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Conference Powerpoint v14 - fibonacci09-10
Using Problem Solving To
Engage Students In Their Study
of Linear Equations
Options: D31 & F34
Presenter: Russell James
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While You’re Waiting …..
Peruse The Discussion Paper For This Option on
pages 96-101 Of the Conference Handbook
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Welcome
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Russell James
Maths co-ordinator
Fairhills High School
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This year …
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I was given 20 days of professional
leave by the education department to
undertake a project that I have
called.
Using Problem Solving To Engage
Students In Their Study of Linear
Equations
1
2
4
Over the course of my project three
units were developed & trialled
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Unit 1: Modelling linear data with
formulae (and working with
equations)
Unit 2: Strategies for finding linear
formulae from tables
Unit 3: Strategies for finding linear
formulae from graphs
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2
4
My Objective Today is to …
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share some of the experiences,
resources and pedagogical
approaches that were trialled during
my Professional Leave project .
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I have given you …
Booklets for each of the following:
• the problem solving activities trialled
• the ICT activities trialled
• the complete activities of unit 1
• sample activities from units 2 & 3
• an Assessment Task from the SNMY project
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Plus a CD with electronic copies
of the majority of the activities trialled
in the project
My Contact Details
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On the CD there is also the file “Presenter Contact
Details” with my school contact details and:
My Email:
[email protected]
Project Wikispace:
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http://fibonacci09-10.wikispaces.com/
Fairhills High School
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We are situated in the City of Knox
near the Dandenongs with a student
population
–
–
–
–
Around 850
Mostly born in Australia
Mainly Anglo/European heritage
Rated in the middle of the Education
Department’s scale for learning potential.
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Two of our major concerns in
Mathematics across the school are..
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1. Learning participation of
2.
Students’
negativity
students with a history
towards
new
problems
of disengagement in the
or
situations
that
Maths.
require more thinking
than just the repetition
of a procedure.
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2
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Hence My Project
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I chose Problem Solving as a key
element of the project because I
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believe well designed activities can:
• Improve students’ thinking resilience
and skills
• Improve students’ engagement if the
activity
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– Gives them a realistic chance to explore a
situation and feel a sense of satisfaction that
they have thought a problem through
– Provides a reference point and everyday
relevance for the study of higher order
mathematical concepts and procedures
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My Design Model For The
Problem Solving Activities
is described on pages 96 of
your conference handbook
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Some features of this model
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Problem solving activities should:
– Have multiple pathways to a solution
providing and entry point and
challenge for students over a range
of abilities
– Provide a rationale for studying
higher levels of mathematics
– Have an investigative stage that is no
more than 40minutes.
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Also ..
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The activities should:
– be presented with a problem solving toolbox to
reinforce students problem solving skills
– include a reflection session to enable students
to develop their metacognitive skills and
reinforce what they have learnt from their
experience.
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Let’s look at a problem
This problem
follows on from a
Multiplicative
Thinking
assessment task
that I will talk
about later
This is just as
important as the
task itself because
a good lesson
structure changes
the task from a
puzzle to a learning
experience
Table & Chairs Follow-Up
Problem
Purpose
To give students:
• A rationale for the study of linear equations
• Develop students thinking resilience and problem
solving skills (including increasing their familiarity with
a problem solving toolbox)
Lesson Structure
1. Introduce The Problem (5-10 min)
2. Solve The Problem (30-40 min)
3. Reflect On The Problem (5-10min)
4. Write a report (next lesson)
Stage 1. Introducing the Problem
After handing out the problem sheet (with the working page
on the back):
– I described Problem 1 to the class
– 1 wrote the Universal problem solving method on the
board
• Understand the problem
• Decide on a strategy
• Solve the Problem
• Check Your Answer
– I invited possible strategies from the class and then
referred them to our Problem Solving Toolbox
A Common Strategy List
Most references and websites I’ve looked up present
between 8 and 10 problem solving strategies. Rob
Vingerhoets (a very good consultant we have worked with)
puts forward the following methodologies:
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–
–
–
–
–
–
–
–
–
Guess & Check
Draw a diagram
Make a table
Make a model
Work backwards
Write a number sentence
Look for a pattern
Act it out
Solve a simpler related problem
List all possibilities
Our Strategy List/Problem Solving Toolbox
• Is an expanded version of Rob V’s
• We broke the table approach into 2
• Included a strategy explicitly mentioning the use of a
graph
• Included 3 algebraic methodologies
• Introduced 4 methods that utilise technology
A hint from Rob Vingerhoets:
Insist On A Starting Strategy From Each
Student Before They Begin The Problem
This sounds like it would take a long time but it is actually
quite quick and it provides an impetus and a starting point for
most students to get underway with the task.
It also provides a way of identifying those students who need
further help, either by revisiting the problem to make sure
they understand it, or by helping them to choose a
meaningful strategy.
Stage2: Solving the problem
As I moved around helping and encouraging students I
constantly reinforced my mantra for all problem solving
activities:
– Record all your working (preferably under a heading)
– Discuss the problem with your friends because this is
a valuable way to learn
– Feel free to change your strategies or combine them
– I am asking you for a prediction with mathematical
evidence. There is no right or wrong way to do the
problem just follow a path that makes sense to you.
