk-talk lectures. We further identify six key characteristic features of these simulations that begin to delineate why these are productive tools. The simulations: support an interactive approach, employ dynamic feedback, follow a constructivist approach, provide a creative workplace, make explicit otherwise inaccessible models or phenomena, and constrain students productively.

Introduction Since the widespread popularization of the microcomputer in the 1980’s educators and researchers have been speaking of the promise of this tool to promote a transformation in student learning opportunities (Turkle, 1997). While there are many examples of effective uses of these new tools, the promise of

computers and computer simulations to reach the broadest populations and to radically transform educational practice in widespread fashion has yet to be realized. In part, we posit that the development and use of computer simulations has lacked a theory and research-base from which it adequately draws. This paper zooms in on the role of computer simulations, and particularly the role of computer simulations in introductory, college level physics to examine when and why these tools can work. In addition to providing novel findings, this paper draws material and summarizes results from several key pieces from the PhET Team (PhET, 2009; Adams, 2009; Finkelstein et al, 2006; Keller et al 2007).

While computer simulations have become relatively widespread in college education (CERI, 2005; MERLOT, n.d.), the evaluation and framing of their utility has been less prevalent. This paper introduces the PhET Interactive Simulations (PhET) project (PhET, 2006), identifies some of the key features of these educational tools, demonstrates their utility, and examines why these are useful. Because it is difficult (and, in this case, unproductive) to separate a tool from its particular use, we examine the use of the interactive PhET simulations in a variety of educational environments typical of introductory college physics. At present, comprehensive and well-controlled studies of the utility of computer simulations in real educational environments remain relatively sparse, particularly at the college level. This paper summarizes the use of the PhET tools in lecture, laboratory, recitation, and informal environments for a broad range of students (from physics majors to non-science majors with little or no background in science). We document some of the features of the simulations (e.g., the critical role of direct and dynamic feedback for students) and how these design features are used (e.g., the particular tasks assigned to students). We find, for a wide variety of environments and uses surveyed, PhET simulations are as productive or more productive than traditional educational tools, whether these are physical equipment or textbooks.

Theoretical Framing Research on learning has demonstrated that students build new ideas based on their prior understanding (Bransford, 2004). Learning is an active process where students are engaged in sense-making, relating new ideas and concepts to prior ideas. As a result, direct instruction, in which students are positioned as passive recipients of knowledge and rewarded for providing answers (rather than reasoning) is less effective than techniques that are interactive, engaging and focussing on reasoning behind answers (Hake 1996). The PhET simulations are designed to be interactive, engage and challenge students so that they may make step-wise progress to a more expert conceptual framework.

At the same time, our work also draws from a socio-cultural perspective of human development and student learning (Cole, 1996; Lemke, 2001; Leontiev 1978; Vygotsky, 1963). A basic tenet of this perspective is that humans learn (and develop) through social interaction. While this perspective attends to the importance of individuals and a neuro-cognitive basis in learning, this theoretical framing forefronts the roles of social interactions and culture in human development. In the present work, we draw from applications of the sociocultural perspective to science and science education (Cole, 1996; Engestrom, 1999; Finkesltein, 2005; Latour, 1987; Lave, 1988; Lemke, 2001), by recognizing that both science and schooling are socially constructed and socially rooted activities, with established norms, rules and roles of participants. From this perspective, we examine the complex interplay among a variety of critical resources that individuals bring to the social environment comprising undergraduate physics education. We explicitly consider student background and what activities we engage students in as they use these new educational tools

Bringing together the learning-sciences and sociocultural perspectives, we consider how computer simulations, as cognitive tools, shape how students might engage with concepts and the conditions that support or inhibit students’ use of these tools. Drawing from prior work (Wartofsky, M., 1973); Cole, M., 1996) we note that the computer simulations, like all tools, are historically rooted, and come with particular predispositions for use. After introducing the PhET simulations, we return to this idea of tool use to identify the particular characteristics of these computer simulations that make them effective for our educational goals.

PhET Simulations PhET Interactive Simulations are a substantial (~85) and growing suite of professional quality simulations (sims) for teaching and learning science. The sims are freely distributed from the PhET website (http://phet.colorado.edu), with roughly 10 million uses in the past year. The majority of PhET sims are for teaching physics but there are a growing number in chemistry, biology, math and other sciences. Considerable research has investigated the use of PhET sims in a variety of educational settings (PhET Team, 2009). The simulations are designed to be highly interactive, engaging, and open learning environments that provide animated feedback to the user. The simulations are physically accurate, and provide highly visual, dynamic representations of physics principles. Simultaneously, the simulations seek to build explicit bridges between students’ everyday understanding of the world and the underlying physical principles, often by making the physical models (such as current flow or electric field lines) explicit. For instance, a student learning about electromagnetic radiation starts with a radio station transmitter and an antenna at a nearby house, shown in Figure 1. Students can force an electron to oscillate up and down at the transmission station, and observe the propagation of the electric field and the resulting motion of an electron at the receiving antenna. A variety of virtual observation and measurement tools are provided to encourage students to explore properties of this micro-world (diSessa, 2000) and allow quantitative analysis.

Figure 1. Screenshot of PhET simulation, Radios Waves & Electromagnetic Fields.

We employ a research-based approach in our design – incorporating findings from prior research on student understanding (Bransford, Brown, & Cocking, 2002; Redish 2003), simulation design (Clark & Mayer, 2003), and our own testing – to create simulations that support student engagement with and understanding of physics concepts. A typical development team is composed of a programmer, a content expert, and an education specialist. The iterative design cycle begins by delineating the learning goals associated with the simulation and constructing a storyboard around these goals. The underpinning design builds on the idea that students will discover the principles, concepts and relations associated with the simulation through exploration and play. For this approach to be effective, careful choices must be made as to which variables and behaviours are apparent to and controllable by the user, and which are not. After a preliminary version of the simulation is created, it is tested and presented to the larger PhET team to discuss. Particular concerns, bugs, and design features are addressed, as well as elements that need to be identified by users (e.g. will students notice this feature or that feature? will users realize the relations among various components of the simulation?). After complete coding, each simulation is then tested with multiple student interviews and summary reports returned to the design team. After the utility of the simulation to support the particular learning goals is established (as assessed by student interviews), the simulations are user-tested through in-class and out-of-class activities. Based on findings from the interviews, user testing, and class implementation, the simulation is refined and re-evaluated as necessary. Knowledge gained from these evaluations is incorporated into the guidelines for general design and informs the development of new simulations (Adams 2008a,b). Ultimately, these simulations are posted for free use on the internet. More on the PhET project and the research methods used to develop the simulations is available online (PhET, 2006).

From the research literature and our evaluation of the PhET simulations, we have identified a variety of characteristics that support student learning. We make no claims that these are necessary or sufficient of

all learning environments – student learning can occur in a myriad of ways and may depend upon more than these characteristic features. However, these features help us to understand why these simulations do (and do not) support student learning in particular environments. Our simulations incorporate:

An Engaging and Interactive Approach. The simulations encourage student engagement. As is now thoroughly documented in the physics education research community and elsewhere (Bransford, Brown, & Cocking, 2002; Hake, 1998; Mazur, 1997; Redish, 2003), environments that interactively engage students are supportive of student learning. At start-up for instance, the simulations literally invite users to engage with the components of the simulated environment.

