Problem Solving - morelandnumeracyaiznetwork

Open-ended Tasks,
Fermi Questions & Problem
Adrian Berenger
27 October 2010
Teaching & Learning Coach
Moreland Network
PoLT 5. Assessment practices are an integral part of teaching
and learning
Assessment contributes to planning at a number of levels.
Monitoring of student learning is continuous and
encompasses a variety of aspects of understanding and
practice. Assessment criteria are explicit and feedback is
designed to support students' further learning and encourage
them to monitor and take responsibility for their own learning.
5.1 Assessment practices reflect the full range of learning program
5.2 The teacher ensures that students receive frequent constructive
feedback that supports further learning
5.3 The teacher makes assessment criteria explicit
5.4. Assessment practices encourage reflection and self assessment
5.5 The teacher uses evidence from assessment to inform planning and
Formative Assessment Examples
• Teacher observation
• Checklists
• Drawings, photos, journal entries
• Interviews, probes
• Portfolios
• Rich tasks and rubrics
• Peer and student self-assessment
Benefits of Group Work
• Everyone has a job to do
• It is easier with many brains ie. ‘Many hands
make light work’
• Different strategies are visible or transparent
to the group
• Mathematical Language can be used
throughout the task
• The group can self-regulate and knows when
they’ve completed the task
Open-ended Questions
• Require more than remembering a fact or
reproducing a skill
• Have several acceptable answers
• Engage learners of different abilities
• Enable students to learn by answering the
• Enable the teacher to learn about each
student from the attempt
Whole Group Activity
A piece of ribbon 1m long is used to tie a present.
20cm of the ribbon is used to form a bow.
How big is my present?
– How can this problem
be modified to cater for
different student
– How do you want
students to record their
Writing an Open Question
• Some Examples
– A rectangle has a perimeter of 16cm. What might
the dimensions be?
– The hour and minute hands of a clock are at 90o.
What might the time be?
– Using the published ladder, what is the chance of
your team winning the grand final?
Other Examples
• What might this be a
graph of?
• Present this
information in
different ways
• The average of three
numbers is 7.1. One
of the numbers is
11.2. What might the
other numbers be?
Sullivan & Lilburn (2004)
Method 1
• Omit enough information so that, although
the answer remains the same, the digits
required to achieve the answer becomes
+ 173
+ *7*
Method 1 (continued)
Two fifths of 250 students
borrow books from the library
each day. Calculate the
number of students who
borrow books each day.
Two fifths of the students in a
school borrow books from the
library each day. How many
students might there be in the
school and how many of them
borrow books each day?
Find the missing angle on this
What might the angles on this
trapezoid be?
Method 2
• Work backwards from the answer. Begin with a
closed task. Calculate the answer, then work
backwards and using the context of the question,
create a question that would allow multiple
responses to achieve the same answer.
The following numbers
represents the temperature of
5 consecutive days in
Melbourne: 44◦C, 42◦C, 36◦C,
22◦C, 29◦C. Find the average
The average temperature over
five consecutive days in
Melbourne was 35◦C. The
highest temperature was 44◦C.
What might the temperature
have been on the other days?
Method 2 (continued)
X 0.5
What is the volume of
the cylinder?
6 cm
The answer is 17.5
What might the
question be?
What might the
dimensions of a
cylinder prism that has
a volume of 300 cm3?
Height of Statue of
• What is the height of the Statue
of Liberty if the length of her
right arm is 42 feet.
• Students work in groups and
then prepare a presentation for
the whole class including the
– An account of strategies that
were used to solve the
– How were concrete
materials used?
– A self-evaluation using a
scoring rubric
Fermi Questions
• How many 10 year olds are there in Australia?
• How many litres of fluid have you consumed in your
• How much toothpaste do Australians use during a
typical week?
• How much money do your parents spend on you in a
• How much money have your parents spent on you up
until now?
• How much money would your parents spend on you by
the time you finish primary school?
Problem solving (& investigations)
• Problem solving strategies
• Problem solving definitions
• Models
• Understanding the process
• Benefits
• Investigations
Problem Solving Strategies
Act it out (role play)
Draw a Picture
Trial & Error
Guess & Check
Make a Model
Make a Table
Make a List
Create a Rule
Make a Chart or Graph
Reduce the Problem
Work Backwards
Use Logic
Re-write the problem
Problem Solving
• find a way where no way is known off-hand. For a
question to be a problem, it must present a challenge
that cannot be resolved by some routine procedure.
Problem solving is a process of accepting a challenge
and striving to resolve it
Polya, 1990, 1965 in Booker et al (2004) p. 43
• Teaching problem-solving is not about the answer to a
problem. It is about thinking and sharing strategies; and
developing processes that solve problems. In other words,
the answer isn’t important - the solution process is. The
journey matters more than the destination.
Berenger, 2010, RMIT TCHE2363
Small Group Activities
• Developing part of a lesson plan on…
– How many handshakes…?
– Matchstick Problems
Mathematical Investigations
• Involves problem-solving but is extended,
often involves research and sharing of ideas,
and allows students to make connections with
other areas of learning.
• Problem solving is integral to the learning
program, and not simply an add-on or
something you do once a week for an hour…
• Problem solving refers to ‘Working
Mathematically’ and ‘Thinking’ within the
• National Curriculum Frameworks will specify
problem solving as a significant component of
any teaching & learning program
The Benefits
• Builds new mathematical knowledge (ideas, sense
making) & connects mathematical ideas…
• Develops a range of strategies…
• Supports the development of positive student selfesteem…
• Supports assessment for learning…
• Caters for the diverse range of student learning
• Engagement! Classroom learning environment…
• E5 Instructional Model
• Principals Of Learning & Teaching (PoLT)
• Open-ended tasks
– Sullivan, P. & Lilburn, P. 1997. Open-ended Maths Activities:
Using ‘good’ questions to enhance learning. Oxford University
Press Melbourne.
• Fermi Problems
– Maths300
– Bobis, J. et al. Mathematics for children: Challenging children to
think mathematically