mathematical thinking 1 - js photography · 2018. 8. 28. · seameo recsam 5 april 2017 ....
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Mathematical Thinking 1
SEAMEO RECSAM
5 April 2017
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18-Apr-17 Assoc. Prof. Dr Kor Liew Kee 2
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18-Apr-17 Assoc. Prof. Dr Kor Liew Kee 3
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Contents
• What is mathematical thinking?
• Types of mathematical thinking
• How do we know mathematical thinking exists?
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What do you know about mathematical thinking?
• If you do not know what mathematical thinking is, how will you know whether or not your students are doing it?!
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What is NOT Mathematical Thinking?
• Tricks and Memorization http://www.pedagonet.com/maths/TrickBusters.pdf
(Sinwell & Dickey, 2015)
http://www.pedagonet.com/maths/TrickBusters.pdfhttp://www.pedagonet.com/maths/TrickBusters.pdfhttp://www.pedagonet.com/maths/TrickBusters.pdf
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What is NOT mathematical thinking?
• Tricks and memorization
• Ability to calculate
• Capability to solve an equation
• Applying procedural rules and formulas
• Using short cuts to get an answer
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Ball ( 2002) https://www.atm.org.uk/write/mediauploads/journals/mt181/non-member/atm-mt181-17-
19.pdf
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A. Watson and J. Mason: Questions and prompts for mathematical thinking, ATM.
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What is mathematical thinking?
Kaye Stacey (2007)
• MT is a process.
• Thinking is triggered during problem solving.
• Possesses skills and abilities in
– Deep mathematical knowledge
– General reasoning abilities
– Knowledge of heuristic strategies
http://www.criced.tsukuba.ac.jp/math/apec/apec2007/paper_pdf/Kaye_Stacey.pdf
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Isoda & Katagiri, 2012
• Mathematical thinking resembles an attitude.
• Can be expressed as a state of “attempting to do” or “working to do” something.
• It happens when one encounters a problem, decides what to do to solve that problem.
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3 variables of thinking
• The problem (situation), the person, and the approach (strategy).
• It is not “the ability to do”, “could or could not do” something.
• MT is used during mathematical activities and is closely associated with content and strategy.
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Lim & Hwa’s (2007) conception
• Three major components of mathematical thinking: • 1) Mathematical content/knowledge • to the specific mathematics subject matter,
mathematical concepts and ideas that one has acquired or learnt
• 2) Mental operations • cognitive activities that the mind needs to perform
when thinking • 3) Predisposition • the tendency to think in certain ways under certain
circumstances e.g. reasonableness, thinking alertness and open-mindedness, beliefs and affects.
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What do you say about mathematical thinking?
• Mental
• Content
• Method
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analyse
create
predict
reason
identify
calculate
propose
estimate
deduct
reflect
explain
evaluate
infer
visualise
sequence
think critically
solve problems
hypothesise
define
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2×(3 apples + 4 bananas )
2× 3 apples + 2×4 bananas
6 apples and 8 bananas
Using pro numerals
2× ( 3a + 4b)
2× 3a + 2×4b
6a +8b
Illustration 1: Algebraic letters do not stand for things
The meaning of letters in algebra
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/St42504P.htmhttp://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/St42504P.htm
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Confusions about the meanings of letters: doughnuts
Students were asked the following question:
Write an equation which describes the situation
» “6 doughnuts cost 12 dollars”
Correct equation, written by nearly all students,
» 6d = 12, but what does it mean?
Instructions to students:
After you have written the equation, say what quantity each
of the numerals and pronumerals represents.
“The meaning of letters in algebra 4.25”
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/St42504P.htmhttp://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/St42504P.htm
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Student 6 d 12
Anna* amount of
doughnuts doughnuts cost
Ben numeral pronumeral numeral
Cath amount of
doughnuts
cost per
doughnut Total price
Dan six doughnuts cost $12
Ellie number of
doughnuts doughnuts price
Interpreting student work 6d=12
* by far most common response
“6 doughnuts cost 12 dollars”.
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Student 6 d 12
Anna* amount of
doughnuts doughnuts cost
Ben numeral pronumeral numeral
Cath amount of
doughnuts
cost per
doughnut Total price
Dan six doughnuts cost $12
Ellie number of
doughnuts doughnuts price
* by far most common response
Only one
correct!
“6 doughnuts cost 12 dollars”
Interpreting student work 6d=12
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Fran wrote this incorrect equation
2 d = 12
2 = cost of each doughnut
d = number of doughnuts
12 = overall cost
Unusual incorrect response, but Fran is one of the few students
who thought carefully about what she really meant.
“6 doughnuts cost 12 dollars”
Interpreting student work 6d=12
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Famous problem
• At a university there are 6 students for every professor.
Let S be the number of students, P be the number of professors,
and write an equation.
• Letter as object misconception again
6S = P
“The meaning of letters in algebra 4.25”
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• A factory makes bicycles and tricycles, using the same wheels
• Supplier provides no more than 100 wheels per day
• Their customer requires at least 4 tricycles for every bicycle.
• Profit is $300 for either a bicycle or a tricycle.
• Aim is to maximise profit
(How many of each should the factory make?)
The meaning of letters in algebra
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Example
• kla_maths_pd_keynote05.pptx
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kla_maths_pd_keynote05.pptx
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Time to reflect
In Lesson 1 I have learnt that . . .
One thing I need to remember . . .
I did know how to . . .
but now I can . . .
The most important thing I learned in this session . . .
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