The Language of Word Problems

1 © N. Rice & P. SwanBond Blocks®

TEACHER NOTES

Solving Word Problems

The Language of Word ProblemsProblem solving is often taught with an emphasis on highlighting key words and assigning a mathematical operation to the

word(s). Attributing word to operations can be mathematically incorrect. For example, in the following problem such words

relate to addition:

“Josh opened a packet of lollies and ate five. There were three left. How many were in the packet to start with?” 5 + 3 = 8

Authors are divided as to the efficacy of using a key word strategy linked to specific operations (+, – , x and ÷) in mathematics. There is more support for linking words to question types. In the research this is called schema based instruction. Different schemes such as “Singapore Maths” and the model advocated by Jitendra (2002) use slightly different

terminology to describe their approach to solving word problems in addition, subtraction, multiplication and division. Most of

these systems follow a diagrammatic approach.

According to First Steps in Mathematics: Number - Understand Operations - Calculate - Reason about Patterns, the

classification system is not the important but rather that students experience a “full variety of ... situations.” (p. 89)

Active and Static Word ProblemsWord problems can be classified into active and static problems • Active problems contain an action representing the operation, expressed verb:

For example, “There were 4 boys sitting on the mat. Six girls sat down to join them. How many children were there

altogether?”

• Static problems do not contain an action expressed as a verb:

For example, “There were 4 boys and 6 girls sitting on the mat. How many children were there altogether?”Typically students are exposed to active problems more often than static problems. Active problems also tend to be easier for students to solve because they can be acted out at different levels (concretely, representationally or abstractly in your minds’

eye). This makes comprehending the problem and identifying the needed operation easier. For these reasons many students benefit from increased exposure to static problems.

Bond® Blocks


Solving Word ProblemsTEACHER NOTES

Part-Part-Whole and Comparison ProblemsWord problems can also be classified into part-part-whole and comparison problems .

Typically students are exposed to part-part-whole problems more often than comparison problems, and find comparison problems more difficult than part-part-whole problems. For these reasons many students benefit from increased exposure to comparison problems.

Part-part-whole and comparison problems can be active or static.

Part-part-whole problems contain two parts that join together to make a whole.

Wholes can be partitioned into more than two parts.

For example, Josh and his sister Grace were counting their pocket money. They wanted to combine it to buy lunch together. Josh had $6. After Grace added her pocket money they had $10 altogether. How much pocket money did Grace have?

6 4

10

Known WholeWhole

$10

Known Part Josh’s Part $6Unknown Part Grace’s Part

$4

Comparison problems compare two sets and use the difference between the two sets.

For example, Josh had $6. This was $2 more than his sister. How much money did his sister have?

Comparison problems use comparative language such as “fewer/more”, “less than/greater than”.

Larger Set Josh’s $6

Smaller Set Sister’s $4

Difference $2



TEACHER NOTES

Missing Number PositionsThe missing number, or ‘unknown’, can be placed in any position within the equation related to the word problem. This one example, “A fish tank contained 10 fish. Six were gold, four were black,” can be used to write word problems where the missing number is placed in a variety of positions.

Typically students are exposed to word problems where the missing number or ‘unknown’ is isolated on one side of the equation. For example, addition as 6 + 4 = ? and subtraction as 10 - 6 = ? These types of questions are shaded darker in the six examples below.

Finding the unknown in a position other than the answer position is often more difficult for students. These questions are shaded lighter in the six examples below. Students benefit from increased exposure to word problems where the missing number is in different positions.

10

6 4

Unknown

6 4

10

Unknown 4

10

6 Unknown

Static Examples

There were 6 gold fish and 4 black fish in a tank. The tank had no other fish.

How many fish were in the tank?

There were some gold fish in a tank and 4 black fish. Altogether there were 10 fish.

How many were gold?

There were 6 gold fish in a tank and some black fish. Altogether there were 10 fish.

How many were black?

6 + 4 = ? 10+ 4 =? 10+6 =?

Active Examples

There were some fish in a tank. Six gold ones were moved to a pond. This left 4 black fish in the tank.

How many fish were in the tank to start with?

There were 10 fish in a tank. Some were gold. The gold fish were moved to a pond. This left 4 black fish in the tank.

How many fish were gold?

There were 10 fish in a tank. Six gold ones were moved to a pond. The fish left in the tank were all black.