Dynamic feedback. These simulations emphasize causal relations by linking ideas temporally and graphically. Direct feedback to student interaction with a simulation control provides a temporal and visual link between related concepts. Such an approach, when focused appropriately, facilitates student understanding of the concepts and relations among them (Clark & Mayer, 2003). For instance, when a student moves an electron up and down on an antenna, an oscillating electric field propagates from the antenna suggesting the causal relation among electron acceleration and radio wave generation.

A constructivist approach. Students learn by building on their prior understanding through a series of constrained and supportive explorations (von Glasersfeld, 1983). Furthermore, often students build (virtual) objects in the simulation, which further serves to motivate, ground, and support student learning (Papert & Harel, 1991).

A workspace for play and tinkering. Many of the simulations create a self-consistent world, allowing students to learn about key features of a system by engaging them in systematic play, "messing about," and open-ended investigation (diSessa, 2000).

Visual models / access to conceptual physical models. Many of the microscopic and temporally rich models of physics are made explicit to encourage students to observe otherwise invisible features of a system (Finkelstein, et al., 2005; Perkins et al., 2006). This approach includes visual representations of electrons, photons, air molecules, electric fields etc., as well as the ability to slow down, reverse and play back time.

Productive constraints for students. By simplifying the systems in which students engage, they are encouraged to focus on physically relevant features rather than ancillary or accidental conditions (Finkelstein, et al., 2005). Carefully segmented features introduce relatively few concepts at a time (Clark & Mayer, 2003) and allow for students to build up understanding by learning key features (e.g., current flow) before advanced features (e.g., internal resistance of a battery) are added.

While not an exhaustive study of the characteristics that promote student learning, these key features serve to frame the studies of student learning using the PhET simulations in environments typical of college and other educational institutions: lecture, lab, recitation, and informal settings.

Research Studies Lecture

Simulations can be used in a variety of ways in the lecture environment. Most often they are used to take the place of, or augment chalk-talk or demonstration activities. As such, they fit within a number of pedagogical reforms found in physics lectures, such as Interactive Lecture Demonstrations (Sokoloff & Thornton, 1998) or Peer Instruction (Mazur, 1997).

In an investigation substituting simulations for real demonstration equipment, we studied a several-hundred student calculus-based second semester introductory course on electricity and magnetism. The class was composed of engineering and physics majors (typically freshmen) who regularly interacted in class through Peer Instruction (Mazur, 1997) and personal response systems. The large class necessitated two lecture sections (of roughly 175 students each) taught by the same instructor. To study the impact of computer simulations, the Circuit Construction Kit was substituted for chalk-talk or real demonstration equipment in one of the two lectures.

Figure 2. Screenshot of Circuit Construction Kit simulation.

The Circuit Construction Kit (CCK) models the behaviour of simple electric circuits and includes an open workspace where students can place resistors, light bulbs, wires and batteries. Each element has operating parameters (such as resistance or voltage) that may be varied by the user and measured by a simulated voltmeter and ammeter. The underlying algorithm uses Kirchhoff’s laws to calculate current and voltage through the circuit. The batteries and wires are designed to operate either as ideal components or as real components, by including appropriate, finite resistance. The light bulbs, however, are modelled as Ohmic, in order to emphasize the basic models of circuits that are introduced in introductory physics courses. Moving blue spheres are explicitly shown to visualize current flow and current conservation. A fair amount of attention has been placed on the user interface to ensure that users may easily interact with the simulation and to encourage users to make observations that have been found to be important and difficult for students (McDermott & Shaffer 1992) as they develop a robust conceptual understanding of electric circuits. A screen shot appears in Figure 2.

In this study, students in both lecture sections first participated in a control activity– a real demonstration not related to circuits followed by Peer Instruction. Subsequently the two parallel lectures were divided by treatment – students in one lecture observed a demonstration with chalk diagrams accompanying a real circuit demonstration (traditional); students in the other lecture observed the same circuits built using the CCK simulation (experimental). Students in both lectures under both conditions (traditional and experimental) participated in the complete form of Peer Instruction. In this method, the demonstration is given and a question is presented. First the students answer the question individually using personal response systems before any class-wide discussion or instruction; then, students are instructed to discuss the question with their neighbours and answer a second time. These are referred to as “silent” (answering individually) and “discussion” (answering individually after discussing with peers) formats.

In the control condition, Figure 3a, there are no statistical differences between the two lecture environments, as measured by their pre- or post-scores, or gain (p > 0.5). In the condition where different treatments were used in the two lectures (Figure 3b) – Lecture 1 using CCK and Lecture 2

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CCK group (Lecture 1) is somewhat lower in “silent” score, their final scores after discussion are significantly higher than their counterparts (as are their gains from pre- to post- scores, p<0.005, by two-tailed z-test). Both sets of data (Figure 3a and 3b) corroborate claims that discussion can dramatically facilitate student learning (Mazur, 1997). However the data also illustrate that what the students have to discuss is significant, with the simulation leading to more fruitful discussions.

Figure 3. Student performance in control (left 3a) and treatment (right 3b) conditions to study the effectiveness of computer simulation in Peer Instruction activities. Standard error of the mean is

indicated by error bars.

While we present data only from a small section of lecture courses and environments, we note that the PhET simulations can be productively used for other classroom interventions. For example, PhET simulations may be used in addition to or even in lieu of making microcomputer-based lab measurements of position, velocity and acceleration of moving objects for the 1-D Interactive Lecture Demonstration (ILD) (Sokoloff & Thornton, 1998). In PhET’s Moving Man, we simulate the movement of a character, tracking position, velocity and acceleration. Not only does the simulation provide the same plotting of real time data that occurs with the ILDs, but Moving Man also allows for replaying data (synchronizing movement and data display), as well as assigning pre-set plots of position, velocity and acceleration and subsequently observing the behaviour (inverting the order of ILD data collection). The utility of PhET simulations has been applied beyond the introductory sequence in advanced courses, such as junior-level undergraduate physical chemistry, where students have used the Gas Properties simulation to examine the dynamics of molecular interaction to develop an understanding of the mechanisms and meaning of the Boltzmann distribution.

In each of these instances, we observe the improved results of students who are encouraged to construct ideas by providing access to otherwise temporally obscured phenomena (e.g., Wave on a String), or invisible models (such as electron flow in CCK or molecular interaction in Gas Properties). These simulations effectively constrain students and focus their attention on desired concepts, relations, or processes. These findings come from original interview testing and modification of the simulation to achieve these results. We hypothesize that it is the simulations' explicit focus of attention, productive constraints, dynamic feedback, and explicit visualization of the otherwise inaccessible phenomena that promote productive student discussion, and the development of student ideas.

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Laboratory

Can simulations be used productively in a laboratory where the environment is decidedly hands-on and designed to give students the opportunity to learn physics through direct interaction with experimental practice and equipment?

In the laboratory segment of a traditional large-scale introductory algebra-based physics course, we examined this question. Most of the details of this study and some of the data have been reported previously (Finkelstein et al., 2005), so here we briefly summarize. In one of the two-hour long laboratories, DC circuits, the class was divided into two groups – those that only used a simulation (CCK) and those that only used real equipment (bulbs, wires, resistors, etc.). The lab activities and questions were matched for the two groups.

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Figure 4. Student performance on final exam questions. CCK indicates student groups using Circuit

Construction Kit simulation; TRAD indicates students using real lab equipment. Error is the std error of the mean.

