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Thinking Mathematically

2Thinking Mathematically

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Thinking Mathematically

This section of the curriculum unpacks the notion of Thinking Mathematically; why it should form an

integral part of the Mathematics Curriculum and some of ways to make sure that it is a key focus for

learning mathematics.

What is Thinking Mathematically and why is it important?

The notion of Thinking Mathematically refers to a broad approach to mathematics learning; it should

form an integral part of learning the mathematics content across the three strands of the curriculum

and be a key focus of all mathematical activity.

Thinking Mathematically is not a separate curriculum strand which exists in isolation; it is a set of

skills and processes to be developed within the context of each curriculum strand. It describes the

actions or processes in which students can engage when learning, using and applying mathematical

content. It can be thought of as the ‘thinking and doing’ of mathematics. For example, making and

justifying decisions about which equipment or unit of measurement to use; describing and reasoning

about properties of shapes; explaining methods of calculation; refining methods of recording

calculations and checking results. These activities all show the integration of Thinking

Mathematically with mathematical content.

Mathematics needs to be thought of as a combination of content and process. How mathematics is

taught and learned is just as important, if not more important, as what mathematics is taught and

learned. The processes encompassed within Thinking Mathematically are needed for students to

develop their mathematical skills, knowledge and understanding of concepts.

In order for a student to develop as a competent reader, they must learn to comprehend the text

alongside developing fluency with reading. When learning mathematics students need to think and

reason and make sense of mathematics in the same way they must learn to comprehend text when

reading. To become competent at mathematics students also need to become efficient with skills

and procedures in the same way that they need to gain fluency with reading.

When students are Thinking Mathematically they are developing understanding and fluency in

mathematics through investigating and connecting mathematical concepts, choosing and applying

problem solving skills and mathematical techniques, communicating and reasoning about

mathematics.

There are three components of Thinking Mathematically:

1. Communicating

Communicating involves sharing and explaining mathematical thinking both orally and in writing,

using the correct vocabulary, notation and pictures or diagrams. It involves explaining methods

and solutions, decisions, reasoning, and developing and refining recording. Communicating with

others about mathematical thinking plays a key role in helping to clarify ideas and deepen

understanding.

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2. Problem solving

Problem solving is a way of teaching and learning. It involves both communicating and

reasoning. Problem solving involves developing skills and strategies to interpret, make decisions,

model, investigate and formulate problem situations. When students solve and formulate

problems they use mathematics to represent situations, they apply strategies to reach a solution

and verify that their solutions are reasonable.

3. Reasoning

Reasoning in mathematics involves thinking logically about mathematics and analysing,

explaining, generalising, evaluating, justifying and proving. It essentially involves answering the

question: Why do you think so? When students investigate statements, they make observations

and give reasons for their observations. They may develop and evaluate mathematical

arguments to prove or disprove a statement. Students are reasoning mathematically when they

justify strategies used and conclusions reached, when they prove that something is true or false.

Whilst three separate components of Thinking Mathematically have been identified, when students

solve problems or investigate a mathematical idea they are required to communicate solutions orally

or in a written form using words or numbers or symbols or diagrams, and are involved in reasoning.

The three components of Thinking Mathematically are linked and tend to overlap but it is intended

they will provide a tool to outline and track progression in mathematical skills and processes.

Thinking Mathematically has a set of level outcomes; this is to ensure that there is progression in the

processes and skills as students learn the content of the curriculum (refer to Appendix A). While not

all aspects of Thinking Mathematically apply to every level outcome for the three content strands,

they indicate the breadth of mathematical actions that teachers should emphasise.

Mathematics is a vehicle for higher order learning and thinking, it encompasses logical reasoning,

problem solving and thinking in abstract ways. It is important to integrate aspects of Thinking

Mathematically to ensure opportunities for students to develop the higher order thinking skills they

need to become lifelong learners and problem solvers.

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The Components of Thinking Mathematically

This section looks closely at the three components of Thinking Mathematically; it explains why they

are important aspects of learning mathematics that should be integrated into the daily mathematics

lesson.

How does communicating about mathematical ideas improve learning?

Communicating thinking about maths through talk is key to helping develop understanding, and in

exposing misconceptions or misunderstandings.

…planned and purposeful talk between teachers and children in its rightful place

at the heart of the drive to improve the quality of learning.

