### Problem Solving

```PROBLEM SOLVING
A PILOT PROGRAMME WITH SECOND-YEAR STUDENTS
SEAMUS KNOX (DES)
FIONA LENNON (PMDT)
TONY KNOX (PMDT)
JOHNNY FLAHERTY (PMDT)
Introduction
Background
• Purpose
• Introduce students to collaborative problem solving
• Motivate curricular content
• Establish student response to the process
• Project Design & Organisation
• Partnership
• Workshops
• Laois Education Centre
• School-Based
• Project Website
mathsprojectlaois.weebly.com
• Content
• Chat
Student Profile
• 18 students in total-9B,9G
• 14 Average or Just above average
• 4 Well above average
• Half of the students listed Mathematics as one of
their two favourite subjects
• 14 enjoyed mathematics ‘most of the time’ in
school
• When asked if they liked mathematics
all agreed or strongly agreed
Workshop Structure
• Minimal Instruction
• Group work
• Unseen Problems
• Collaboration
• Discussion
• Internet Access
• Presentation of Solutions
• Homework
• Research Topics
• Feedback
Workshop Content
Workshop 1
• Introduction to GeoGebra
• Car Park Problem
• Paper Folding
Workshop 2
• Constructions
• The Euler Line
• Napoleon’s Theorem
Role of GeoGebra in facilitating a problem-solving approach
• Strengths
• Weaknesses
Expanding Car park
A client wants to double the area of his car park.
He insists on retaining a square shape and as many of
the trees as possible.
• What is the best solution?
Why Choose This Problem?
• Interesting.
• Many approaches.
• Accessible.
• Layers.
• Lends itself to group work and promotes
discussion.
• Requires students to explain their thinking.
Workshop Content
Workshops 3 &4
• Basic Number Theory
• Properties of number
• Divisibility Tests
• Sequences
• Building on the students’ prior learning
• Algebra
• Factors
• Place Value
• Basic Expansions
School Workshops
Four in Total
• Designed and delivered by students
• Content
• Delivery Model
• Outcomes
• Positives
• Student Engagement
• Response from the Delivery Teams
• Response from Schools
• Shortcomings
• Too ambitious
• Too much content
• Too little exploration
Student Responses
Strongly
Agree/
Agree
I enjoyed participating in the project
Disagree/
Strongly
Disagree
(%)
18
100%
0
0%
Participating in the project increased my
understanding of mathematics
17
94.5%
1
5.5%
Participating in the project increased my
appreciation of mathematics
18
100%
0
0%
My approach to school work and homework
in mathematics is different now
14
78%
4
22%
I liked the way the workshops were
organised (group work etc.)
18
100%
0
0%
I enjoyed using GeoGebra to solve problems
16
89%
2
11%
Student Responses
Strongly
Agree/
Agree
I enjoyed solving problems in number
theory (residue classes, divisibility tests
etc.)
Disagree/
Strongly
Disagree
(%)
17
94.5%
1
5.5%
16
89%
2
11%
I enjoyed delivering the workshops to my
fellow students
17
94.5%
1
5.5%
Presenting the material in the workshops
helped me to understand the material better.
17
94.5%
1
5.5%
I would be prepared to participate in a
further project in problem solving
18
100%
0
0%
Participating in the project will help me
when I meet difficult mathematics problems
in future
Student Responses
Using GeoGebra (7)
Doing the Sierpinski Patterns
Group work, sharing opinions, collaborating (6)
Problem solving
Food (2)
Discovering new material
Divisibility Tests
Describe the part (or parts) of the project you
enjoyed most
Applying the skills outside of school
Creating worksheets for our class mates
Looking at maths in a different way
Presenting, (more time should be allowed)
The Puzzles
Residue Classes (simple and useful)
The presenting, the puzzles, the use of GeoGebra
Student Responses
Waking up on Saturday morning (2)
Nothing
Euler Line (2)
Solving very difficult algebra problems
Number theory
When it Ended
Describe the part (or parts) of the project you
enjoyed least
Having to miss a class due to my brother’s
confirmation
There wasn’t any part I didn’t enjoy
Having to present to my class mates
I Didn’t enjoy the modular arithmetic that much
Residue Classes-but I still enjoyed them
Presenting to the class (2)
GeoGebra (2)
Student Responses
Napoleon’s theorem, prime numbers, Circumcentre etc.
How to work out problems to the full extent
I learned how to use algebra to prove things
My problem-solving improved as a result
How GeoGebra works and just thinking about numbers
Not to overreact when faced with a problem but to take my time
and figure it out
What did you learn most as a result of participating in the project
That with determination and hard work along with group work
you can solve problems easily
Have a better understanding of Maths
How to approach problems