Celia Hoyles

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Engaging students with mathematics the challenge
of the twenty-first century

• Celia Hoyles
• Institute of Education
• University of London, UK

_at_ Mathematics education in Flanders and Europe A
state of the art and future perspectives
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Congratulations on your 100th birthday!
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structure of talk

• Research themes
• learning to prove mathematically
• activities to promote structural engagement with
  mathematics
• using computers
• using discussion
• learning mathematics collaboratively over the web

overall mission structural reasoning with
engagement
policy initiatives
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uniqueness of mathematics

• multiple faces of mathematics
• core skill for all
• subject in its own right
• service subject for science engineering ...
  more and more subjects careers
• each face has different demands in terms of
• content skill
• language structure
• pedagogy trajectory of learning

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uniqueness of mathematics

• Dual nature of mathematics
• procedures calculation
• concepts structures
• mathematics is much more than procedures and
  calculation but
• procedures and calculation are the visible face
  of mathematics for most people
• example Grade 6 Korean student

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procedures
concepts
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Longitudinal Proof Project 1999-2003(Hoyles
Küchemann http//www.ioe.ac.uk/proof/)

• analysing students learning trajectories in
  mathematical reasoning over time (age 12/13-15)
• annual surveying of high-attaining students from
  randomly selected schools
• 3000 students (age 13) from 63 schools tested
  June 2000 in number/ algebra geometry
• same students tested again in 2001 in 2002
• some questions from the previous test
• some new or slightly modified questions
• followed by co-design proof activities with
  teachers to research scaling out

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some general findings

• performance consistently better in algebra than
  geometry
• modest progress over 3 years (12/13-15 years)
• turbulent progress didactical curriculum
  influences as well as cognitive
• importance of developing ownership of structural
  reasoning, explanations proof even successful
  students lacked conviction
• girls performed better than boys (if account
  taken of baseline mathematics attainment)
• two teacher variables significant in geometry but
  not algebra
• length of teaching experience
• recent involvement in continuing professional
  development

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Generalising structure
  Question A1 age 13
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results similar question asked in Yr 9 same
in Yr 10 (with request for algebraic model)

• cross-sectional data
• Yr8 A1 Yr9 A1 Yr10 A1
• Pattern spotting 35 22 22
• Correct answer 46 67 68
• modest improvement -- then plateau

• longitudinal data
• turbulence with some regression from correct
  answer to pattern spotting

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Greg aged 15yrs
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• G Yes. First I worked it out as if all the 60
  white tiles were in one straight line so
  there are 60 white tiles so you have to times
  that by 10 to get to that one. And then there are
  18 grey tiles round there so I times that by 10
  to get 180 grey tiles round the outside.
• CH And then when you came to do it with n,
  youve got 2n 6?
• G I randomly guessed that... I'm rubbish at that
  kind of thing.
• CH Well funnily enough thats absolutely right.
• G Youre joking?
• ---I came back to change that at the end
  pointing to algebra because I was thinking
  about it through the rest of the paper... about
  the first question ... then realised and came
  back to it and I changed it in the end.

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a way forward?

• G was beginning to use algebra to communicate the
  structure that previously had alluded him
• expressing in algebra kept connection with sense
  of situation while expressing in arithmetic did
  not
• this is unusual....
• need to support students to

• interconnect representations of structure
  calculations
• construct a symbolic language that
• retains meaning for them
• allows them to explain communicate, rather than
  simply appropriate a representation from
  outside
• but abstractions are situated...

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using computers to engage with structure

• iteratively co-designed a teaching sequence
• investigate with different software
• make informal conjectures
• test verify conjectures
• explain and prove in algebra

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designing tools activities

• In design - need to focus on core (may look
  simple?) ideas
• a generic example
• When is the sum of n consecutive numbers a
  multiple of n?

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investigating sums of consecutive numbers
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(No Transcript)
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What about the sum of 4 consecutive numbers?