Robin Alexander (2004)

It is the nature of mathematics that makes communication a vital part of learning mathematics.

Derek Haylock and Anne Cockburn (1989) suggest that effective learning takes place when the

learner makes cognitive connections. They devised a model of learning based on connection, which

has been represented diagrammatically.

Haylock and Cockburn suggest that it is when students make connections between these

experiences that learning is more deeply embedded. In order to link the four different elements

dialogue needs to be a central part of the maths lesson. The example below illustrates how the four

aspects can be linked in a classroom. It describes the scene from a Grade 2 classroom:

Students had a box of toy cars, paper and pencil, a collection of cards with various words written on

them – shared, between, is each, sets, of, makes, altogether, three, six, nine, twelve - and a

calculator.

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Their first task was to share six cars between three of them. They discussed the result. Then they

selected various cards to make up sentences to describe what they had discovered. The children drew

pictures of their sharing and wrote their two sentences underneath. One of the students picked up

the calculator and interpreted the first sentence by pressing the keys 6 ÷ 3 =

Haylock and Cockburn, 1989

In the example given, students are involved in manipulating concrete materials (sharing cars), using

mathematical language to discuss and make sentences, using pictures to represent mathematics and

communicating using mathematical symbols with a calculator.

The connective model highlights why students experience difficulties learning mathematics if they

are introduced to abstract ideas too quickly. For example, if formal methods of written calculation

are introduced too early three of the crucial parts of the model would be missing from their learning

experiences. It is important to continue to connect symbols and language with concrete materials

(progressing to real contexts) and images as students progress through their primary education. The

connective model should be incorporated into the planning and teaching of the daily maths lesson.

The example below shows the model being used in a Prep classroom as a ‘think board’. The students

were learning to represent numbers up to 20.

Planning for talking and thinking is crucial to improve the quality of learning. To enable students to

enquire, reflect, reason and evaluate i.e. engage in higher level thinking, regular opportunities for

discussion need to be planned. The internal dialogue that goes on inside students’ heads involves

thinking and reasoning; this is developed and honed by discussion.

Discussion is also a valuable assessment tool for teachers to find out where students are in their

learning, identify any misconceptions they may have and plan the next steps for learning. Paired,

group or whole class discussion should form part of every lesson. It is vital that teachers plan ways to

integrate a variety of appropriate co-operative learning strategies into lessons, such as talk partners

or learning partners, think-pair-share etc. to allow opportunities for high quality talk. Further

information about the importance of high quality talk in enhancing learning in mathematics is

provided in Teaching Mental Calculation Support Document, Part A.

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The plenary is an ideal part of the lesson in which to plan for talk. Students should always be

informed in advance what they need to think about whilst they are involved in a learning activity, if it

is expected that they will participate in discussion. If the discussion planned is for the whole class,

students need to be given time before the plenary to reflect and discuss with a partner prior to

participating in a whole class discussion.

Why should problem solving be at the centre of mathematics learning?

The primary goal is that students make sense of the mathematics they are learning. They need to

make sense of mathematical concepts and procedures so they can use them flexibly and

appropriately in unfamiliar situations. When students are told things rather than figuring them out

they are less likely to remember them or be able to apply the ideas in the future. A teaching

approach, in which students are confronted with a problem, supported to solve the problem and

helped to discuss and consolidate learning helps students to make sense of mathematics. Allowing

mathematics to be problematic does not mean that the teacher needs to search for lots of extra

problems, instead it means allowing students to grapple with everyday challenges rather than

stepping in and telling students exactly what to do before they have been given a chance to think.

Children start school being able to solve many problems. Teaching problem solving should always

build on what students already know. Students can begin solving problems from level 1, they do not

need to be skilful at computation before engaging in problem solving.

What is problem solving?

A problem is something a person needs to figure out, something where the solution is not

immediately obvious. Problem solving is directed towards a goal; the statement of the problem

indicates what type of goal is being aimed for e.g. if the problem is to design a timetable for the

classroom, the goal is to complete a useable timetable. Although there is a goal in problem solving it

is possible to have more than one answer, for example there could be more than one possible class

timetable. Problem solving can therefore involve problems where there is a single solution, or

problems which are open ended and there is more than one possible solution.