How many were black?

10 - 4=?4- 6 =? 10 - 6 = ?



Polya’s Problem Solving ProcessThe Bond Block system uses the problem solving process outlined by Polya (1954) to scaffold problem solving:

i. Understand the problem.

Many problems are given as ‘word problems’. In order to comprehend word problems students need to be able to:• Read (including mathematical vocabulary).

• Interpret graphics (diagrams, graphs, tables).• Interpret symbols.

Problem solving strategies such as draw a diagram (for example, part-part-whole diagram) assist in comprehending.

ii. Devise a plan.

Students represent the relevant known and unknown numbers. Students can use:

• Bond Blocks or a diagram, such as part-part-whole, that is then translated into a number sentence.• A written equation. It is important that students write an equation that represents their understanding of the problem

even if this results in an equation where the unknown is not isolated on one side of the equation.

iii. Carry out their plan.

Students calculate the unknown, recording the calculation used.

• Bond Blocks can be used to support the calculation strategy. For example,

• If students write an equation where the unknown is not isolated on one side of the equation use this as a teaching opportunity. Arrange Bond Blocks in a part-part-whole representation. Then use part-part-whole to teach students to rearrange the equation, isolating the unknown. For example,

iv. Look back to check their answer.

• Bond Blocks are self checking so can be used to check the calculation was correct.

• Reread the word problem to check the answers the question that was asked by the problem and makes sense. Students

write a sentence that answers the question asked in the word problem.

“ I think 8 - 3 is 6. Oops! 6 is one too big.” “ It must be 5. I’ll check. Yes, 8 - 3 is 5.”

“There were 5 birds in a tree. Some more flew into the tree and joined them. Then there were 8. How many birds joined the tree?”

Unknown not isolated.

5 + ? = 8

Student initially solved by adding

3 onto 5 because Part + Part = Whole

Unknown isolated.

8 - 5 = ?

Teacher then models solving by subtracting 5 from 8 because

Whole - Part = Part



TEACHER NOTES

Supporting Students Who Have Difficulty

Identify the stage of difficultyNewman (1983) identified five stages students progress through when solving word problems. Different errors arise at each stage and are identified by interview cues.

Newman Error Analysis Stage Newman’s Interview CueRelated Stage in Polya’s

Problem Solving Process

Reading and Decoding

The student reads the problem and decodes words and symbols.

Please read the question to me. If you don’t know a word, leave it out.

Understand the problem

Comprehending

The student make sense of what they have read.

Tell me what the question is asking you to do.

Transforming

The student ‘mathematises’ the problem; that is, works out what maths is needs to be done.

Tell me how you are going to find the answer.

Devise a plan

Processing

The student does the maths.Show me what to do to get the answer.

‘Talk aloud’ as you do it, so that I can understand how you are thinking.

Carry out the plan

Encoding

The student records their final result appropriately.

Now, write down your answer to the

question.Look back and check



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TEACHER NOTES

Identify the stage of difficultyProblem solving, particularly of less commonly encountered problem types, is a complex process. For this reason it is

advisable for students to solve problems using numbers that are slightly less than they can confidently use when purely calculating. Further reducing the complexity of the numbers used in word problems is one adjustment that can be made to

support students who are experience difficulty.

Types of problemsThis process is especially important for atypical problem types. These question types are more difficult and students typically have less exposure to them. Increased exposure and opportunity to solve these problems is required. Teachers who select limited problem types, that are easier to solve, as an “adjustment” for students are impeding learning.

Devising a planIdeally, students read a problem and represent it abstractly with numbers, in an equation where the unknown is isolated on one side of the equation. For example,

When solving this problem not all students will immediately arrive at 10 – 4 = ? Other students may understand the problem as ? + 4 = 10. The teacher’s role is to support these students to move forward in their understanding. For example,

There were some gold fish in a tank and 4 black fish. Altogether there were 10 fish. How many were gold?

Step 1: Student’s current concrete additive understanding.

Step 2: Moving to a part-part-whole representation.

Student, “There was some gold fish and 4 black fish. This is 10 fish altogether. I had to draw 6 gold fish.”

Teacher, “Use your drawing to tell me about part-part-whole for this number story.”

?



Step 3: Moving to abstract part-part-whole.

Step 4: Moving to isolating the unknown on one side of the equation.

