Advocating for mathematically highly capable students (Primary years) presented by Linda Parish Contrary to popular belief students who are mathematically highly capable or gifted are not a ‘privileged’ group, they are simply children who learn differently and therefore may require a different type of teacher support. This session explores some of the unique learning needs of mathematically highly capable students, and suggests some important ways teachers may be able to support this learning in the regular mathematics classroom. The associated webinar and resources can be found at the Connect with Maths Engaging All Students community - http://connectwith.engaging.aamt.edu.au Connect with Maths~ supporting the teaching of mathematics ONLINE

1. Advocating for the Mathematically Highly Capable
Linda Parish
Lind.Parish@acu.edu.au
from Rosie Revere, Engineer
by Andrea Beaty (2013)

2. Who are the mathematically highly capable or gifted?
• Often students who are mathematically highly capable or gifted are
viewed as being privileged, as being at an enviable place where learning
comes quickly and easily.
• Children who are mathematically highly capable or gifted are students
who possess unusually high natural aptitudes for constructing
mathematical concepts and who consequently learn differently to their
age peers. They therefore require a different type of support.

3. Who are the Mathematically Highly Capable or Gifted?
Profound cognitive
impairment,
or dyscalculia
Severe
cognitive
impairment,
or dyscalculia
Mild to
moderate
cognitive
impairment
or dyscalculia
Average capability
Moderate to
highly
capable
Gifted Profoundly
gifted
Normal Bell Curve
Distribution of Variance in Mathematical Capabilities

4. (Very) Simplified version of Gagne’s Differentiated Model of Giftedness and Talent (DMGT)
There is a difference between being ‘gifted’ and being ‘talented’
Our responsibility as educators of mathematically highly capable and
gifted students is to encourage and help facilitate talent development.

5. Identifying Mathematically Highly Capable or Gifted
Have a “mathematical cast of mind” (Krutetskii, 1976):
• Readily grasp the structure of a problem
• Tend to generalise easily
• Develop chains of reasoning
• Use symbols and language accurately and effectively
• Think flexibly - backwards and forwards, switching between strategies
• Are efficient problem solvers. They naturally strive “for the cleanest,
simplest, shortest and thus most ‘elegant’ path to the goal” (Krutetskii)
• Mathematically gifted people look at life through a mathematical lens

6. Identifying Mathematically Highly Capable or Gifted
• There are no tests for ‘mathematical giftedness’
• Not necessarily high achievers (and some high achievers are not
necessarily ‘gifted’).
• Not necessarily ‘fast finishers’ – some are actually quite slow and
deliberate in their work, wanting to be precise.

7. Using problem solving to identify mathematically gifted
students:
• “In one task, the researcher gave K [a 9-year-old careless, not highly
motivated, average maths student] one sheet from a newspaper with
pages numbered 35, 36, 109, 110. From this K was able to quickly
work out how many pages there were in the newspaper.”
(Haylock & Thangata, 2007)
Identifying Mathematically Highly Capable or Gifted

8. Australian Curriculum
• Gifted and talented students are entitled to rigorous,
relevant and engaging learning opportunities drawn from
the Australian Curriculum and aligned with their
individual learning needs, strengths, interests and goals.
[Australian Curriculum: Student diversity]
All

9. Learningasacontinuum,fromwhatis
knowntowhatisnotyetknown.
What is already known and understood
What is not yet known and/or understood
ZPD – What is too difficult to be known/understood by the
student on their own, but can be learnt with guidance and
encouragement from a knowledgeable other.
Vygotsky’s Zone of Proximal Development –
gifted learner versus typical learner
(“Zone of Confusion”)

10. In order for a butterfly to have strong
wings and a solid body it needs to
struggle and fight it’s way out of the
cocoon.
Learning takes place when there is
cognitive conflict and our brains need to
make sense of new information.
Learning takes effort.
Embrace the struggle – “Zone of Confusion”

11. “Students need to know
that even the best
mathematicians in the
world spend most of
their time frustrated and
confused.”
(Math: An Integral Part of Happiness)

12. Melanie (1985)
12
• 10/10 every week in the customary ‘Friday test’
• She could have got 10/10 with most of the questions
last year, so what has she learnt this week??
• What will happen the week she gets 9/10??

