Scaling student

access to

dynamic

mathematics

technology in

classrooms

@aliclarkwilson

Dr Alison Clark-Wilson

UCL Knowledge Lab

University College London, UK

Linz STEM Education Conference: 18-19 May 2017, JKU

Overview…

1• Learning from history

2

• Known barriers to wide-scale technology integration

3

• Introducing the case example and conceptualising teacher knowledge

4

• Implications – and where next for design based research

@aliclarkwilson

Overview…

1• Learning from history

2

• Known barriers to wide-scale technology integration

3

• Introducing the case example and conceptualising teacher knowledge

4

• Implications – and where next for design based research

@aliclarkwilson

Significant

attempts at

technology

integration in

education since

1900

1

1800s • Slate and slate pencil

1900s • Paper and pencil

1910 • Movie film

1920 • Radio broadcasts

1950 • Television broadcasts

1972 • Computers

@aliclarkwilson

Two early visionaries

1

@aliclarkwilson

1

@aliclarkwilson

1

@aliclarkwilson

Two early visionaries

1

@aliclarkwilson

Burrhus Skinner (1904-1990)o Psychologist (Behaviourism)o Inventor of ‘teaching machine’ (1954) and

‘programmed instruction’ through a response/reward mechanism.

o Digital resources of this type are the most prevalent worldwide.

Seymour Papert (1928-2016)o Mathematician, Computer Scientist and

Educator.o Co-invented the programming language

LOGO (paving way to NetLOGO, Scratch).

@aliclarkwilson

1985 – ‘The Influence of Computers and Informatics on Mathematics and its Teaching: The 1st ICMI Study’

2009 – ‘Mathematics Education and Technology - Rethinking the Terrain: The 17th ICMI Study’

2014 – ‘The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development’ 1

@aliclarkwilson

• Dynamic mathematics technology [DMT]– ‘computational tools through which students and teachers (re-)express their mathematical understandings’ (Clark-Wilson et al 2013)

• Landmark activities – disruptive but carefully designed technologies lead to a cognitive breakdown, or a ‘situation of non-obviousness’ (Winograd and Flores 1986 p. 165)

• But for teachers this leads to ‘…an explosion of techniques which remain relatively ad hoc, and pose a didactic obstacle to the progressive building of mathematical activity instrumented in an efficient way.’ (Artigue, 2002)

1

Tensions

@aliclarkwilson

designing 21st

century TEL to provide access to

21st century curricula

designing 21st

century TEL to support 19th

century curricula

41

@aliclarkwilson

1

designing TEL to solve problems that

teachers/lecturers don’t recognise

working with teachers/lecturers to design TEL that takes account of prevailing

pedagogies, whilst still challenging these…

Tensions

Overview…

1• Learning from history

2

• Known barriers to wide-scale technology integration

3

• Introducing the case example and conceptualising teacher knowledge

4

• Implications – and where next for design based research

@aliclarkwilson

@aliclarkwilson

2

- Supportive models for collaborative prof dev: lesson study; coaching; (online) communities…

- Realistic amount of time (years rather than months...)

- Careful resource design withteachers (design based research)

- Weak alignment with institutional constraints (curriculum, exams, prevailing practices)

- Conflicting views of simplicity and complexity for teaching with digital tools

- Top-down initiatives that attempt to implement ‘at scale’

@aliclarkwilson

MEXICO

(primary school)

• ‘teachers, who have only received training on the general use of the software, without a hint of how to introduce them into specific lessons, often develop teaching strategies where technology is used as replacement or amplification’. (Trigueros, Lozano and Sandoval, 2014)

NEW ZEALAND

(secondary school)

• ‘strong correlation between confidence in using technology in the mathematics classroom and pedagogical technology knowledge’ (Thomas and Palmer, 2014)

CANADA

(undergraduate)

• ‘Overall it seems that the great majority of tutors who integrate technology into their mathematics teaching do so by their own volition’. (Buteau and Muller, 2014)

2

@aliclarkwilson

Most secondary mathematics

students do not use transformative

technologies in lessons

Most schools are very well equipped with technology –

so access is not the problem!

Most secondary mathematics

teachers have had limited training to use technology in

lessons

2

In England…

@aliclarkwilson

• Funded by Li Ka Shing Foundation and Hutchison Whampoa Europe Ltd

• Planning Phase 1 (Jun-Jul 2011)

• Phase 1 (Jul – Dec 2011) – Pilot phase (unit 1)

• Phase 2 (Jan – Jul 2012) – Pilot phase (unit 2)

• Phase 3 (Dec 2012 – Nov 2014) – Ongoing (re)design cycles with ‘Design’ schools alongside scaling to 100+ ‘Focus’ schools (units 1, 2 and 3).

• Funded by Nuffield Foundation (Dec 2014- April 2017) Scaling to 209 teachers in 42 London schools.

Project phases

3

@aliclarkwilson

3

Research questions• What is the impact on teachers’ knowledge around the

mathematical concepts, algebraic generalisation; geometric similarity, and linear functions, of their engagement with cycles of professional development and associated teaching that embeds DMT?

