Contextual Problem-PosingSeeing The Mathematics Around Us

I’m Kate. I’m a primary educator, with a passionate interest in how the practice of

contextual problem-posing can increase the engagement, confidence and enjoyment

found in mathematics for both students and teachers alike.

Hi!

Table of Contents

What is Contextual Problem-Posing?

Teachers as Problem-Posers

Students as Problem-Posers

Problem-Posing in Practice

– Georg Ferdinand Ludwig Philipp Cantor (1867)

“In mathematics the art of proposing a question must be held of higher

value than solving it.”

What is Contextual Problem-Posing?

What do I mean by ‘contextual problem-posing’?

The development of a problem within a given

context

What do I mean by ‘contextual problem-posing’?

Context:

- Physical Setting- Idea or Question- Artifact

What do I mean by ‘contextual problem-posing’?

Meaningful, relevant, real

Real ≠ Real Life

Open, context given, content given, or content & context given

Context Given

Pose as many mathematical problems as you can about

the book we just read.

Context Given

What mathematical questions do we need to answer in

order to plan a class party?

Content Given

Pose as many problems as you can involving division with a

remainder.

Context & Content Given

Pose as many multiplication problems as you can about

designing a treehouse.

Hello!

“A good problem is something you don't know how to solve. That's what makes it a good

puzzle and a good opportunity.”― Paul Lockhart

Teachers as Problem-Posers

Problem-Solving Resources

How are problem solving resources sourced for your

school?

Benefits to Teachers

Improve content knowledge

Improve ability

Improve confidence

Improve problem-posing capacity & self-efficacy

Benefits to Teachers

Improves beliefs and attitudes toward

mathematics

Professional Development for Teachers

Allows teachers to see the benefits of problem posing, & encourages implementation with students

Allows teachers to build their own resources for problem-solving sessions

Improves content knowledge, and problem-posing capacity

Increases confidence & improves attitude toward mathematics

Promotes rich mathematical discussion

Teacher Professional Development1. Identify a content area2. Choose a context3. Teachers generate as many

mathematical problems as they can4. Discuss

1. Multiplicative Thinking2. Out to dinner3. Teachers generate as many

mathematical problems as they can

4. Discuss

For Example...

Example Response...I went to dinner with two friends, Tim and Hala, and we shared

some dumplings. For every dumpling I ate, Tim ate 2, and Hala ate 3.

How many dumplings did my friends each eat by the time I’d had 4?

How many each would Hala & I have eaten if Tim had eaten 14 dumplings?

How many each would Tim & I have eaten if Hala had eaten 18?

How would we have shared a plate of 12 dumplings?

Discussion

MathematicalIs the problem

mathematical in nature? Is it a

problem?

Concept SpecificDoes the problem primarily address

the concept?

Suitabilityopen/closed;

language; difficulty;

relevance etc.

1. Choose a concept that you think would benefit the teachers at your school

2. Pose as many mathematical problems as you can about a context that would be engaging for your colleagues

For example...

Discussion

MathematicalIs the problem

mathematical in nature? Is it a

problem?

Concept SpecificDoes the problem primarily address

the concept?

Suitabilityopen/closed;

language; difficulty;

relevance etc.

Collegiate Planning for Problem Solving

1. Problem-pose for students2. Open/context/content

3. Discuss the problem & anticipated responses

What questions will you ask the students who need scaffolding?

What questions will you ask the students who need extending?

What responses do you anticipate, and how will you address/assess them?

Collegiate Planning for Problem Solving1. Identify a context/content area2. Teachers generate problems and

anticipate responses3. Discuss and choose

Hello!

“The real satisfaction from mathematics is in learning from others and sharing with others.”

- William Thurston

Students as Problem-Posers

Hi!

Benefits to Students

Encourages creative, flexible & reflective thinking

A deeper understanding

of the concepts and structures of

problems

Benefits to Students

Increases engagement and

enjoyment.Improves

attitudes and opinions

Demonstrate understanding

Highlight misconceptions

Benefits to Students

Play

Messy

Fun

Authentic, rich assessment of

content, thinking,

reflection, communication

& strategy

Students as Problem-Posers

Many benefits

Small scale: based on a book, toy, game or school location

Large scale: real-world, authentic problems

Solving their own or their peers’ problems is more engaging

Hello!