Numberless problemsExposure to “numberless problems” (Bushart, 2014), prior to solving word problems with numbers, can significantly increase understanding.

There are two distinct steps we need to take when solving a word problem. First we must work out the mathematical

relationship between the quantities and, second, we must operate on the quantities. The first step is totally independent of the actual size of the quantities. (Askew, 2019, p. 56)

By engaging with word problems that are numberless students focus on understanding the relationships between the numbers. Numbers are introduced in gradual steps so as the students “talk about the relations between quantities without knowing the actual quantities. As specific numbers are gradually introduced so the reasoning about what to do with those numbers emerges” (Askew, 2019, p 56).

Student, “ 6 + 4 = 10”

Student, “How many were gold?”

Student, [Touches the block that represents the previously unknown part of 6.]

Student, “ 10 - 6 = 4”

Student, “ 10 - 4 = 6”

Teacher, “Correct, part add part equals whole.”

Teacher, “Point to the related Bond Block.”

Teacher, “Correct. Write a number sentence where this is the answer.”

Teacher, “Correct, when you started you didn’t know the part of 6. Whole subtract part equals part.”

Teacher, “Correct, whole subtract part equals part.

Teacher, “Look at your Bond Blocks. Write a related subtraction number sentence?”

Teacher, “Write a number sentence to match your part-part-whole diagram.”

Teacher, “Read aloud the question asked by the word problem.”



TEACHER NOTES

For example, this word problem could be scaffolded, beginning with a numberless word problem, using the following steps:

Steps:

1. There were fish in a tank. Some were gold. The other fish were black.2. There were 10 fish in a tank. Some were gold. The other fish were black.

(This is an ideal place to pause and investigate possibilities.)3. There were 10 fish in a tank. Some were gold. The other 4 were black.

How many fish were gold?

Concrete-Representational-AbstractWhen supporting students to solve word problems first identify which stage of the problem solving process they are finding difficult. Then support with materials and diagrams. The level of scaffolding provided when using materials and diagrams can be adjusted, from most support to least, using the Concrete-Representational-Abstract framework. Answers to all Bond Block word questions have been modelled at each of these levels to assist teachers when providing support.

For example,

? + 4 = 10

There were some gold fish in a tank and 4 black fish. Altogether there were 10 fish.

How many were gold?

Yassmine is at her local library. Each person is allowed to borrow 7 books. She has 5 books on loan at

home. How many can she borrow today?

Concrete

Representational

Abstract

Whole

PartPart

Unknown

7

5 2

5 + 7=2

7 - 5 = 2

1 2 3 4 5 6 7



References

Askew, M. (2019). Help students understand the structure of maths problems before introducing numbers. Teachwire. Advance online publication. https://www.teachwire.net/news/help-students-understand-the-structure-of-maths-

problems-before-introducing-numbers

Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Harvard University Press.

Bushart, B. (n.d.). Numberless Word Problems. Teaching to the beat of a different drummer. https://bstockus.wordpress.com/numberless-word-problems

New Zealand Ministry of Education. (n.d.). Problem Solving Strategies. NZMaths.

https://nzmaths.co.nz/problem-solving-strategies

Polya, G. (1954). How to solve it. Princeton, NJ: Princeton University Press.

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D., & Treacy, K. (2004). First Steps in Mathematics Number: Book 2. WA: Department of Education

Reflection looking backStudents learn problem solving by solving problems. The final stage in Polya’s problem solving process, looking back and reflecting, is often overlooked to the detriment of learning. Students benefit from looking back (i) individually at the end

of each problem to check their own answer and (ii) with other students to reflect on one common problem solved. Whilst

reflecting with others, students explain their strategies used to:

• Solve the problem. For example, using a diagram.

• Calculate. For example, using ? + 4 = 10 or 10 – 4 = ?

Looking back and reflecting at a group level provides students who experienced difficulty the opportunity to evaluate the efficiency of their own strategies and learn new ones.

The New Zealand Ministry of Education (NZ Maths, n.d.) provides a helpful list of common strategies synthesized with examples for primary school aged students:

• Guess (guess and check, guess and improve)

• Act It Out (includes use equipment)• Draw (includes pictures and diagrams)

• Make a List (includes making a table)

• Think (includes using known skills, looking for patterns, working backwards)Students can create their own strategy guides, recording examples of problems under the headings of specific strategies.

The Language of Word Problems

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