Transcript Slide 1

CENTRE FOR EDUCATIONAL DEVELOPMENT
Students making the connections
between algebra and word
problems
http://ced.massey.ac.nz
Teacher to Adviser
Team Leader, Numeracy
and Mathematics
Centre for Educational Development
Massey University College of Education
Palmerston North
New Zealand
[email protected]
Palmerston North (New Zealand)
NZAMT-11 conference
New Zealand schools
Years 1- 6
Primary
Years 7 & 8
Intermediate
Years 9 -13
Secondary
Full
primary
Year 7–13
Issues in education in New Zealand
• Numeracy and literacy
• Curriculum
• Assessment
– NCEA
– Technology
– National testing
You didn’t tell me it was a word problem
..\little league movie_WMV V9.wmv
Difficulties with word problems
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Educators frequently overlook the
complexity of Mathematical English
Vocabulary
Half of the sum of A and B,
multiplied by three
Connectives
Word order
Syntactic structure Half of the sum of A and
B
multiplied
three
Punctuation
Context is complicated
Contextualising maths creates another layer
of difficulty – the difficulty of focusing on the
maths problem when it is embedded in the
‘noise of everyday context’
(Cooper and Dunne, 2004, p 88)
Placing mathematics in context tends to
increase the linguistic demands of a task
without extending the mathematics
(Clarke, 1993)
The national standard in NZ
• “use algebraic strategies to investigate and
solve problems… Problems will involve
modelling by forming and solving appropriate
equations, and interpretation in context”
• “must form equations…at least one equation”
(assessment schedule, NZQA)
Algebra word problems in NAPLAN
Skills assessed in NAPLAN 2008
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Identifies the pair of values that satisfy an
algebraic expression.
Solves a multi-step algebra problem.
Solves algebraic equations with one variable
and expressions involving multiple
operations with negative values.
Determines an algebraic expression to
model a relationship.
Algebra word problems in NAPLAN
What is it about algebra word
problems?
• What are algebra word problems?
• Why do students find them difficult?
• What can teachers do to help their
students tackle them with more success?
Solve this word problem
A rectangle has a perimeter of 15 m
Its width is 2.2 m
Calculate the length
of this rectangle
It is a word problem…
A rectangle has a perimeter of 15 m
Its width is 2.2 m
Form and solve an equation to
calculate the length
2.2 + 2.2 = 4.4
of this rectangle
15- 4.4 =10.6
10.6 / 2 =5.3
It is a word problem
… but is it an algebra word problem?
What makes an algebra word problem?
What solution strategies are we expecting?
Is this algebra?
Is this an equation?
2.2 + 2.2 = 4.4
15- 4.4 =10.6
10.6 / 2 =5.3
Algebra word problems in NAPLAN
Methods of solving word problems
• Do you have a preferred way of solving word
problems?
• What do you consider when you are deciding
how you will tackle a word problem?
• What makes you decide to use algebra to
solve a word problem?
• Can you write a word problem that all your
students use algebra to solve?
Solving algebra word problems
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Experts tend to solve algebra word problems
using a fully algebraic method. They
translate into algebra and use algebra to find
the answer.
Students commonly use a variety of informal
solution strategies. They work with known
numbers to find the answer.
Informal methods
Trial and error, guess and test, or guess,
check and improve, involve testing numbers in
the problem. These methods involve working
with the forwards operations.
Logical reasoning methods involve first
analysing the problem to identify forwards
operations, then unwinding using backwards
operations.
Informal methods work well
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
Forwards : multiply by 5, add 3
Backwards: subtract 3, divide by 5
Focus
on
translation
Four problems
Focus
on
translation
Four problems (cont)
(Stacey & MacGregor, 2000)
Informal methods have limitations
Informal methods can be effective for simple
word problems.
More complex problems such as those with
‘tricky’ numbers as solutions and those
involving equations with the unknown on both
sides are not readily solved by informal
methods.
The expert model
The expert model
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
1. Comprehension - Read and understand
problem
2. Translation - Write as an algebraic equation
5 x +3 = 50
3. Solution - Manipulate equation to find x
Comprehension
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
1. Comprehension - Read and understand
problem
2. Translation - Write as an algebraic equation
5 x +3 = 50
3. Solution - Manipulate equation to find x
Translation
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
1. Comprehension - Read and understand
problem
2. Translation - Write as an algebraic equation
5 x +3 = 50
3. Solution - Manipulate equation to find x
Translation
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
1. Comprehension - Read and understand
problem
2. Translation - Write as an algebraic equation
5 x +3 = 48
3. Solution - Manipulate equation to find x
Solution
When 3 is added to 5 times a certain
number, the sum is 48. Find the number.
