using technology to support mathematics education and research

Using technology to support mathematics

education and researchDr. Christian Bokhove

13 July 2017

Hong Kong

Who am I

• Dr. Christian Bokhove• From 1998-2012 teacher maths, computer science, head of

ICT secondary school Netherlands• National projects Maths & ICT at Freudenthal Instituut,

Utrecht University• PhD 2011 under Prof. Jan van Maanen and prof. Paul Drijvers• Lecturer at University of

Southampton– Maths education– Technology use– Large-scale assessment– Computer Science stuff

Contents

WORKING WITH THE FREUDENTHAL INSTITUTE

ALGEBRA AND TECHNOLOGY

DIGITAL MATHEMATICAL BOOKS

ENGASIA

CONCLUSION

Wisweb and WELP

• Wisweb: collections of (Java) applets

• WELP: integrate the use of the applets in lessons

Galois and sage projects

• Government grant programme for teacher innovations

• Make integrated version

• First version of the ‘Digital Mathematics Environment’ (Peter Boon)

• Sage: prize money

More on FI

With thanks to Prof. Paul Drijvers

Hans Freudenthal (1905-1990)

„Mathematics as human activity“

• construct content from reality

• organize phenomena with mathematical means

The iceberg metaphor

RME Key Characteristics

• Meaningful contexts as starting point for learning

• Progressive mathematization from informal strategies and (horizontal and vertical)

• Intertwinement of content strands• Interaction• Room for students’ own

constructions(Treffers, 1987)

What do we mean by “Realistic”?

“Realistic” may have different meanings:

• Realistic in the sense of feasible in educational practice

• Realistic in the sense of related to real life(real world, phantasy world, math world)

• Realistic in the sense of meaningful, sense making for students

• Realistic in the sense of “zich realiseren” = to realize, to be aware of, to imagine

• HF: “How real the concepts are depends on the conceiver”

11

Key RME design heuristics

A. Guided reinvention

B. Didactical phenomenology

C. Horizontal and Vertical Mathematization

D. Emergent Modeling

A. Guided reinvention

• Reinvention:Reconstructing and developing a mathematical concept in a natural way in a given problem situation.

• Guidance:Students need guidance (from books, peers, teacher) to ascertain convergence towards common mathematical standards

• Reinvention <-> guidance: a balancing act

B. Didactical phenomenology

The art to find phenomena, contexts, problem situations that …

• … beg to be organized by mathematical means

• … invites students to develop the targeted mathematical concepts

These phenomena can come from real life or can be ‘experientially real’

‘Realistic’ context Mathematical model

Mathematical objects,structures, methods

Horizontal mathematization

Translate

Verticalmathematization

Abstract

C. Mathematization

D. Emergent modeling

• View on mathematics education which aims at the development of models

• Models of informalmathematical activitydevelop into models formathematical reasoning

• Level structure byGravemeijer

situational

referential

general

formal

Some debate

• Implementation ok? (Gravemeijer, Bruin-Muurling, Kraemer, & van Stiphout, 2016)

• Influence of contexts? (Hickendorff, 2013)

• Procedural skills and conceptual understanding go hand in hand (Rittle-Johnson, Schneider, & Star, 2015)

• Not enough emphasis on procedural skills e.g. algorithms (Fan & Bokhove, 2014)

I don’t see a contradiction doing both. Combined in subsequent PhD work

Contents




ENGASIA

CONCLUSION

Part of PhD

Algebraic skills Year 12 students Netherlandsoftendisappointing

But even when rewriting skills are OK…

…many things can go wrong

So conceptual understanding and pattern recognition are important!

Use of ICT

So can’t we use ICT for acquiring, practicing and assessing algebraic

expertise?

Equations: in-between steps, multiple strategies allowed

Store student results, and use these as a teacher to study misconceptions and for starting classroom discussions

students

Design principles

(i) students learn a lot from what goes wrong,

(ii) but students will not always overcome these if no feedback is provided, and

(iii) that too much of a dependency on feedback needs to be avoided, as summative assessment typically does not provide feedback.

These three challenges are addressed by principles for crises, feedback and fading, respectively.

Crisis-tasks“students learn a lot from what goes wrong”

Feedback: worked examples and hints

IDEAS feedback, webservice with Jeuring et al

Fading“too much of a dependency on feedback needs to be avoided”

Hands-on: Equations

http://is.gd/hkeng1

These are HTML5 versions of those applets.

http://is.gd/hkeng1

Contents




ENGASIA

CONCLUSION

Towards digital textbooks

• Digital textbook: theory, examples, explanations

• Interactive content (in MC-squared widgets)

• Interactive quizzes (formative assessment, feedback)

• Integrated workbook

FP7 EU projectDesigning creative electronic books

for mathematical creativity

The environment

stores student work.