13. Fixed v Growth Mind Sets

14. Fixed mindset
(self-limiting)
Growth mindset
(self-actualising)
Need to look smart even at the cost of sacrificing
learning by avoiding challenging tasks
Wants to learn new things even if hard or risky
Failure is seen as an indication of low intelligence Failure is seen as an indication of poor strategy and/or
low effort
Effort is seen as an indication of low intelligence Effort activates and uses intelligence
Less effort the typical response when faced with a
difficulty
More effort typical response when faced with a
difficulty
Self-defeating defensiveness high: not willing to face
ignorance and to risk mistakes
Self-defeating defensiveness low: eager to learn and
open to feedback about mistakes
Performance after facing a difficulty impaired Performance after facing a difficulty equal or improved
Dweck, C. S. (2006). Mindset: The new psychology of success. New York: Random House.
Result: May plateau early and not reach full potential Result: Can reach ever higher levels of achievement
Dispositions for Learning:

15. Fostering a Growth Mindset
https://www.pinterest.com/
search: teaching growth mindset

16. 16
Alex was “a little bit happy” with this solution but not
really because “there was too much crossing out”
He was much happier with his second solution
because he was able to do it quickly with “no
crossing out”… There was also a lot less
mathematical reasoning.
Alex – Year 1

17. 1. The learning process, when perceived as incorrect, was highly distressing.
[Gifted children are often hypersensitive - they not only learn differently they also often feel
differently (Sword, 2008). Dabrowski & Piechowski (1977) call this ‘over-excitabilities’].
2. Any subsequent learning opportunities in that lesson were destroyed..
12 = 1x12
12x1
2x6
6x2
3x4
4x3
12 = 3x3+3
2x3+2x3
18=3x5+3 15=3x3+3+3
Sammy – Year 3

18. Adding Corners – Fred (Grade 5)

19. Types of fixed mindset statements Re-training for growth mindset self-talk
I’m no good at maths.
(if the answer is not obvious, or takes a bit
of thinking to work out)
Hang on…I need to think about this a bit more.
This is too hard for me.
(if the task requires thinking and effort to
complete)
Remember learning takes effort.
I need to be working through a zone of confusion’ if I am to learn something
new.
I’m finished!
(indicating a need to be first finished)
Learning is not a race.
There is always something more to learn, what can I explore now?
This is easy! / I know how to do this.
(making sure people know they are smart)
This is easy for me, how can I challenge myself further?
To learn I need to be working in my ‘zone of confusion’.
This is taking too long.
(thinking they should be able to work
quickly and easily)
This is a good challenge for me. I’m needing to think long and hard about this
problem. I wonder who I can discuss my thoughts with.
I’m making too many mistakes. How can I learn from these trials? Where have I gone wrong?
Why didn’t this work? (‘Mistakes’ are an integral part of success. The most
successfully innovative people in the world are often those who have ‘failed’ the most)

20. “I think a lot of people think that with maths problems you
should be able to just read them and solve them, and if you
can’t solve them then you’re not good at maths.
All the real maths problems, the problems that are worth
solving, aren’t the ones you can solve as soon as you see
them. They’re the ones you may need to let sit and let your
brains do a little background processing over a period of
time. The solution to these types of problems is much more
satisfying.” (unknown)

21. Advice to parents to allow their children ‘mental health
days’…
“…days on which gifted kids are given an opportunity to
stay home to learn more. They don’t have to sit in a room
waiting for the other kids to catch up. They can unfurl their
wings and fly” (Bainbridge, n.d.).
http://giftedkids.about.com/od/socialemotionalissues/qt/mental_day.htm
There is something inherently wrong with having to give children
regular ‘mental health days’ so they can explore, engage and
participate in their own learning…