Solving problems involves effort and higher level thinking. If a student immediately sees how to get

the answer to the problem, then it is not really a problem for that student. Teachers need to think

carefully about whether the task being set is really a problem or just an exercise. For example, a

page in a traditional textbook often begins with an exercise for students to practise followed by

some story problems. Often the story problem is not a true problem, it is an exercise with words

around it. Such an approach is ineffective in developing problem solving skills in students, as little

decision making is required; this is evident when students perform poorly in assessment tasks that

involve problem solving or in the real world where problems are not conveniently grouped by

operation.

There have been many attempts to classify problems, the importance is not the classification but the

fact that there is a range of different types of problems and all need to be experienced by students

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because they involve using and developing different skills and strategies. Some of the key categories

for problems include:

Logic problems

Finding rules and describing patterns

Finding all possibilities

Diagram problems and visual puzzles

Word problems

There are a number of structured approaches to teaching problem solving that have been devised to

help students identify steps they might work through, many of these focus on solving word

problems. Students who are effective problem solvers identify what the problem is asking, ask

themselves if what they are doing makes sense, adjust their problem solving strategies when

necessary and look back afterwards to reflect on the reasonableness of their solution and their

approach. Examples of general problem solving processes to use with students at different levels can

be found in Appendix B.

Students can also engage in investigational work, commonly referred to as investigations.

Mathematical investigations are inquiry based and a part of problem solving. Investigations are

about processes and the emphasis should be on the process rather than finding the correct answer.

Students are presented with a mathematical situation which they are told to investigate. It is

expected that first students generate results using a systematic approach, look for patterns in their

results, make generalisations and justify their statements. An example of a piece of student

investigational work annotated with the processes listed above is provided in Appendix C.

Suggestions for investigations for students to conduct in the context of the three curriculum strands

can be found in the outcome elaborations.

Generic Problem Solving Strategies There are generic problem solving strategies that should be explicitly taught. Below is a list of seven

strategies that can help students solve a range of problems:

Act it out (use concrete materials to model the problem)

Draw a diagram or make a model

Look for a pattern

Guess and check

Work backwards

Solve a similar or simpler problem

Construct a table

Explanations and examples of each strategy can be found in Appendix D.

Students need to be taught strategies and be shown how they can apply these systematically to problem solving (being systematic is about first deciding what is being asked, what information can be used and what data can be generated). Through exposure and discussion students should begin to recognise that some strategies are more appropriate than others for certain problems. They

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should also be encouraged to adapt strategies and to make them appropriate for a problem being solved. Problem solving needs to be integrated into mathematics lessons on a regular basis if students are to

become confident and competent problem solvers. Allocating specific lessons to problem solving will

help but embedding problem solving into daily lessons will provide the regular practice and

consolidation that students need.

The planning overviews for each term include the integration of problem solving strategies; this can

be found in the planning section. It is recommended that no more than two generic strategies are

introduced in each term, with students being given opportunities to select and apply strategies

learned to a range of problem solving situations on a regular basis.

What is reasoning and why is it important?

Being able to reason mathematically is essential to making sense of mathematics. Reasoning

involves observing patterns, making generalisations, thinking about them and justifying why they

should be true.

An example of reasoning at level 2b:

The student used concrete materials to represent and

sort various numbers and explained their reasoning as

follows:

All even numbers are just rectangles because you can

make them to two by two. But all the odd numbers

can’t. Odd numbers make rectangles with chimneys. If

you put two rectangles with chimneys together, the

new one doesn’t have a chimney. So two odds makes

an even.

Making generalisations involves applying reasoning skills. To make a generalisation is to make an

observation about something that is always true, or always the case for all the members of a set of

numbers or shapes. For example, to make a generalisation that when you count every other number

is even is to make a true generalisation about the set of counting numbers. It is possible to make

generalisations that may not be true.

When solving a problem or investigating, a student can use their observations to make a conjecture.

A conjecture is a guess, it is a statement which may be true based on the observations and results

generated at that point in the investigation or problem solving process, but has not yet been proven

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to be true. The process of conjecturing and checking is fundamental to reasoning in mathematics.

Investigating odd and even numbers is an example of a good context for students to learn to

formulate conjectures and justify their reasoning. Suggestions for ways to investigate odd and even

numbers can be found in the outcome elaborations for number.