Separate ‘schools’ can have several

classes.

This is the ‘edit’ mode of the environment : this c-book is

about planets

c-books can have several pages: each circle indicates a page. Other

options are available as well

C-book pages can have random elements, like random values.

Pages consist of ‘widgets’, which can range from simple text to simulations (here: Cinderella). Some widgets can

give automatic feedback.

The MC-squared project aims aims to design and develop a new genre of creative, authorable e-book, which the project calls 'the c-book

MC-squared platform based on Utrecht University’s ‘Digital Mathematics Environment’ (now Numworx).

https://app.dwo.nl/en/student/

Authorable

Feedback

Also geometry

Creativity: fluency

http://is.gd/kheng5

http://is.gd/kheng5

Bokhove, C., & Redhead, E. (2017). Training mental rotation skills to improve spatial ability. Online proceedings of the BSRLM, 36(3)

Flexible environment

Hands-on: Cube Buildings

http://is.gd/hkeng2: Cube Buildings

http://is.gd/hkeng3: Planets

These are HTML5 versions of those applets.

http://is.gd/hkeng2

http://is.gd/hkeng3

Contents




ENGASIA

CONCLUSION

enGasia project1. Compare geometry education in England, Japan and

Hong Kong → some shown now.

2. two digital resources (electronic books) will be designed. They are then implemented in classrooms in those countries.

3. The methodology will include a more qualitative approach based on lesson observations and a quasi-experimental element.

This could be a geogebra widget but perhaps not necessary. More important is feedback.

Challenges

• Differences in curriculum regarding geometry

• School and teacher participation

• Software: Java

Now writing the findings in several articles.

Flowchart

• Prof. Miyazaki and team

http://engasia.soton.ac.uk

Contents




ENGASIA

CONCLUSION

Technology-added value of the c-books

• Creative and interactive activities made by designers (creative process authoring)

• Collaboration within CoI between designers, teachers and computer scientists. Feeds into DA component (see later section)

• Interactivity: feedback design

• More than one widget factories used

• All student data stored

• Sum is more than the parts…

Bokhove, C., (in press). Using technology for digital maths textbooks: More than the sum of the parts. International Journal for Technology in Mathematics Education.

Thank you

• Contact:

–[email protected]

– Twitter: @cbokhove

–www.bokhove.net

• Most papers available somewhere; if can’t get access just ask.

• I’ll add the references and post on Slideshare

mailto:[email protected]

http://www.bokhove.net/

Bokhove, C., (in press). Using technology for digital maths textbooks: More than the sum of the parts. International Journal for Technology in Mathematics Education.

Bokhove, C., &Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45-62. Online first.

Bokhove, C., & Drijvers, P. (2012). Effects of a digital intervention on the development of algebraic expertise. Computers & Education, 58(1), 197-208. doi:10.1016/j.compedu.2011.08.010

Bokhove, C., & Redhead, E. (2017). Training mental rotation skills to improve spatial ability. Online proceedings of the BSRLM, 36(3)

Fan, L., & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development. ZDM-International Journal on Mathematics Education, 46(3), doi:10.1007/s11858-014-0590-2

Fischer, G. (2001). Communities of interest: learning through the interaction of multiple knowledge systems. In the Proceedings of the 24th IRIS Conference S. Bjornestad, R. Moe, A. Morch, A. Opdahl (Eds.) (pp. 1-14). August 2001, Ulvik, Department of Information Science, Bergen, Norway.

Freudenthal, H. (1991). Revisiting Mathematics Education. China Lectures. Dordrecht: Kluwer Academic Publishers.

Gravemeijer, K., Bruin-Muurling, G., Kraemer, J-M. & van Stiphout, I. (2016). Shortcomings of mathematics education reform in the Netherlands: A paradigm case?, Mathematical Thinking and Learning, 18(1), 25-44, doi:10.1080/10986065.2016.1107821

Hickendorff M. (2013), The effects of presenting multidigit mathematics problems in a realistic context on sixth graders' problem solving, Cognition and Instruction 31(3), 314-344.

Jaworksi, B. (2006). Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187-211.

Rittle-Johnson, B. Schneider, M. & Star, J. (2015). Not a one-way street: Bi-directional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27. doi:10.1007/s10648-015-9302-x

Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction – The Wiskobas project. Dordrecht: D. Reidel Publishing Company.

Wenger, E. (1998). Communities of Practice: Learning, Meaning, Identity. Cambridge University Press.