Stage 3: The Reflection Phase
This is perhaps the most crucial phase with respect
to:
a) Producing a long term improvement in
students thinking skills
b) Providing a meaningful rationale for the
subsequent study of related maths skills.
For this problem I asked selected students to
describe how they solved the problem and recorded
their approaches on the board.
Solving with a diagram
A small number of students solved the problem by
drawing 30 tables (no one opted to build it with tokens
and rectangles)
………
Using a table
Lots of students liked this
method because they
appreciated the efficiency of
getting an accurate answer with
less effort than drawing 30
tables.
Tables
Chairs
1
6
2
10
3
14
4
18
5
22
..
..
..
..
30
Writing number sentences
A number of students tried writing number sentences to get
an answer. Not all were right but there was some good
discussion about the correct answer
1 + 30 x 4 + 1 = 122
1 + 4 + 4 + 4 + …… + 1 = 122
30 x 6 = 180
I am always very conscious
about my language when
presented with answers that do
not give the “right answer”. E.g.
“there is some good maths
there, can you explain why you
think this is right”
Look For A Pattern
A limited number of students recognised a pattern
2 + 4 x number of tables
Or
1 + number of tables x 4 + 1
Building A Unit Around The Starting
Problem
From the primary and follow
up problems I was able to give
credibility to the statement
“If we know how to find and
work with rules (formulae) we
might be able to make our life
easier in the future”
Formulae & Equations Skills Development &
Assessment Tasks
Problem Solving Activity 2
Word problems
ICT Activities
The Linear Equations Continuum
Students who have progressed quickly through the skills
and assessment tasks were directed to an exercise called
“Year 8-9 Linear Equations Continuum”.
Portfolio page for Unit 1
As well as including more problem solving into our
teaching one of my other goals in my TPL project was
to investigate alternative ways teaching and assessing
linear equations so that students were offered some
decisions about what and how they learned.
Currently much of our curriculum could be classified as
passive learning since it relies on the more traditional
textbook approach that focuses on “mastery through
practice” (rather than active learning).
Intermission
• Eat a frog
• Have a read of the booklets I have
supplied
• Formulate some questions
Evaluating our project
This year Fairhills had seven classes at year 8
and they were all involved in the project.
It is important to note that not all of these
classes have the same course design.
At year 7 & 8 students our classes are
divided into two pathways
An accelerated pathway, with two
Maths classes aiming to attain a
A second
VELS
level pathway
of at leastwith
5.5 five
by the end
ofMainstream
Year 8 andMaths
henceclasses
be able aiming
move to
attain
a VELS
levelthird
of atyear
leastat5.0 by
into
Year
10 in their
the end of school.
Year 8 and then move into
secondary
Year 9.
Data we examined
To evaluate our project we collected and
analysed



Student achievement data
Student feedback sheets
Teacher interview notes
In general faster &/or motivated
learners …





appreciated the ability to move through the topics at
their own pace
performed at least as well as in these topics as with
other topics. In many cases they were attempting
higher order questions that they would not have
been doing otherwise
became more adept and resilient with problem
solving situations as they undertook more such
tasks
become more active learners in other topics
become more adept at making connections between
their existing knowledge and the mathematics
encountered in new topics.
In general learners with a history of
disengagement …..
 found the problem solving tasks very difficult and
tended to give up early or asked for guidance at
every step. They did show some improvement in
their problem solving resilience but were still heavily
reliant on the teacher for instruction and motivation
 seemed to prefer the textbook approach because of
its structure and the fact that they are lead into each
question with a recipe for finding a solution.
 did not show any marked difference in interest or
progress relative to other units undertaken.
Teachers felt:
 a degree of satisfaction seeing their
motivated students becoming more active
learners and having the opportunity to
progress to levels of understanding
appropriate to the abilities.
 at times it was a lot of work managing the
units, particularly when there was a spread
of activities underway. Overall it was
survivable if the units were kept to less than
3 weeks.
They also felt …
that the lack of success with hard to motivate
students was largely due to frustration caused
by their lack of thinking resilience and skills
Compounded by the fact that these students were
unaccustomed to this type of task because in the past
they have been directed more to repetition, rather
than cognition. Thus to achieve more success with
the hard-to-engage students, we felt that we need to
give them a lot more experience with cognitive and
meta-cognitive exercises.
Was the Project A Success?
Overall, we would say Yes
Because
 There were observable benefits for a
significant proportion of students with no
disadvantage to the remainder of the cohort
 It further highlighted the need for us to
develop the thinking resilience and skills of
the learners with a history of disengagement.
So in 2010 we will …..
 provide Year 8 students with some
preliminary thinking tasks before we run
slightly revised versions of the units trialled in
2009.
 incorporate more Problem Solving activities
into the Year 7 curriculum so as to provide
students with more cognitive and metacognitive experience for Year 8.
SNMY Multiplicative Thinking
Assessment Task
Scaffolding Numeracy in the Middle Years
(SNMY) refers to a research and evidencebased project under the direction of
Professor Di Siemon, in conjunction with
DEET, RMIT and the Tasmanian Dept Ed.
The project aimed at identifying and refining
a learning and assessment framework for the
development of multiplicative thinking at this
level using rich assessment tasks
For more details Google SNMY project.
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