On the final exam, three DC-circuits questions probed students’ mastery of the basic concepts of current, voltage, and series and parallel circuits. For a given series and parallel circuit, students were asked to: (1) rank the currents through each of the bulbs, (2) rank the voltage drops across the bulbs in the same circuit, and (3) predict whether the current through the first bulb increased, decreased, or remained the same when a switch in the parallel section was opened. In Figure 4, the average number of correct responses for the DC circuits and non-DC-circuit exam questions are shown. The average on the final exam questions not relating to the circuits was the same for the two groups (0.62 for CCK, with N = 99; σ=.18, and 0.61 for TRAD, N = 132; σ=.17). The mean performance on the three circuits questions is 0.59 (σ=.27) for CCK and is 0.48 (σ=.27) for TRAD groups. This is a statistically significantly difference at the level of p<0.002 (by Fisher Test or one-tailed binomial distribution) (Finkelstein et al., 2005).

We also assessed the impact of using the simulation on students’ abilities to manipulate physical equipment. During the last 30 minutes of each lab class, all students engaged in a common challenge worksheet requiring them to assemble a circuit with real equipment, show a TA, and write a description the behaviour of the circuit. For all CCK sections, the average time to complete the circuit challenge was 14.0 minutes; for the Traditional sections, it was 17.7 minutes (statistically significant difference at p<0.01 by two tailed t-test of pooled variance across sections). Also, the CCK group scored 62% correct on the written portion of the challenge, whereas the traditional group scored 55% – a statistically significant shift (p<0.03 by a two-tailed z-test) (Finkelstein et al., 2005).

These data indicate that students learning with the simulation are more capable at understanding, constructing, and writing about real circuits than their counterparts who had been working with real circuit elements all along. In this application the computer simulations take advantage of the features described above – they productively engage students in building ideas by providing a workspace that is simultaneously dynamic and constraining, and allows them to mess about productively.

Informal Settings

During interviews of students outside the classroom, we have found that for students to engage in self-guided exploration of the PhET sims, only minimal guidance need be provided. A controlled study to explore these findings more carefully using two specific types of interview guidance was performed: Open Conceptual Questions and Gentle Guidance. “Open Conceptual Questions” uses one or two challenging conceptual questions to trigger a self-driven exploration, and “Gentle Guidance” uses a carefully–designed activity that asks students to investigate particular controls or features of the simulation. (Adams, 2009; Paulson, 2009). This study used Faraday’s Electromagnetic Lab (Figure 5).

Figure 5: Screenshot of Faraday’s Electromagnetic Lab. Here the ‘Pick up Coil’ tab is shown when the magnet is moving to the right through the coil. The red and white diamonds represent the magnetic field

and the blue spheres represent electrons.

An interview with open conceptual questions includes questions such as: “Can a magnet affect an electron?” and “What are some ways you can make a magnet?” These questions are asked before the student sees the simulation. After answering the questions, students are asked to play with the simulation and think out-loud as they do so.

When students explore a simulation with only these open conceptual questions, we observed them explore many different things, choosing the most inviting items first. During this engaged exploration, about half of the interviewees spontaneously revise their answers to the open conceptual questions as they explore, while the other half appear to have forgotten the questions. Again, if the simulation is too complicated or too intimidating, students do not spontaneously explore the simulation.

Gently guided interviews include a series of questions, typical in educational activities, along the lines of “In the ‘Bar Magnet’ tab, identify the things on the screen and in the controls in the control panel (at the right.) A. What does the ‘Strength’ slider do? B. What does the ‘Field Meter’ do? etc…” With this sort of activity we see that student exploration is limited to looking just enough into the specific aspect that has been asked about to answer the question. Then they wait for the next question. They rarely explore beyond the bounds of the question.

Figure 6: Graph shows the average number of the three sim elements that were noticed by each group. Explored means students carefully investigated the element, just noticed means they clicked on it but did not show any further recognition that it existed and not noticed means there was no indication that

they saw the element as existing. MP* is the results for the missing pieces group of one anomalous student is removed.

Based on these results we hypothesized that by adding and removing particular questions we could direct which aspects of the simulation the students notice. We called this the “Missing Pieces” (MP) treatment. Two questions that mentioned three simulation features were removed from the gently guided activity. When students were interviewed using this protocol, they did not notice the three features; however nearly all students who had the open conceptual questions (which do not explicitly mention any sim items) did notice these elements (Fig. 6).

Conclusion This paper has introduced a new suite of computer simulations from the PhET Interactive Simulations project and demonstrated their utility in a broad range of environments typical of instruction in undergraduate physics. Under the appropriate conditions, we demonstrate that these simulations can be as productive, and often more so, than their traditional educational counterparts, such as textbooks, live demonstrations, and even real equipment. We suspect that an optimal educational experience will involve complementary and synergistic uses of traditional resources, and these new high tech tools.

As we seek to employ these new tools, we must consider how and where they are used as well as for what educational goals they are employed. As such, we have started to delineate some of the key features of the PhET tools and their uses that make them productive. The PhET tools are designed to: support an interactive approach, employ dynamic feedback, follow a constructivist approach, provide a creative a workplace, make explicit otherwise inaccessible models or phenomena, and constrain students productively. While not an exhaustive list, we believe these elements to be critical in the design and effective use of these simulations.

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Acknowledgement

We would like to thank Carl Wieman, Archie Paulson, and Jordan Brown for their many contributions to the work discussed. This work would not have been possible without the expert simulations created by The PhET Team, particularly its software developers Sam Reid, Chris Malley, John Blanco, Mike Dubson, and Jonathan Olson. PhET is supported by the National Science Foundation, the William and Flora Hewlett Foundation, The Excellence Center of Science and Mathematics Education at King Saud University and the University of Colorado.

Science, mathematics and technology education research unlocking barriers

across lived world and expert mediated frontiers of learning for a sustainable future

Rob O’Donoghue Rhodes University

Abstract: Preliminary work with the recent Critical Realist oeuvre, currently shaping a ‘weight of evidence’ turn in research, exposes research paradigms as an elusive fiction and artefact of earlier struggle where the critical tools for mediating opposing propositions were inadequate. This insight points to extending a Critical Realism under-labouring of the mediating theories of difference in Worldview Theory1. Here differences that could not adequately be reconciled at the time have been interpretatively theorised as a matter of differing (cultural) vantage points on the world. There is little doubt that such differences exist after Michael Kearney (1984) but the mediating default of an escape-valve in science education, where difference is read as an alternative and standing against the hegemony of Western Science, is inadequate. An interpretative circle with ambivalence such as this points to inadequate tools for critically mediating cultural differences in relation to the real. This study works with critical realism in a search for a more refined lived world perspective for engaging modern science as a source of object congruent tools to mediate difference. It notes how a numerical grasp of things and better access to the objectified knowledge of modern scientific institutions can be significant. To probe this, three exploratory cases of situated learning and change are examined:

(1) A numerical exposing of the loan shark (Mathematics Education) (2) A situated critical engagement with modern science (Science Education) (3) Exploratory use of a design and assessment tool (Technology Education)

The experience and evidence reported in these cases is of a preliminary nature but the insights are providing some useful perspective on how difference might better be brought into critical engagement with the real in learning interactions. The study concludes by making a case for reducing the current ontological terror at the heart of problem centred pedagogies of environment and sustainability. The sustainable practices move proposed will bring fewer problems and terrors to children, rather working on a critical engagement with situated cultural practices and the modern sciences to enhance or re-imagine current sustainability practices. Context of the study

1 A comprehensive engagement with worldview theory after Aikenhead and Cobern, for example, must remain beyond the scope of this study other than to raise the question of reorientation to an ontology of being a teacher at the interface of lived world and the ordered hierarchies of more object congruent knowledge in modern scientific institutions.