22. "At university they get you to actually learn things
yourself, instead of school where they tell you
everything and get you to do it a certain way…"
Jacob Bradd (on acceleration to university at age 14)
SMH, Dec 27, 2014
http://www.smh.com.au/national/education/study-gifted-children-benefit-from-bypassing-school-for-university-20141227-12cnf0.html
There is something inherently wrong with having to accelerate
children through school so they can explore, engage and
participate in their own learning…

23. Establish an understanding that learning requires hard thinking, and that is what
we expect. Hard thinking is a good thing, not a sign that you are not good at
maths.
Establish that when I (the teacher) ask a question I am posing a problem I want
you to think about. I don’t want a quick answer (I am not testing you). What I
require is a well thought out explanation, the answer is the by-product of this.
Modell that there is always more you can explore (teaching them how to think
deeper; there is a skill in learning how to learn).
Give permission and encourage students to run with their own ideas. Give them
time to do this.
Constantly ask questions like “How are you challenging yourself?”, “Are you
working in your ‘zone of confusion’?”, “What’s next?”, “How can you be creative
with this?”
Be aware of, and challenge negative mindset statements.

24. Scaffolding Creativity:
Adding Corners
?
Draw a triangle. Choose a number to write in the centre
of your triangle and then split (partition) your number –
putting a number at each corner of the triangle – so that
the three numbers add up to the number in the centre.
Be creative and challenge yourself!
Adapted from Adding the Corners in Downton, Knight, Clarke & Lewis (2006)

26. Adding Corners – before and after…
Fred - July Fred - November

27. Explore the mathematics further some examples:
 Can I solve this problem a different way?
 Can I find another solution (for an open-ended task); how many different
solutions are there; how will I know I’ve found them all?
 What if I try the same problem but make it more complicated (e.g., larger
quantities, smaller quantities (fractions), more components)?
 How can I adapt the rules of this game to improve it?
 What is the best strategy to use to ensure the greatest chance of winning
this game?
 What other components of this investigation look interesting, are worth
exploring? (Permission to use computer search engines for investigations
may be part of this).

28. Budgeting Worksheet
• Complete a budget for your ‘Rubbish Knight’ business
with a partner…
• What profit margin have you planned for in the second
month?

30. • “We don’t want students to be third-rate computers; we
want them to be first-rate problem solvers.”
(Wolfram, 2013. Stop teaching calculating, start learning mathematics!)
• This requires an ability to know how the problem was solved,
and be able to explain this, as well as knowing what the
solution is.

32. The skill of explaining solutions…
• If we are encouraging our students to be creative then they
will need to know how to report, record, and share their ideas
otherwise some of the best innovations from the best
innovators of this century could be completely lost to us.
• Imagine if people like Einstein or Newton couldn’t explain
their thinking or record their discoveries in a way that could
be replicated by others!

33. How do you know someone is good at maths?
Sammy Before …
“They always finish their work in time. They’re
always going ‘done!’, and always get the right answer”
(Sammy, May)

34. If being good at maths means you can “do the work quickly and get the
right answer” there is a big risk in tackling tasks you may not be able to
complete quickly and easily…
This is a skewed understanding of what learning is, and it develops a
false view that effort is equated with a lack of ability, or that the work
is too hard for them. Mistakes are perceived as failure, as
something to be avoided at all costs.

35. Sammy After…
• “I know I'm good at maths because I did that
[pointing to a task she’d persevered with for over
30 minutes] and I thought it was too hard but I
did it!”
Sammy (Nov)

36. If mathematically highly capable and gifted students are expected to
think creatively, and are given permission and time to explore their
own curiosities, could we be seeing even more amazing ideas from
our students before they even finish school…
• Elif Bilgin (16y.o. Turkish girl) - created a bio-plastic from banana peels as
an alternative way to make plastic without using oil, which is extremely
harmful to the environment.
• Ciara Judge (15y.o. Irish girl) - in order to combat the global food crisis,
investigated the use of diazotroph bacteria as a cereal crop germination and
growth aid.
• Jack Andraka (15y.o. American boy) - developed a cheaper, more sensitive
cancer detector test for early diagnosis of pancreatic cancer
• Roni Oron (13y.o. Israeli girl) - invented a satellite system for the production
of oxygen in space.