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 MathematicsCentre for Educational DevelopmentMassey University College of EducationPalmerston NorthNew Zealanda.lawrence@massey.ac.nz

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

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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

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

• 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

of this rectangle2.2 + 2.2 = 4.415- 4.4 =10.610.6 / 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 = 4.415- 4.4 =10.610.6 / 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

• 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 translationFour problems

Focus on translationFour 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, non-algebraic 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… BUTthe 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 • 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”…. BUTMost 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

a.lawrence@massey.ac.nz

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!