When I received the request from SAARMSTE for a plenary paper, I had just returned from a meeting on the core curriculum for the Advanced Certificate in Education. Here, when I asked how education for sustainable development had been dealt with, I was told that environmental education was an elective course in its own right with its own experiential and problem-solving methodologies that did not lend themselves to being included in a core module for all learning areas. This response brought a realisation that despite the centrality of environment and sustainability in the education project of the 21st Century, conventions of paradigmatic bracketing as specialist fields militated against a deeper and more cross cutting engagement with some of the pressing questions of our times. The experience had me approach this study to clarify environment and sustainability concerns in Science, Mathematics and Technology Education and to explore some of the barriers to relevance and knowledge access. The research evidence examined here emerged in three small method projects with PGCE student teachers; one in Natural Science, another in Technology Education and the third in Mathematics Education during interactions with a colleague teaching on the same course. Lise Westeway, a mathematics specialist who convenes the PGCE (IP), was doing some work on critical mathematics literacy that was intriguing and novel, so I asked her if I could work with her and her data to probe the question of knowledge access and relevance. This had been at the forefront of our work for some years. The research materials on science were plentiful and on the same theme of critical innovation towards a more just, equitable and sustainable future. For data on a technology education focus I reviewed work with an assessment innovation from SAARMSTE’09. Here Andrew Stevens had been supporting us to strengthen the technology education activities in the Environmental Education and Sustainability Unit. This gave me sufficient data on the contemporary challenges and barriers/boarders focus with which to engage the SAARMSTE conference theme. What I still needed, however, was a theoretical framework with which to approach the work. Having been involved in the supervision of some MEd students working with Critical Realism (Bhaskar, 1989; 1993; Sayer, 2000), I decided to undertake the challenge of the plenary paper as a process of working with this oeuvre. The fortunate break that I then had was convening a PhD Research Week where Leigh Price provided an overview of Bhaskar’s Critical Realism. This considerably broadened my insights into the emerging Critical Realism oeuvre and gave me a guiding framework and a hand to hold for the venture into the extremely dense and challenging texts of Roy Bhaskar who is now at the University of London and working in Education. This paper is the outcome of this developing challenge to probe some of the barriers at the globalising frontiers of environment and sustainability concerns that Science, Mathematics and Technology educators will be engaging in the coming years. Undertaking the study was not an easy matter on my part, particularly where some of the conventional wisdom guiding our research and practice surfaced as ground-clearing challenges as one begins to chart a way forward. The paper thus opens with these matters of conceptual orientation and initially presents as somewhat of a challenging read. This work gave me a clearer grasp on questions of knowledge access and relevance where a politics of difference has often been an insurmountable problem. The layered ontology and open model for scientific processes of Critical Realism provided the tools necessary to open up issues and to chart a positive way forward in Science, Mathematics and Technology Education. An attempt to model the necessary changes concludes the paper Critical Realism ground clearing to situate the study In response to the conference theme, this paper approaches education research as a social practice unlocking barriers towards a sustainable future. The purpose of the study is to reveal and disrupt barriers in conventional wisdom, particularly where this stands in the way of

improving knowledge (epistemological) access to science, mathematics and technology as intermeshed disciplinary fields for engaging sustainability in a globalising world of and at risk. In approaching this task, some strongly held conventions in education research were disrupted to derive better orientation in science, mathematics and technology as expert mediated disciplinary fields. The re-orientation opened a terrain beyond a legacy of institutional appropriation and a failing induction that is still notable in most science curriculum settings today. To achieve the necessary ground clearing and reorientation, the study draws on critical realism. With this vantage point, empirical reductionism recedes to foreground an historicised ontology of being in the world (1). This move exposes contradictions in expert-mediated orthodoxy and prevailing perspective on paradigms and cultural worldviews are revealed as barriers to a better mediation of differences in emergent practice (2). Finally, the developing insights beyond assumed difference give effect to a situating shift in orientation that opens contemplation of a purposeful primacy of situating practice in meaningful learning with epistemological access (3).

(1) An historicised ontology of being in the world At a recent seminar on Critical Realism, Leigh Price2 reported how Bhaskar started out in economics, attempting a PhD dissertation in economic theory on the problem of under development. The thesis proved impossible since there is no relation to be found between the mathematical formulae of modern economic theory and the real world issues of underdevelopment. It became clear that modern economic theory after Milton Friedman had immunised itself from any direct reference to the world, developing as a self-referential cycle of predictive equations to inform its theoretical formulations. Being in the world of economics was not a worldly ontological process but an interpretative theoretical act of modelling to predict trends and to inform predictive modelling of capital economic gain. Price described how this experience had Bhaskar probe the question of reference to the world. His realist critical project uncovered how science cannot be reduced to empiricism and that there are three dimensions in reality: the real, the actual and the empirical (a layered ontology). This oeuvre served to re-vindicate ontology (historicised purposeful practice) as a key dimension for re-orientation and agency in the modernist education and sustainability project. (2) Disrupting prevailing orthodoxy of paradigms / world views During the modern period of living in what (Beck, [1986] 1992) termed a Risk Society, new social movements shaped pedagogy and research to foster awareness and change. In this way a failing modernist project gave rise to and shaped Environment Education as one of many responses initiating education activities in relation to emerging concerns like environmental degradation, peace, human rights and HIV for example. The education research culture of the times and an ecological sciences focus in early environmentalism initially demanded a positivist, quasi-experimental (empirical analytical) design in environmental education research. This structural functionalist approach centred on measuring awareness and change soon gave rise to the so-called ‘paradigm wars’ in environmental education research during the early 1990s. An uneasy truce was variously achieved with the mediating hand of Thomas Kuhn’s paradigms and the knowledge-constitutive interests of Juegen Habermas. Positivist, interpretative and critical research paradigms are now inscribed in most research methodology texts so that the co-opted notion of parallel paradigms is still taught to science, mathematics and technology education researchers wishing to investigate aspects

2 These orientating narratives were developed with workshop notes reporting on a Realism and Education conference (University of London 18-20 July 2008) that were produced by Leigh Price, current co-ordinator of the Africa Desk for Critical Realism.