• What knowledge is desirable for teachers to integrate DMT in their teaching of these concepts?

• What are the design features of professional development activities for lower secondary mathematics teachers that support them to use DMT in ways that become embedded in their practice and lead to effective learning?

@aliclarkwilson

2

in technology-enhanced mathematics classrooms, it is anticipated in the design that the technology will disrupt routine practices in a transformative sense, and ensuing breakdowns would promote:

• rethinking of the mathematics or

• extend previously held ideas by a process reflective inquiry

3

@aliclarkwilson

Technology enhanced

units of work

Professional development

‘hard to teach’ topics

3

@aliclarkwilson

Let's work on a game with robots. We need to set upthe mathematics to make our robots move at differentspeeds.

3

@aliclarkwilson

Context: Developing games for mobile phones: using mathematics to analyse and create simulated motion games.

Design principles: dynamic simulation and linking between representationscontrol the simulation from the graph or the functionshow/hide representations, as appropriate

Big ‘mathematical’ ideas Coordinating algebraic, graphical, and tabular representationsy= mx+c as a model of constant velocity motion – the meaning of m and c in the motion contextVelocity as speed with direction and average velocity

3

@aliclarkwilson

so, have we found a way?

• we’ve worked with over 300 teachers…

• ..from over 100 diverse schools across England…

• …and nearly 8000 students..

• …with positive impacts on learning outcomes (reported by teachers/schools)

• BUT have we done enough?

• have we ‘scaled’ our innovation?

• what impact are we having on teachers’ knowledge and practices?

3

@aliclarkwilson

Espoused knowledge

as evidenced by: lesson plans

questionnaires

interviews

self-accounts

Enacted knowledge

as evidenced by: lesson observation

stimulated recall interviews

Conceptualising teacher knowledge

4

Mathematical Pedagogical Technology Knowledge (Thomas and Hong 2013)

4

@aliclarkwilson

4

•“Ask students to discuss their thoughts on how the equations relate to the different speeds. Give them 5 minutes and thereafter come up with their final conclusion to write down in their workbook. Helicopter around classroom and introduce the word gradient to represent the steepness of a line. Introduce the word coefficient of x to represent the number in front of x that represents gradient.”

Teacher A

•“Stop the class and discuss Q1 – use keywords, focus particularly on Q1G”

Teacher B

• 1. Describes teachers’ actions/questions.

• 2. Describes pupils' actions on DMT.

• 3. Supports pupils in their instrumental genesis of the DMT, as appropriate to the activities.

• 4. Refers to the mathematical concept at stake (i.e. variables, functions, geometric objects).

• 5. Describes acting on and connecting mathematical representations.

• 6. Uses mathematical vocabulary.

• 7. Uses technological/contextual vocabulary.

• 8. Includes planned teacher use of the DMT

Eight desirable features of lesson plans

4

4

Feature of lesson plan (espoused knowledge) Frequency % (n=42 plans)

1. Explicit descriptions of teachers’ actions/questions 29 69%

2. Explicit descriptions of pupils' actions on DMT during the lesson

19 45%

3. Appreciation of pupils’ instrumental knowledge (i.e. prior skills with software, progression of skills in lesson)

16 38%

4. Explicit reference to variables (i.e. creating, naming, acting on)

24 57%

5. Explicit reference to acting on reps (i.e. dragging/moving sliders)

11 26%

6. Explicit use of mathematical vocabulary 26 62%

7. Explicit use of technological/contextual vocabulary 13 31%

8. Includes planned plenary phases that involved teacher use of software

10 24%

4

• High emphasis placed on the need for pupils to make sense of the ‘hotspots’ that facilitated the graph to be edited.

• Attention to the graphs axes, and discussion of the effect of changing the scale.

• Emphasising the multiplicative/additive relationships in the table to justify the meaning of the equation.

• Highlighting the invisible variant m within the table for equations with a non-zero value of c.

• Strategies to gather back the pupils’ multiple responses as particular cases in order to support the overarching generalisation that the greater the value of m, the faster that Shakey will move.

• Extending Shakey’s journey time such that his final position could not be read from the graph nor the table to provoke pupils to use the equation to calculate its position after a given time and thus highlighting the power of the mathematical equation as a generalisation.

Effective practices from the lesson observations

4

• process of integrating DMT within secondary mathematics teaching presents considerable potential for learning but also a significant challenge for teachers.

• easy access to the technology is necessary but not sufficient; and ongoing and sustained support over time is needed to address issues of mathematical knowledge and pedagogy:

• exploiting the dynamic features of the software in:

• whole class teaching to highlight key concepts.

• interactions with pupils to probe and extend their emergent understandings.

• developing a mathematical language that supports the classroom discourse: in relation to the new dynamic mathematical objects within the technology (sliders, 'hotspots', dynamic images) and being able to engage in descriptive and analytical conversations stimulated by these objects.

• exhibiting the perseverance that is required to go beyond 'first lessons’.