“It is clear that the chief end of mathematical study must be to make

the students think.” – John Wesley Young

1. Identify content area, and/or2. Choose a context3. Students pose problems4. Discuss

1. 2-digit numbers2. An excursion to the zoo3. Student pose problems4. Class discussion

For Example...

Example Responses (anticipated)...We visited the elephants, then the zebras and the giraffes.

Which animal did we visit first?

There were 12 penguins, 18 meerkats, 23 birds, and 15 lemurs. Which animal were there most of? Put the animals in order

from least to most.

At the zoo, we saw 10 penguins and 3 monkeys. How many animals did we see altogether?

We saw 35 animals at the zoo.

We saw 36 animals’ legs at the zoo. Which animals and how many of them might we have seen?

Student Group Discussion

ProblemHas a problem been

posed?

StructureIs the problem open-ended?

MathematicalIs it a mathematics problem?

EngagingIs the problem interesting to you? Do you want to find the solution?

Example Responses (anticipated)...We visited the elephants, then the zebras and the giraffes.

Which animal did we visit first?

There were 12 penguins, 18 meerkats, 23 birds, and 15 lemurs. Which animal were there most of? Put the animals in order

from least to most.

At the zoo, we saw 10 penguins and 3 monkeys. How many animals did we see altogether?

We saw 35 animals at the zoo.

We saw 36 animals’ legs at the zoo. Which animals and how many of them might we have seen?

No problem has been posed →

← Not mathematical

←Not open ended…Engaging?

Not open → ended…Engaging?

Open ended →Engaging?

1. Imagine your students are going on an excursion to the zoo. You invite your students to pose problems during the excursion.

2. Anticipate as many problems as you can that your students might pose

What misunderstandings are you likely to come

across?

Addressing Anticipated Responses

Misunderstandings

LearningHow will you facilitate

student learning?

ResponseHow will you address misunderstandings?

ScaffoldingHow can you scaffold

students who are having difficulty getting started?

Structural ErrorsWhat errors might you see

in students' problems?

ExtensionHow will you extend

students who need it?

1. Identify content area, and/or2. Choose a context3. Students pose problems4. Discuss

1. 2-digit numbers2. World’s most dangerous animals3. Student/s pose problems4. Discuss

For Example...

Have a go!

1. Choose a difficult content area OR a student who is difficult to engage in mathematics

2. Think of a context to make the content more engaging to your student/s

3. Anticipate problems your student or students may pose

What misunderstandings are you likely to come

across?

Addressing Anticipated Responses

Misunderstandings

LearningHow will you facilitate

student learning?

ResponseHow will you address misunderstandings?

ScaffoldingHow can you scaffold

students who are having difficulty getting started?

Structural ErrorsWhat errors might you see

in students' problems?

ExtensionHow will you extend

students who need it?

Hello!

“It’s fine to work on any problem, so long as it generates interesting

mathematics along the way – even if you don’t solve it at the end of the day.”

– Andrew Wiles

Problem-Posing in Practice

Implementing Problem-Posing

Professional LearningImprove content

knowledge, confidence, engagement and attitudes

Students Posing ProblemsImprove content

knowledge, confidence, engagement and attitudes

Planning for Problem-SolvingTeachers creating

relevant, engaging problems

Student EngagementRelevance for

previously disengaged students

For Students and Teachers

Many benefits

Planning for problem-solving

Engaging in authentic mathematics

Many options for embedding in learning

Authentic Assessment

Problem-posing skills

Mathematics proficiencies

Mathematical content knowledge

Community EngagementProvides opportunity to engage in community

context

What is important to your students’ community?

What opportunities for problem-posing are there?

Who could you collaborate with from the community?

Have a go!1. Choose a content area that you will be focusing on with your students in the next few weeks

2. Pose as many mathematical problems as you can about a context you think will engage your students

Problem Posing as a Platform for Social Equity

CulturalE.g., Potential to

engage Indigenous students with

community and Country

SocioeconomicMathematics that is real and relevant to

all students

VocationalEngaging students in mathematics for

a wide range of futures

Hello!

Engage your students in the maths around , the maths in real world, the

maths in everyday livesthem their

theirthem their

their

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ThanksDo you have any questions?

kate.eastcott@education.vic.gov.au

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