of schooling and disciplinary knowledge acquisition, for example. Here paradigms have become ‘regimes of truth,’ (Foucault, 1972: 131) providing security and identity in a field and often acting as a fortress to insulate upholders of the faith from the assaults of outsiders. Setting aside paradigms as the crumbling ramparts of earlier times is not a trivial matter as these flimsy demarcations of territory and dominance that could not be reconciled at the time become barriers to change. As the barriers crumble, a new and more open terrain emerges and what was behind the barriers and defended as research tradition will undoubtedly become exposed but should not be discounted as these practices need to be recognised at the flawed foundation of much existing knowledge as well as worked with to be extended and re-orientated in emergent research practice so that the blind spots of past science and research are not repeated. Unfortunately few researchers have ever critically read the substance and qualifying detail of Kuhn’s scientific revolutions (or the communicative rationality of Habermas for that matter) to probe for possible contradictions and to note how competing understandings were incommensurable at that time. The inability to reconcile competing cultural orientations had research fall headlong into the blinkered calm of plural perspectives and often a relativism of assumed equal value. An inability to adequately mediate amongst competing positions within the individualising trajectories of the times had shaped a populist muddle of parallel research paradigms and cultural worldviews (different but of equal merit) that persists in education today. Critical Realism after Bhaskar elegantly notes how the transitive dimension in rival positions must refer to the same intransitive dimension of reality or the practices would not be rivals. The perspective also notes that when theory or perspective changes there is little likelihood that the phenomenon in question also changes. These realist insights allows competing propositions to be reconciled with a mediating weight of evidence to resolve the impasse, effectively consigning paradigms and parallel worlds to historical artefact, restoring dialectic (the pulse of freedom) to a catalyst for purposeful engagement in meaning-making and change practices. (3) Re-orientating within a primacy of practice Not surprisingly, an ontology of being in the world had been consigned to the margins in an expert mediated orthodoxy of propositions (philosophy) for bringing order to a modernist landscape in paradigmatically ordered universes of research practice. Norbert Elias waged a relentless war against analytical philosophy for blind spots that obscure an historicised and emergent interplay between a close ontological engagement in the world (involvement) and a standing back that accompanies more formal symbolic representation (detachment). He documented this as an epistemology of the modern sciences a ‘detour via detachment.’ Bhaskar found a flaw in the detaching empiricism of science, noting a hidden ontology implicitly smuggled in, yet left out in the positivist ideal of empiricism after Kant, Hume and Popper. Without this oeuvre philosophy was always about theory and practice (sacred / profane) with modern education philosophy immersed in this dislocating and incomplete orthodoxy. The modern education move to decentre the subject in child-centred-education came with the notable imperative to ascertain what is known for meaning-making engagement (Prior knowledge after David Ausubel, 1963) but this was little more than a surface matter in a science, mathematics and technology orthodoxy of expert mediated concept-to-context pedagogy that gave lip service to contextuality and cultural sovereignty against service to the scientific method and an ordered hierarchies of concepts. Reading this science as Western in the cultural relativism of the recent postmodern condition (Heller and Feher, 1988) has done little but ferment epistemological confusion mediated as a matter of

difference in worldview, the latter (as mentioned earlier) being rooted in the individual without adequate reference to emergence (history) or a real world of objects and processes in and against which ideas were read and tested in the realising of human purpose. Bhaskar locates science as involving both intransitive (being in a human-independent world - ontology) and transitive (with a cultural capital of purposeful propositions - epistemology) where ontology exists and the nature of ontology is layered. Here science is a process that cannot be reduced to empiricism and might best be understood as having been mapped in a world of structures and regularities where human purpose and practice has constituted a hierarchical capital of propositions within modern, expert mediated institutions.

A critical realist ‘underlabouring’ of modern empiricist conventions consigns research paradigms to an interesting artefact of historical struggle. In this way, work with critical realism opens the way to education research that is historically informed and purposefully engaged in evidence generating practice to inform social mediation for the common good. An emergent and situated perspective such as this necessitates a reframing of worldview theory as lived world ontology in a real world and open to mediation for the common good. The attendant re-orientation surfaces a primacy of situated practice where historicised ontology fuels expansive learning practices (with dialectic: the pulse of freedom) amidst the symbolic capital of the times and the axes of tension of the day. Positioning the study Against this and arising from the ground clearing and reorientation process (above), the study is positioned around two intermeshed propositions, namely:

• An emerging sensitivity to mediating cultural preconditions in sustainability practices (culture and discourse coupled with material and economic orientations that Sen (2002) probes in economics as ‘capability,’ the beginnings of an engagement with the complexities of sites of so called ‘under development’ in a global capital economy.)

• A primacy of practice for knowledge access in science education. (Lived world reflexive practice of working with purpose and science)

At one level these point to a situating primacy of cultural practice for meaningful learning with epistemological access in modern curriculum settings. Noting an inadequate sense of the primacy of the mediating cultural preconditions in meaning-making interactions appears to have produced a pedagogical barrier of cultural exclusion, the reification of concepts ordered in detached hierarchies that have been accompanied by a process reduction in science (empiricism as the scientific method). The latent exclusion of being in the world has constituted science as a field of detaching exclusivity where expert communities of practice mediate the induction others. Currently expert mediation to induct modern African children is failing with scientific abstractions proving of little relevance and the field being inaccessible to all but a small elite. Catherine Odora-Hoppers coherently pointed to this when she noted from a cultural practices perspective, that what had been theirs as Africans and remains meaningful to them in lived world experiences and for providing orientation in the world, is not recognisable as belonging to their world or of carrying any meaning with which they can identify. This insightful remark has been variously engaged with some useful affirmation in research evidence and with some movement towards a resolution in recent research after Hanisi (2006), Kota (2006) and Mandikonza (2007) to mention a few local projects. This at times, seemingly insurmountable problem that is currently bedevilled the empiricism of the scientific method and a muddle of parallel cultural worldviews, is surprisingly easily resolved within the ontology of being in a real world and coming to know it within emerging practices of purposeful engagement in developing context and over time. Working with a layered ontology means that being scientific and learning science can now be modelled as engaging processes of purposeful description (1) that open the way to a situating retroduction

(2). From this situated stepping off point, it is possible to engage in mediating hypothesis testing to enable the elimination (3) of ambivalence and the identification (4) of mediating mechanisms that provide an explanatory power to steer our continuing livelihood practices (with correction (5) in the real world). The 5-stage process model of an open and iterative process, DREI(C) of science is reflected by Bhaskar (1989:109) as:

There is the epistemically progressive movement after (1) Referential detachment - This is when an absence or anomaly or contradiction generated by, say, a relevant incompleteness (or a tedious homology induced by replication of already well established results) leads to the (2) retroduction of explanatory hypothesis.

Here, the hypothesis is not the starting point of science as empirical method but a detaching turn out of emergent ontology of being closely engaged in a real world. Just as in the case of unravelling the barrier of contemporary economic theory not being adequately constituted to refer to the problem of underdevelopment, the same insights can help resolve the problem of relevance and knowledge access in a modern curriculum of hierarchically ordered concepts. This simple tool allows us to take up the meaning-making challenge of being in an emergent world of plural representations. It can allow us to reconcile ambivalence and to better mediate sustaining practices at the nexus of who we are, what we know from being in the world and what we can find out by coming to recognise mediating mechanisms that might enable a corrective steering of sustainability practices for the common good. The intergenerational ontology of being in the world and this critical process tool became a framing touchstone for this study to review pedagogical practices so they might be more closely aligned with productive meaning-making in relation to enhancing the human condition and our emergent grasp of the realities of the world we live in. The study was thus located in the emergent globalising field of environment and sustainability education (below) and directed at a preliminary reading of the case evidence to contemplate the question of epistemological access to modern disciplinary knowledge for meaningful learning in a real world. Locating the question of pedagogical orientation in the ontology of being a teacher mediating learning at the plural interface of lived world experience, formal institutional concepts and the issues of the day allowed the respectful setting aside of the political histories of paradigms and any a priori assumptions that come with Worldview Theory. The brief historicised orientation to the field that follows serves to set an emergent context and then each case is examined for perspective that may be useful for continuing work. An emerging environment and sustainability position for the study In the early 1980s, education research in an emerging field of environmental education set out to change values, attitudes and behaviour. This was notable in environmental awareness approaches where experiential learning and problem solving encounters on environmental degradation and the greenhouse effect (global warming) for example, emerged as education practice in nature reserves, schools and local community settings. Overshadowed at the time, by the hole in the o-zone that was causally linked to CFC refrigeration technologies, the science of the greenhouse effect of the 1980s has only recently escalated to global prominence once again. It is now top of the pops on the global hit parade, primarily thanks to ‘an inconvenient truth’ recently shared by Nobel Laureate Al Gore. The new environment and sustainability risk of global warming and global climate change has now briefly been overshadowed once again by the recent economic downturn reflected as ‘the crime of our times,’ only to reappear as an escalating challenge of potentially catastrophic global social-ecological change into the mid 21st Century. The National Research Foundation (NRF) in South Africa is currently mounting a national strategic plan on global change for transition to sustainable livelihood practices centred on democratic governance that ensures sustainable ecological services, economic production and appropriate, low impact technologies. The emerging field of environment and sustainability education was for some time stuck in a

