mathematical thinking 1 - js photography · 2018. 8. 28. · seameo recsam 5 april 2017 ....

Mathematical Thinking 1

SEAMEO RECSAM

5 April 2017

[email protected]

18-Apr-17 Assoc. Prof. Dr Kor Liew Kee 2

Contents

• What is mathematical thinking?

• Types of mathematical thinking

• How do we know mathematical thinking exists?


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?!


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

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


Ball ( 2002) https://www.atm.org.uk/write/mediauploads/journals/mt181/non-member/atm-mt181-17-

19.pdf


A. Watson and J. Mason: Questions and prompts for mathematical thinking, ATM.


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


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.


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.


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.


What do you say about mathematical thinking?

• Mental

• Content

• Method


analyse

create

predict

reason

identify

calculate

propose

estimate

deduct

reflect

explain

evaluate

infer

visualise

sequence

think critically

solve problems

hypothesise

define

14

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

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

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”.

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”


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”


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”

• 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

Example

• kla_maths_pd_keynote05.pptx


kla_maths_pd_keynote05.pptx

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 . . .

16

mathematical thinking 1 - js photography · 2018. 8. 28. · seameo recsam 5 april 2017 ....

Documents