94 Students making the connections between algebra and word problems

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

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

a.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

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

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 rectangle 2.2 + 2.2 = 4.4 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

• 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)

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

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

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

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

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… 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

• 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

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

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.

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 cm 2 , what is the width of the rectangle?

This one is not hard You know that 21

is 7 times 3 so

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!