nature experience genre but soon progressed into more problem orientated and problem solving approaches, often drawing these pedagogies from science education. In science, mathematics and technology it contributed to the shaping of learning outcomes that refer to the restoration of natural systems and processes for sustainability and the common good. We have now reached a stage that the conditions of our times are cut through by risk and fear for future sustainability. This is driving a curriculum of fearful alienation and wrath with little chance of redemption because the necessary change is complex, contested and unclear. Critical realism redirects an ontology of intergenerational being in the world, potentially allowing us to reduce the current ontological terror at the heart of problem / issue-centred environment and sustainability education, pointing to a methodological move from taking problems to children to a curriculum of engagement in sustainability practices for the common good. Drawing on Habermas, Abbinett, (2003) notes the ‘objectifying’ power of the modern institutions that are the drivers of this ontological terror and steering mechanisms to mediate social re-orientation, with communication, regulative intervention and education for the common good. He notes that: ….considered as an ‘expert culture’ whose rationality is increasingly distant from the interpretative structures of everyday life, and increasingly powerful in its influence over the steering mechanisms which control the techno-economic future of society, science enters into the political dynamics of the lifeworld as a force whose methodological, epistemological and theoretical procedures always require communicative reinterpretation (Abbinett, 2003:164). Beck, noting this role of science in modernity, also finds evidence of an increasingly ethical and critical modern science noting how this may be borne of the increasing scientisation of risk awareness that he found evidence of in his ‘Risk Society.’ Working from a 20th Century reconstituting of earlier imperial sciences into modern scientific institutions, the study is directed at examining three small-scale experiments in mathematics, science and technology education. This is undertaken so as to surface pedagogical practices that are better situated in lived world experience and better directed to critically access disciplinary knowledge and skills for enabling better cultural orientation in the world. The three cases of ‘Loan Shark mathematics,’ ‘fermentation science for healthier foods’ and ‘creative garden design technologies’ are probed with these critical processes of situating relevance and knowledge access in mind. Each case is reviewed as a small-scale innovation undertaken with student teachers invited to step into the modernist pedagogical space of a ‘risk society’ and open up situated curriculum practices for critical knowledge access towards enhanced relevance and agency for the social mediation of a sustainable future. Case 1: Loan shark mathematics After an informal discussion on house prices and access to the housing market Lise Westeway the mathematics lecturer and PGCE(IP) course convenor developed an exploratory task to investigate how much is paid for a house over the period of a mortgage bond. Being a critical task that could be replicated in the primary school classroom, the complexity of compound interest was excluded. The study was made more locally relevant with a case example and a newspaper article questioning the practice of negotiating a bond extension to reduce monthly payments. The students had little grasp of the issues until they started ‘doing the numbers’ on buying a house over 20 years @ prime minus 0.5% with prime being 12.5%. One of the most interesting tasks was calculating what it would mean to extend a 20-year bond to a 30-year period. Interestingly this practice proliferated in the USA housing market with ‘refinancing’ in the global financial market to, in no small measure, contributed to the current global downturn that has been recently labelled ‘the crime of our times’ (mentioned above). One group put a positive spin on the extension to 30 years as a necessary risk that allowed access to the housing market that is now beyond most young families. The others, however, represented the practice as somewhat of a moneymaking racket that gave rise to teatime

references to ‘loan-shark mathematics.’ A formal analysis of the learning was not undertaken at the time but the tasks and letters retained for review argue various positions and were revisited for this paper. Most notable was how the topic grabbed the students, firing them up as the numeric representations emerged and were contested within divergent interpretations of and views on the matter. The case was not only relevant but the numerical representations both gave the students a grasp of the issue and these data became grist for the mill when arguing out details and taking a position on the matter. The students already had a grasp of the mathematical concepts and processes involved but did not intuitively deploy this when confronted with the problem. They thus remained somewhat in the dark until the mathematical representations exposed the issue in a tangible and graspable form around which it was possible to take up a position and to argue a case for and against the practice. With hindsight, the importance of the deployment of mathematics to gain a grasp of these matters elevates this small case of critical mathematical engagement in response to a relevant issue. The key point here is that a numeric representation of the issue provides a closer hold on an issue and can open the way to an evidence-based re-imagining of possibilities with the agency necessary to see beyond the ambivalence of uncertainties borne of matters that were beyond immediate grasp without the tools of mediating critical representation. This recent focus in critical mathematics and the mathematics literacy curriculum is not only of relevance to the discipline but to many other disciplinary areas engaging environment and sustainability issues. After this sort of experience, few mathematics teachers are prepared to leave the application of numerical representation of issues to other disciplinary fields. It has been notable in Eco-Schools, for example that water use audits are seldom adequately conducted and interpreted without input from the mathematics teacher. Hofmann (2008) for example noted how auditing of problems was often undertaken as a surface verification of what was already believed to be the case. This approach did not allow the students to gain access to a firm grasp of the issue so that they could work with the complexity of the knowledge necessary for an explanatory grasp that often opened the way to doing something about the issue. This suggests that becoming aware and experiencing an issue is of a different order and that the use of mathematical representations as a process that opens the way to knowledge access is critical to processes of reflexive change. Despite it being self-evident that formal numerical representations contribute to an enhanced grasp of most contemporary issues, this critical dimension of mathematics has not been coherently taken up into the school curriculum manifestation of the discipline. Somewhat sadly, what has happened is a split so that mathematics literacy can be done by the less mathematically able and the curriculum is playing out as an arena of less demanding numerical operations rather than an arena for critical mathematical engagement in the issues of a world risk society. The next case in science is more closely located in a modern African context and at the interface between a lived world of intergenerational, everyday knowledge and the abstract hierarchies of disciplinary knowledge of the curriculum. The challenge here for the student teachers was to work with the co-presence of situating orientation that provides relevance (Everyday culture) and abstract propositions that can add explanatory insight and enhance our grasp of emergent risk (Expert culture). Case 2: Accessing scientific knowledge Over a seven-week module and into teaching practice, the PGCE(IP) science students worked to differentiate key propositions in the science curriculum and to align these with the everyday knowledge practices and prior knowledge of school learners. The students worked in groups to unpack the science curriculum in relation to daily life and the issues of the day. Their tasks was to explore continuities and contradictions with the assistance of some simple questions:

• What are the big ideas we have to teach? • What are the connecting practices in the everyday? • How do these relate to the issues of the day?

