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Advocating for the Mathematically Highly Capable
Linda Parish [email protected]
from Rosie Revere, Engineer by Andrea Beaty (2013)
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.
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
(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.
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
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.
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
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
Lear
ning
as a
conti
nuum
, fro
m w
hat i
s kn
own
to w
hat i
s not
yet
kno
wn.
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”)
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”
“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)
• 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??
Fixed v Growth Mind Sets
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:
Fostering a Growth Mindsethttps://www.pinterest.com/
search: teaching growth mindset
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
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 = 1x1212x1 2x6 6x2 3x4 4x3
12 = 3x3+3 2x3+2x3
18=3x5+3 15=3x3+3+3
Sammy – Year 3
Adding Corners – Fred (Grade 5)
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)
“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)
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…
"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…
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.
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)
Adding Corners – before and after…
Fred - July
Fred - November
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).
Budgeting Worksheet
• Complete a budget for your ‘Rubbish Knight’ business with a partner…
• What profit margin have you planned for in the second month?
• “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.
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!
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)
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.
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)
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.