• investigate
• make informal conjectures
• divisible by 2
• not divisible by 4
• test verify conjectures
• explain and prove in algebra

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Christine's generalisation
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applying a theorem

• after much investigation of the sum of 3 then 4
  consecutive numbers, students wrote
• The sum of 4 consecutive numbers is even but not
  divisible by 4
• Then they were asked
• Predict whether you can find 4 consecutive
  numbers that add up to 44. If yes, write them
  down, if no, explain why it cannot be done
• All but one of students tried to find 4 numbers

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design for mathematical discussion
take students voices as starting point
Amy, Bonnie, Ceri, Duncan, Sam, Tom, Yvonne,
Ursula and Eric were trying to prove whether the
following statement is true or false When you
add any 2 even numbers, your answer is always
even
Proof Projects Hoyles, Healy and
Kuchemann Institute of Education University of
London
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Duncans answer Even numbers end in 0 2 4 6 or
8 When you add any two of these the answer will
still end in 0 2 4 6 or 8. So Duncan says its
true.
Bonnies answer 2 2 4 4 2 6 2 4
6 4 4 8 2 6 8 4 6 10 So Bonnie
says its true.
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Ceris answer Even numbers are numbers that can
be divided by 2. When you add a number with a
common factor, 2 in this case, the answer will
have the same common factor. So Ceri says its
true.
Sams answer I chose two arbitrary even numbers,
say 245224 and 543876. When I added them, I
obtained 245224 543876 5685100 which is
even. So Sam says its true.
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Yvonnes answer ????? ???? ?????
???? ?????????
????????? So Yvonne says its true.
Toms answer Here is what I would do if the given
numbers are 12 and 22 12 6 6 22 11 11 12
22 (6 11) (6 11) I can do this with any
two given even numbers. So Tom says its true.
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Amys answer a is any whole number b is any whole
number 2a and 2b are any two even numbers 2a 2b
2(a b) So Amy says its true.
Ursulas answer Let a be any even number, so a
2k Let b be any even number, so b 2L a b
2k 2L a b 2 (k L) which is even. So
Ursula says its true.
Erics answer Let x any whole number, y any
whole number x y z z x y z y x z z
(x y) x y 2z So Eric says its true.
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From the above answers, a) choose one that would
be closest to what you would do if you were asked
to answer this question.
Duncans answer Even numbers end in 0 2 4 6 or
8 When you add any two of these the answer will
still end in 0 2 4 6 or 8. So Duncan says its
true. Bonnies answer 2 2 4 4 2 6 2
4 6 4 4 8 2 6 8 4 6 10 So Bonnie
says its true.
Erics answer Let x any whole number, y any
whole number x y z z x y z y x z z
(x y) x y 2z So Eric says its true.
b) choose the one to which your teacher would
give the best mark
hugely effective in promoting discussion among
students and teachers
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Weblabs (Hoyles Noss) tools and activities for
students to construct models of their
mathematical knowledge a web-based system,
WebReports for sharing discussing what they
know, and working models of what they know
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design to promote discussion within and between
classrooms

• focus on process not just on answer by
• sharing critiquing each others
• conjectures models
• commenting, explaining, counter examples

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New roles for the teacher
same complexity as before and more need to
sustain interaction across distributed
communities what can be devolved to system?
new project (2007-2010) a collaboration between
mathematics educators computer scientists
(Hoyles Noss)
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Lessons from research
situated abstraction crucial role of
representations tools role of discussion
construction/ proof teachers are
crucial! other dependencies (diversity gender)
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consistent picture

• participation in mathematics increasingly
  dominated at every stage by bright, white males
  coming from socially advantaged backgrounds.
• Other groups of learners with comparable high
  grades more likely to drop out at every point
  where choices are made
• Attrition linked with three factors
• self-efficacy or confidence in ones ability to
  achieve in the subject
• exposure to appropriate curriculum content and
  challenge (bored)
• insufficient time to master and reflect upon
  content and processes.

not engaged in structural reasoning images of
mathematics mathematicians that alienate
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Some figures of mathematicians in popular culture
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more images of mathematics mathematicians
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implications for policy practice