Some of the topics that we have worked on together have been particularly interesting, especially probing the scientific propositions that they as learners in school did not find relevant or understand fully themselves. The emphasis on knowing disciplinary content coincides with one of the PGCE curriculum imperatives of teaching students knowledge of the discipline. Here a subtext of thinking about ‘how I came to learn this content’ seemingly plays out to help open up the double hermeneutic necessary for professional practice i.e. thinking about learning and scaffolding learning processes towards outcomes that are assessed. Experience and evidence from our science curriculum module on fermentation has been briefly represented here to develop this case study. A concern for addressing problems of relevance and epistemological access in school curriculum settings is significant as successive waves of institutional appropriation (colonial and modern) and socio-economic marginalisation has characterised both the colonial era of western scientific imperialism and the modern market economy of techno-scientific objectification. This legacy is still with us as barriers to learning continue to exclude young African learners today. A patchwork of fermentation practices were described as we opened up the topic together, with side notes on patterns of change and risk. Here evidence was gathered on an Nguni mastery of fermentation that preceded the colonial era. Nguni knowledge practices were not static and notable innovations are amaRewu being made with cooked maize meal and later with a wheat ferment starter. A similar trajectory of innovation was tracked in dumplings and steam bread, with isonka (oven bread) being mastered in Boer kitchens to later be deployed in resisting apartheid relocations, locally notably in amaHlubi oral histories in the Herschel District. These issues and ideas were further explored and worked with in a Mother Tongue demonstration of fermented foods by Gladys Tyatya where isiXhosa knowledge practices were brought into our science classroom and were related to the diverse cultural practices amongst of all of the participating student teachers. The demonstration of Xhosa fermented foods was followed by a brainstorming and web search activities to map out ideas and open up information on health risk issues that had arisen with the advent of the modern diet. Obesity, diabetes and gluten allergies were highlighted here, as it was noted that bread is now the staple diet of modern South Africa. The group then did a ‘back-analysis’ of changing food production practices to identify how changed fermentation time was shaping modern health risks. The third arena of engagement was the modern science curriculum. The purpose here was to search for how abstractions might enable an additive explanatory logic to either shed light on what was common sense knowledge and what change practices were contributing to emerging issues and risk. Some of the things to surface here were modern baking practices and the industrial appropriation of rural baking as multinational monopolies like Tiger Brands3 developed, particularly after 1994. (This company had recently been fined millions of Rand after recent Competition Commission hearings on collusion and price fixing). The emergent complex mix of indigenous knowledge practices, modernising patterns of change, curriculum concepts in modern science and emerging issues of health and sustainability was near impossible to work with. We thus mapped these out on a model of three intersecting dimensions, cultural practices, issues of the day and formal scientific knowledge (See slide 1 below). The concept maps allowed the students to locate themselves as teachers having to

3 This company had recently been fined millions of Rand after recent Competition Commission hearings on collusion and price fixing

operate at the nexus of intergenerational everyday practices, social ecological risk of the day and the ordered hierarchy of modern scientific objectifications of a real world. And then, ‘how to teach this’ became the agenda. The task was to develop a learning programme that is of relevance in the everyday as well as providing knowledge access to the explanatory power of modern scientific propositions (concepts) for informing future social-ecological sustainability. Here an active learning framework (Start-up story (DR) leading to finding out and trying out (EI) dimensions and deliberating synthesis (C)) was used to scaffold the learning programme assignment. The two-stage process of generating materials and beginning to order these in science teaching practices is provided in the two power-point slides below.

Learning as a situating interplay amongsteveryday and expert cultures

Everyday /indigenousknowledgepractices

Intergenerationalnexus of events &representations

of order

Illustrativeactivity /

experiment

Abstractingnexus of events &representations

of order

Modern context of social - ecological risk

(C)IERD

Processes

Context(Proximity of Being)

Expert Culture (Curriculum concepts)

Everyday Culture(History / practices)

In relationto situatedpractices

Situating Engaging Reflection

Start-up:Authenticstory oflearning &change

Materials for change orientated learning in relationto sustainable practices in community & school

Findingout aboutthings &

Story sharingshapes learnerpurpose forÉ

(experiences, ideas,concerns, questions

and possibilities)

Prior knowledge &information

Enquiry & action Reporting &reflection

(Curriculum Learning Areas & outcomes) (Assessment standards Š scope & depth)with

Tryingnew

things out&

Deliberatingchanged

sustainabilitypractices

Developingstory of change

to moresustainablepractices

In a first attempt at structuring materials using start-up stories, although creative and meaningful in their own learning, were not very effective as scaffolding for classroom learning interactions. We thus deepened the analysis by mapping the critical nexus of events at the interface between the everyday cultural knowledge and the expert culture of modern scientific propositions of the curriculum (slide 3 below).

Mapping key nexus of events forepistemological access with relevance

� Fermentation practiceby cracking grain� smell of malting in� warm calabash with� visible bubbles asevidence of success A microscope reveals cell division of yeast

Abstract concepts allow us to test & deduce:� Bubbles are carbon dioxide being released� Yeast (enzymes) break down starch/glucose

Abstract modelling allow us to formulate how:�Maltase and other enzymes break down starchC6H12O6 with yeast -> 2 C2H5OH + 2CO 2 (glucose) (alcohol) (carbon dioxide)

Abstracting theory enables the use of concepts, modelsand insights in context and other nexus of events

Concepts, models andtheory in intergenerationalknowledge practice offermenting sorghum .

LIVED WORLDCULTURAL CONTEXT

EXPERT FIELD CULTURAL CAPITAL

(C)IERD

ProcessesModern context of social - ecological risk

The more detailed mapping of the critical nexus of events implicitly structured using the layered ontology (real; actual; empirical) of Critical Realism opened the way for relating each critical nexus to the object congruent constituents of the modern sciences. This allowed a co-validating of the knowledge systems and a mapping out of access routes into the disciplinary field of the modern sciences as this is represented in the current school curriculum. The now more visible lines of constructive knowledge access became clearer and this allowed the

students to scaffold activities and design illustrative experiments to enhance knowledge access. What I subsequently did was insert the Critical Realism DREI(C) model of science. This provides a further enhancement that is allowing us to contemplate DR that leads to hypothesis-led EI processes with deliberative (C) to unlock the explanatory power of causal mechanisms for deployment in the creative re-imagining of sustainability practices that might mitigate some of the pressing environmental concerns of our times. This work is still ongoing. The key point out of the emergent experiences and evidence represented here is that the situated description and retroduction processes appears to correct a misalignment / gulf between the intergenerational cultures in the everyday and modern empirical science disciplines. It also opens up insights into how modern disciplinary knowledge might be used to more explicitly unlock causal dimensions for a better explanatory grasp of the complex environment and sustainability challenges of the day. A Critical Realism approach also detours the current interpretative paradox of plural cultures and worldviews apart, opening a prospect of plural processes of cultural co-engagement towards innovative change to reduce the impact of many modern livelihood practices on our shared life-support systems. Here it is notable that far from being cultural process that is unitary in character, the student’s ways of working with learners and science were creative and diverse. In a similar way the next case illustrates that our assumptions about the creative design process cannot exclude the need for an explanatory grasp of causal knowledge as a pre-requisite for effective design. Case 3: Working with an assessment framework for technology innovation When I wanted to include more of an environment and sustainability focus in my PGCE science and environmental education courses, my colleague Andrew Stevens emphasised that the design and technology curriculum required more than the students simply reproducing technologies that are healthier and more environment friendly. Getting the balance right was not easy as the sustainability science and environmental knowledge would always drive the design towards rather narrow parameters. After consulting with Richard Kimbell, I asked Andrew if he would assist us to work with the assessment template that Richard presented in his 2009 plenary at SAARMSTE. We did some preliminary work on waste but then switched to creative garden design when the opportunity came to work on a ‘Handprint for Change’ resource book for environmental learning. Creating the Handprint resource involved working with an authentic case of environmental learning and change from an Eco-School project, in this case a science teacher working with learners to design and implement tower gardens for food production using less water. An element of the earlier waste management focus (compost bins and worm farms) was retained as an integral part of the low-cost, low-impact, organic food gardening design task. The emerging design and technology case study involved mapping out a design task that could be assessed for the extent to which the design criteria were successfully met. We worked with the 16-stage design process on the technology assessment template4 (Kimbell et al., 2004) and did a detailed mapping out of the background knowledge that was necessary to inform the design task. We also decided to run the design workshop at Sci-Fest and as a pilot project towards a more detailed study. The design task was tied to reducing our ecological footprint to mitigate climate change and was thus presented as a challenge, “Don’t pick and pay the BIG FOOT way, but plant and pick

