CENTRE FOR EDUCATIONAL DEVELOPMENT Students making the connections between algebra and word problems

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CENTRE FOR EDUCATIONAL DEVELOPMENTStudents
making the connections between algebra and word
problemshttp//ced.massey.ac.nz
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Teacher to Adviser

• Team Leader, Numeracy and MathematicsCentre for
  Educational DevelopmentMassey University College
  of EducationPalmerston North
• New Zealand
• a.lawrence_at_massey.ac.nz

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Palmerston North (New Zealand)
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NZAMT-11 conference
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New Zealand schools

• Years 1- 6 Primary
• Years 7 8 Intermediate
• Years 9 -13 Secondary

Full primary
Year 713
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Issues in education in New Zealand

• Numeracy and literacy
• Curriculum
• Assessment
• NCEA
• Technology
• National testing

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You didnt tell me it was a word problem

• ..\little league movie_WMV V9.wmv

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Difficulties with word problems

• Educators frequently overlook the complexity of
  Mathematical English
• Vocabulary
• Connectives
• Word order
• Syntactic structure
• Punctuation

Half of the sum of A and B, multiplied by three
Half of the sum of A and B multiplied three
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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)

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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 equationsat least one equation
• (assessment schedule, NZQA)

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Algebra word problems in NAPLAN
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Skills assessed in NAPLAN 2008

• 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.

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Algebra word problems in NAPLAN
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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?

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Solve this word problem

• A rectangle has a perimeter of 15 m
• Its width is 2.2 m
• Calculate the length
• of this rectangle

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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
• of this rectangle

2.2 2.2 4.4 15- 4.4 10.6 10.6 / 2 5.3
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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
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Algebra word problems in NAPLAN
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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?

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Solving algebra word problems

• 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.

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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.

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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

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Focus on translation
Four problems
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Focus on translation
Four problems (cont)
(Stacey MacGregor, 2000)
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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.

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The expert model

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The expert model

• When 3 is added to 5 times a certain number, the
  sum is 48. Find the number.
• Comprehension - Read and understand problem
• Translation - Write as an algebraic equation
  5 x 3 50
• Solution - Manipulate equation to find x

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Comprehension

• When 3 is added to 5 times a certain number, the
  sum is 48. Find the number.
• Comprehension - Read and understand problem
• Translation - Write as an algebraic equation
  5 x 3 50
• Solution - Manipulate equation to find x

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Translation

• When 3 is added to 5 times a certain number, the
  sum is 48. Find the number.
• Comprehension - Read and understand problem
• Translation - Write as an algebraic equation
  5 x 3 50
• Solution - Manipulate equation to find x

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Translation

• When 3 is added to 5 times a certain number, the
  sum is 48. Find the number.
• Comprehension - Read and understand problem
• Translation - Write as an algebraic equation
  5 x 3 48
• Solution - Manipulate equation to find x

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Solution

• When 3 is added to 5 times a certain number, the
  sum is 48. Find the number.
• Comprehension - Read and understand problem
• Translation - Write as an algebraic equation
  5 x 3 48
• Solution - Manipulate equation to find x
• x 9

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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)

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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)

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Students use informal methods

• Many students rely on informal, non-algebraic
  methods even in problems where they are
  specifically encouraged to use algebraic methods
  
• (Stacey MacGregor, 1999)

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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)

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Focus on translation

• Expert blind spot is the tendency
• to overestimate the ease of acquiring formal
  representations languages, and
• to underestimate students informal
  understandings and strategies
• (Koedinger Nathan, 2004, p. 163)

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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)

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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.

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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 dont know
what to say
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Key words
Key words are something I do use but I am not
sure how well they work
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Problems with the key word strategy

• Keyword focus tends to bypass understanding
  completely so when it doesnt 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.

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So what strategies are effective?

• Explicit expectations

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The algebraic problem solving cycle

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Effective strategies

• Explicit expectations
• the problem solving cycle
• Focus on translation
• from English to algebra (encoding)
• from algebra to English (decoding)

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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.
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Effective strategies

• Explicit expectations
• Focus on translation
• from English to algebra (encoding)
• from algebra to English (decoding)
• Create the press for algebra

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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.

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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 its got to be 3.
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Its obvious

Once you see it, its 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.
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Effective strategies

• Explicit about expectations
• Focus on translation
• Create the press for algebra
• problems with tricky numbers
• problems that dont unwind
• Focus on the whole problem
• the complete problem solving cycle

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Focusing on the whole problem
Knowing what to let the variable be is critical.
Initially it seemed like it didnt matter.

I understood what I was doing because I had
translated it into words first.
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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.
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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?

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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

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Hells library
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Thank you

• a.lawrence_at_massey.ac.nz

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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.
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Getting the point

They understood the point of algebra. I had
students answering in class with confidence who
normally dont and seemingly enjoying what they
were doing!
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Student improvement

I feel a lot better about algebra now. Before I
didnt know how to write equations and now I do.
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More focus on solving for a few

I can write equations but I still dont know what
to do with them. Its really good but its like
What do I do next? - like, I dont 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!