• my role as UK Government Chief Adviser for
  Mathematics since 2004

who engages with mathematics is there a problem?
Yes in England.... here?
what are the data?
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A level entries (specialist mathematics
examination, 18years)
Prov
Entries increased in 2006.Still below the
numbers seen prior to 2002
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increasing uptake in mathematics at every level
is a central Government objective

• what is the evidence?
• following the publication of Making Mathematics
  Count, 2004) there are policies actions
• raise attainment at every phase
• identify promote different pathways for
  mathematics
• support at transition from school to university
• enhance motivation for mathematics
• promote the importance of CPD

mathematics has a voice
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participation in mathematics consistent picture

• interventions to advance mathematics raise
  awareness of mathematics do make a difference!

• need
• good teaching that is personalised
• curricula reforms
• good resources

what has the Government put in place to date?
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intervention 1 promoting Active Learning post-16
incorporating discussion the need to explain
and justify in nationwide initiative
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intervention 2 Further Mathematics Network
National network of 46 Further Mathematics
Centres. http//www.fmnetwork.org.uk/ provide on
line mentoring for students who are unable to
receive face to face tuition
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The National Centre for Excellence in the
Teaching of Mathematics
funded by Government who accept need for
CPD launched by Secretary of State June
2006 put in place for the first time a national
infrastructure for CPD for teachers of
mathematics... all phases regional national
events and portal
visit www.ncetm.org.uk
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NCETM the vision

• we have in place
• expert committed team
• powerful infrastructure to
• identify needs
• broker ways to fill gaps local national
• enhance participation visibility of
  professional
• development communities
• an innovative portal for mathematics CPD that
  is evolving
• with teacher use www.ncetm.org.uk
• and Government support
• Maths is high profile but needs support!

• to develop a sustainable national infrastructure
  for subject-specific professional development of
  teachers of mathematics that will
• enable the mathematical potential of learners to
  be fully realised
• raise status of the profession
• to achieve this we need to promote a blend of CPD
  approaches that
• are effective
• convince key stakeholders of effectiveness

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the portal
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teachers can develop a personal CPD journey
Communities and Blogs
Portfolio Personal learning space
Mathemapedia
Resource case studies research groups
PD Opportunities News events
Professional self-assessment (individual
group)
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• conjecture to be successful
• portal must
• support shape CPD provision
• have the functionality interactivity demanded
  by teachers, schools, regions
• by engagement, teachers will develop
• ownership of portal
• create opportunities for new networks
• plan their own personal learning space

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risks challenges

• NCETM and the idea of a portal is new
• still problems of access to ICT
• aims potential yet to be fully understood
  across all the boundaries
• hard to capture our value-added

• help us to develop the National Centre
• register www.ncetm.org.uk
• tell others about NCETM

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scaling out successful design experiments

• conjectures about design experiments
• initiatives end when experiment ends
• initiatives are transformed by ritualising and
  neutralising
• what are the counter examples?
• what can and does survive scaling out process?
• need a national infrastructure teacher
  communities

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Bowland Mathematics Initiative for Key Stage 3
funded by Bowland Trust with matched support from
Government (launched by Secretary of State) aims
to develop thinking, reasoning problem solving
skills in mathematics ( beyond!) cool,
interdisciplinary, in/out of school 23 case
study problems with 15 - initially- collaborative
development teams -some from abroad www.bowlandmat
hs.org.uk
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these will be shared nationally through NCETM
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Science and Innovation Investment framework
2004-2014, March 2006

• ambitions include
• 95 of mathematics lessons in schools to be
  delivered by a mathematics specialist (compared
  to 88 in 2005)
• and

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....and ambitious target for A level entries in
mathematics
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stop press trends in mathematics (Aug 2007)
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recall consistent picture of trends in
participation in mathematics
Interventions make a difference CPD for teachers
makes a difference importance but challenge of
structural reasoning centrality of co-design
clip from Andrei!
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Mathematics as empowerment(CEO of multinational
packaging company)

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Congratulations again
Look forward to the next 100 years ... developing
research-based professional development for
teachers of mathematics together