4 Kimbell reported that his project that had set out to more rationally assess design tasks had a two-fold outcome, one being the refinement and verification of the design process and secondly, the development of a rational assessment process. My interest was in the design process where his fear of being too prescriptive had an overwhelmingly response from learners that they has sufficient scope to be able to report that they had experienced a supported sense of design autonomy.

the Handprint way!” The content was presented as a mix of information sheets summarised on a 10-minute power point presentation. The workshop participants then had an hour and a half to come up with their designs, interacting with others and documenting progress as the design process unfolded. Participants could either draw or model their design and the workshop was concluded with teach-backs on the key features (structural / explanatory) of their designs. An analysis of the pilot study data pointed to a need for a better situating of the design process, notably a description of the context for which the garden is to be designed. Whereas we had concentrated on informing the design process in relation to:

• Sourcing seeds and plants for free • Producing your own low-cost plant nutrients • Using a source of clean, low-cost water • Keeping pests away without costly poisons,

There was clearly a need for more situating work in relation to materials and capabilities in context. Although we were unable to keep track what the participants did with their designs, some clearly developed gardens as I had some vegetables arriving at my office with notes of appreciation and one participant sent me some heirloom seed that she had acquired. This pilot study will be more fully explored next year. The key point here is that the closer and unfolding scaffolding of the design process shaped an arena of innovation that was directed to purposeful change and practical application in the world. Synthesis and discussion The ground clearing work at the opening of this study was both a demanding analysis and an unsettling process. Whereas I was reasonably comfortable with moving beyond the strictures of paradigms, I was not prepared for the ripple effect into the foundational assumptions of World View Theory constituted within an interpretative mediating amongst clearly different cultures. I thus had to leave the bigger question of ‘where to from here’ and merely point to an ontological re-orientation into lived world experience. However, as I worked through the problem in the research design, I felt more comfortable with a reorientation to a more object congruent pluralism, namely co-engagement amongst differences to constitute and work on a diverse range of alternatives. This goes against an earlier alienating resistance to Western Science for an apparent trajectory towards one dominant culture with an attendant loss of diversity. This problem seems to have originated in modernist scientific discourse towards the mediating of change in an institutionally defined direction. My finding to the contrary is that a Critical Realist ontology provides a closer object congruence around which a critical re-imagining of sustainability practices is far from unitary and, if anything, may actually shape a widening range of practices if the innovative orientations in the pedagogy of the students are anything to go by. Perhaps this is best illustrated from within the notion of a primacy of practice. As one probes the critical nexus of events in emergent practices and these are carried into the more object congruent knowledge capital of the modern sciences, the range of options appears to correspond more with the diversity of the socio-ecological than with the steering narrative of the mediating expert culture of modernity. This, in turn, calls for a sceptical resistance to any one-size to fit everyone mentality and a corresponding disposition to regulate everyone within the same rules. The closer one get to the real the more diverse things are from context to context, suggesting that a directive rule of a universal real by science is a failing project. This study suggests that an alternative of cultural co-engagement without the more object congruent foil of the sciences is equally flawed. What Critical Realism offers is better tools to get real and to teach in really engaging ways that provide knowledge access with relevance and agency, something that is a worthy enterprise for science, mathematics and technology education in a 20th Century facing cataclysmic climate change unless we can reinvent our modernist

sustainability practices in the medium term. References Ausubel, D. (1963). The Psychology of Meaningful Verbal Learning. New York: Grune & Stratton Abbinnett, R. (2003) Culture and Identity: Critical Theories. London, Sage. Aikenhead, G.S. (2006) Science Education for Everyday Life: Evidence-based practice. New York, Teachers College Press. Bhaskar, R. (1989) Reclaiming Reality: A Critical Introduction to Contemporary Philosophy. London: Verso. Bhaskar, R. (1993) Dialectic: The Pulse of Freedom. London: Verso. Cobern, W. W. (1991) Worldview Theory and Science Education Research. National Association for Research in Science Teaching Monograph, Number 3, 1991, Manhattan Kansas 66506. Cobern, W. W. (1996) Worldview Theory and Conceptual Change in Science Education. Science Education, 80, 51-59. Cobern and Aikenhead, G. (1998) Cultural aspects of learning science. In Fraser, B. and Tobin, K. (Eds.) International Handbook of Science Education. (pp. 39-52) UK, Kluwer Academic Publishers. Elias, N. (1987) Involvement and Detachment. Oxford: Basil Blackwell. Foucault, M. (1972) Power/Knowledge: Selected interviews and other writings 1972-77. Bury St Edmunds, Harvester Press. Green, B. (2009) (Ed.) Understanding and Researching Professional Practice. Rotterdam, Sense Publishers. Habermas, J. (1994) The Philosophical Discourse of Modernity. Trans. Frederick Lawrence. Cambridge, Polity Press. Hanisi, N. (2006) Nguni Fermented Foods: Mobilising Indigenous Knowledge in the Life Sciences. Unpublished M.Ed study, Department of Education, Rhodes University, Grahamstown. Heller, A. and Feher, F. (1988) The Postmodern Political Condition. London, Polity Press. Kearney, M. (1984) World View. Novato Ca, Chandler and Sharp Publishers. Kimbell, R., Miller, S., Bain, J., Wright, R., Wheeler, T., Stables, K., (2004) Assessing Design Innovation: a research and development project for the Department for Education & Skills (DfES) and the Qualifications and Curriculum Authority (QCA), TERU, Goldsmiths College, London UK. Kota, L. (2006) Local food choices and nutrition: A case study of amaRewu in the Consumer Studies Curriculum. Unpublished M.Ed study, Department of Education, Rhodes University, Grahamstown. Kuhn, T. S. The Structure of Scientific Revolutions. Chicago, Chicago Press. Mandikonza, C. (2007) Mobilising Indigenous Technologies and Intergenerational Ways of Knowing in Science Curriculum: Case of Mutare Teacher’s College, Zimbabwe. Unpublished M.Ed study, Department of Education, Rhodes University, Grahamstown. Sen, A. (2002) Rationality and Freedom. London, Harvard University Press. Somekh, B. and Lewin, C. (2005) Research Methods in the Social Sciences. London, Sage. Acknowledgements: I am grateful to Lise Westaway, Andrew Stevens and Leigh Price for their support and access to guiding ideas, materials and research data for this project. I must apologise at the outset if I have run roughshod over any of these materials or have not adequately probed their depths. All conceptual features of the piece are my responsibility for which I alone am accountable.

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Documents

Programme · presentations, e.g. short papers and snapshots). Notify speakers when (i) 15 mins remain and when (ii) 5 minutes remain (for long presentations, e.g. long papers, symposia