Project findings

4

Component Content

Getting startedVideo introduction to the CM projectOverview of CM and its design principles Overview of the toolkit - and suggestions for how it might be usedLinks to the software and teaching resources

Curriculum UnitPD Design

MKT items Instrumentation tasks MPTK tasks to analyse students’ digital productions.Video introduction to the landmark activity

Leading to a…Lesson planning task of the landmark activity

Landmark activityLinks to the software and pupil taskVideo clips of ‘typical’ classroom enactmentsImages of typical student responses from CM classrooms (written, screenshots, etc.)Video clips of teachers mediating students’ activity with technology

Suggestions for assessing pupils’ mathematical understanding

Departmental Case

Studies

Outline of their approach for scaling/sustainability of CMExamples of within-school PD activities

Evidence baseResearch summary for senior leaders and headteachersMore detailed research summary for Heads of Dept. (to refer to departmental case studies)Links to published research

Project communityLink to the online project community - where teachers can share resources they create and discuss their implementations

• Limited opportunities for ‘within-school’ PD.

• Existing school cultures for the format and content of PD, which mainly focus on pragmatic rather than epistemic goals.

• Time for (all/some/most) teachers in a mathematics department to pilot.

• The nature of the ‘evidence’ needed to convince different members of the department.

• Locus of control; who actually decides whether DMT should be embedded within the scheme of work?

Unresolved challenges

4

Results (a teacher’s)I enjoyed observing how students took to the idea of manipulating on screen characters readily and how positive learning attitudes were quickly fostered.

The written student book accompanying these lessons put great emphasis on students explaining what they were learning in written form throughout the investigations. This promoted deep learning.

My favourite lesson was when I could show the traditional abstract representation of a straight line on a grid and then work with the class to derive its equation, drawing and linking what they understood about these line graphs in the context of their cornerstone [maths] experience.

For example how words such as coefficient could be understood, linking gradient and speed and connecting position and distance.

The unit had strong extracurricular links with science as it presented the concept of velocity and displacement through the simulation exercises.

4

References

Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245-274. doi:10.1023/A:1022103903080

Buteau, C., & Muller, E. (2014). Teaching Roles in a Technology Intensive Core Undergraduate Mathematics Course. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (pp. 163-188). Dordrecht: Springer.

Churchhouse, R. F., Cornu, B., Howson, A. G., Kahane, J.-P., van Lint, J. H., Pluvinage, F., . . . Yamaguti, M. (1986). The Influence of Computers and Informatics on Mathematics and its Teaching (Vol. ICMI Study 1). Cambridge: Cambridge University Press.

Clark-Wilson, A., Aldon, G., Cusi, A., Goos, M., Haspekian, M., Robutti, O., & Thomas, M., O. J.,. (2014). The challenges of teaching mathematics with digital technologies - The evolving role of the teacher. Paper presented at the Proceedings of the Joint Meeting of PME 38 and PME-NA 36, Vancouver, Canada.

Clark-Wilson, A., Robutti, O., & Sinclair, N. (2014). The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (Vol. 2). Dordrecht: Springer.

Cordingly, P., Bell, M., Rundell, B., Evans, D., & Curtis, A. (2003). The Impact of Collaborative CPD on Classroom Teaching and Learning: How does collaborative Continuing Professional Development (CPD) for teachers of the 5-16 age range affect teaching and learning? Retrieved from London:

Hong, Y. Y., & Thomas, M. O. J. (2006). Factors influencing teacher integration of graphic calculators in teaching Proceedings of the 11th Asian Technology Conference in Mathematics (pp. 234-243). Hong Kong: Asian Technology Conference in Mathematics.

Hoyles, C., & Lagrange, J. B. (Eds.). (2009). Mathematics Education and Technology - Rethinking the Terrain: The 17th ICMI Study. Berlin: Springer.

References

Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. Bishop, M. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Second international handbook of mathematics education. Dordrecht: Kluwer Academic.

Hoyles, C., Noss, R., Vahey, P., & Roschelle, J. (2013). Cornerstone Mathematics: Designing digital technology for teacher adaptation and scaling. ZDM Mathematics Education, 45(7), 1057-1070. doi:10.1007/s11858-013-0540-4

Hung, D., Lim, K., & Huang, D. (2010). Extending and scaling technology-based innovations through research: The case of Singapore. In Organisation for Economic Co-operation and Development (Ed.), Inspired by Technology, Driven by Pedagogy: A Systemic Approach to Technology-Based School Innovations (pp. 89-102): OECD Publishing.

Papert, S. (1972). Teaching children to be mathematicians versus teaching about mathematics. International Journal of Mathematical Education in Science & Technology, 3(3), 249–262.

Skinner, B. F. (Producer). (1955). Teaching machine and programmed learning. Retrieved from http://www.dailymotion.com/video/x2pvqa3

Thomas, M. O. J., & Palmer, J. (2014). Teaching with digital technology: Obstacles and opportunities. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (pp. 71-89). Dordrecht: Springer.

Trigueros, M., Lozano, M.-D., & Sandoval, I. (2014). Integrating Technology in the Primary School Mathematics Classroom: The Role of the Teacher. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (pp. 111-138). Dordrecht: Springer.

Winograd, T., & Flores, F. (1986). Understanding Computers and Cognition: A New Foundation for Design: Ablex Publishing Corp.