1. Comprehension - Read and understand
problem
2. Translation - Write as an algebraic equation
5 x + 3 = 48
3. Solution - Manipulate equation to find x
x=9
In the expert model
“Equation solving is a sub-problem of
story problem solving, and thus story
problems will be harder to the extent that
students have difficulty translating stories
to equations”
(Koedinger & Nathan, 1999, p. 8)
Few students use the expert model
Even after a year or more of formal
algebraic instruction, many students find
word problems easier than algebraic
problems
(van Amerom, 2003)
Students use informal methods
Many students rely on informal, nonalgebraic methods even in problems
where they are specifically encouraged to
use algebraic methods
(Stacey & MacGregor, 1999)
Difficulties with translation and solution
Students who do try to follow the expert
model may have difficulties at any of the
three stages… BUT
the major stumbling blocks for secondary
students are the translation and solution
phases.
(Koedinger & Nathan, 2004)
Focus on translation
Expert blind spot is the tendency
•
to overestimate the ease of acquiring
formal representations languages, and
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to underestimate students’ informal
understandings and strategies
(Koedinger & Nathan, 2004, p. 163)
Symbolic precedence view
Secondary pre-service teachers prefer to use
an algebraic method regardless of the nature
of any given word problem. They tend to use
formal methods regardless of the problem and
view the algebraic method as “the one and
only ‘truly mathematical’ solution method for
such application problems”
(Van Dooren, Verschaffel, & Onghena, 2002, p. 343)
Mismatch between approaches
• The mismatch between teachers’ and
students’ approaches is reinforced by
textbooks which commonly portray methods
that do not align with typical students’
algebraic reasonings.
• Teachers need to critically view tasks and
create or select activities and problems that
are appropriate.
Teachers lack explicit strategies
I am not even sure I
know how I tackle
word problems.
I have never been taught how to go about
problems myself. I just seem to know what
to do, so when it comes to teaching kids,
well, I don’t know what to say…
Key words
Key words are
something I do use… but
I am not sure how well
they work
Problems with the key word strategy
• Keyword focus tends to bypass understanding
completely so when it doesn’t work students
are at a total loss.
• Key words are only able to be identified in
simple word problems.
• Key words can be misleading with more
complex problems.
So what strategies are effective?
• Explicit expectations
The algebraic problem solving cycle
Effective strategies
• Explicit expectations
– the problem solving cycle
• Focus on translation
– from English to algebra (encoding)
– from algebra to English (decoding)
Focusing on translation both ways
I liked how we learnt from both
views - putting it into word
problems and taking a word
problem and putting it into
algebraic. I understand it much
better now.
Effective strategies
• Explicit expectations
• Focus on translation
– from English to algebra (encoding)
– from algebra to English (decoding)
• Create the ‘press for algebra’
Tasks encourage informal strategies
Teachers commonly start with problems that
are easy for students to do in their head in
order to demonstrate the “rules of
algebra”…. BUT
Most students only see a need to use
algebra when they are given problems that
they cannot easily solve with informal
methods.
A common problem
A rectangle is 4 cm longer than it is wide.
If its area is 21 cm2, what is the width of the
rectangle?
This one is not hard.
You know that 21
is 7 times 3 so
it’s got to be 3.
It’s obvious
Once you see it, it’s obvious…
Why would a student use
algebra? But algebra is what I
would always do first. At least
now I know I will have to be so
careful with the problems I use.
Effective strategies
• Explicit about expectations
• Focus on translation
• Create the ‘press for algebra’
– problems with ‘tricky’ numbers
– problems that don’t ‘unwind’
• Focus on the whole problem
– the complete problem solving cycle
Focusing on the whole problem
Knowing what to let the
variable be is critical. Initially it
seemed like it didn’t matter.
I understood what I was
doing because I had
translated it into words
first.
Making sense
Translating into words was really
helpful before we had to solve
the equations… It made it easier
to solve them and it made it
make more sense.
Questions raised
• What are algebra word problems?
• Why do students find them difficult?
• What can teachers do to help their
students tackle them with more success?
Teachers can make a difference
• Make explicit connections between algebra
and word problems
• Develop skills of encoding and decoding
• Use tasks which press for algebra
• Focus on the full problem-solving cycle
• Emphasise flexible approaches to solving
problems
Hell’s library
Thank you
[email protected]
Connecting with algebra
It is glaringly obvious that it has worked. The
whole idea of starting with the word problems
and working on how to translate it and then
develop the skills from that. I think that whole
way of them understanding the use of algebra
made them connect much better with the topic.
Getting the point
They understood the point of
algebra. I had students answering in
class with confidence who normally
don’t… and seemingly enjoying what
they were doing!
Student improvement
I feel a lot better about
algebra now. Before I didn’t
know how to write equations
and now I do.
More focus on solving for a few
I can write equations but I still don’t know
what to do with them. It’s really good but it’s
like “What do I do next?” - like, I don’t even
know the steps. What do you do after that,
and what do you do after that? I really
needed teaching for solving ’cos then I
would have been done!