In level 5 students develop their reasoning skills to formulate hypotheses. A hypothesis is a

statement which gives an explanation to a series of observations and is therefore believed to be

true. A hypothesis can be tested and rejected; it is not necessarily true until proven. A hypothesis is a

statement that has an ‘if’ and ‘then’ part e.g. If all four sides of a quadrilateral measure the same

then the quadrilateral is a square.

If a convincing argument and evidence can be given to prove that the hypothesis is true in every case

this is said to be a proof. A proof involves deductive reasoning. A proof is written using the ‘if…then’

language leading to a series of logical deductions. Students up to level 4 are not expected to produce

proofs; they should however be encouraged to formulate explanations as to why a generalisation

must be valid.

Keeping track of thinking or the steps involved in solving problems or puzzles or investigating a

statement, requires some systematic recording. This is often helped by organising notes in a table,

diagram or structured list. Students will also use pictures, diagrams, statements, mathematical

notation to help them record their explanations and reasons.

Making Thinking Mathematically Part of Every Lesson

This section looks at ways to create a learning environment that allows various aspects of Thinking

Mathematically to be integrated into every lesson.

How can a teacher integrate Thinking Mathematically into every lesson?

In order to make Thinking Mathematically an integral part of every lesson, students need a learning

environment that encourages them to think and question mathematics. This involves thinking about

the following:

Learning culture of the classroom

Role of the teacher

Nature of the learning tasks

The factors listed above are all interrelated. It is not just the source of the problem, or the type of

task that is important in encouraging students to develop Thinking Mathematically processes and

skills, but the way students are expected and allowed to treat the tasks.

The learning culture of the classroom needs to be established with students to allow students to

learn from each other and engage in meaningful discussion to enhance learning. The classroom

needs to be an environment where ideas and methods are valued and reflected upon thoughtfully.

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Students should be given opportunity to choose and explain strategies and methods. Mistakes

should always be treated as valuable learning opportunities and used constructively to move

forward students with their learning. Fundamental to the learning culture is that both the teacher

and students embrace challenge and struggle as a positive part of learning; and where learning

about learning is part of the curriculum.

The teacher plays a significant role in engineering a learning environment that allows opportunities

for students to engage in key processes and skills to enrich learning. Students need to be given

opportunities to construct their understanding, rather than relying on the teacher as the main

source of information. The role of the teacher needs to change from one where they confirm if the

answer to a solution is correct or incorrect. Students need to be questioned to explain their solution

regardless of whether it is seen to be correct or incorrect. The teacher needs to actively engage in

dialogue with students about solving problems, asking questions to initiate further thinking, rather

than simply confirming if solutions are correct. Challenging students to explain their mathematical

thinking develops language and clarifies ideas. A culture needs to be created where students are

asked to follow up and justify solutions; analyse the appropriateness of methods used and search for

better ones; and be prepared to listen to and learn from their class members.

Teaching mathematics in a way that allows it to be problematic encourages students to be reflective

learners and helps to develop higher order thinking skills such as evaluation and reasoning. One

approach to help promote the integration of problem solving, a key component of Thinking

Mathematically, into lessons is to use a question or statement to start a lesson.

Using Statements and Questions to Start a Lesson

Using questions or statements linked to level outcomes to start a lesson is one way to integrate

Thinking Mathematically into lessons. The use of a question or statement to investigate puts the

onus on the students to find solutions or prove whether the statement is true or false, essentially it

problematizes the learning intention.

For example, in Level 3a students are expected to learn to:

Draw and complete shapes with reflective symmetry and draw the reflection of a shape in a mirror

line along one side.

To achieve the outcome students could experience a lesson that involves identifying and drawing

lines of symmetry on given shapes, which should include quadrilaterals. As an alternative to simply

sharing this learning focus with students at the beginning of a lesson this could be presented as a

statement for the students to investigate:

All four-sided shapes have two lines of symmetry. Is this always, sometimes or never true?

Investigating whether the statement is true will lead students to learning much more about

symmetry and shape than if they had experienced labelling lines of symmetry on given shapes.

Research has shown that students become much more involved in the lesson and stay focused on

the purpose of the activity; they know that by the end of the lesson they need to be able to say

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whether the statement is true or false and explain how they know (justify their answer), or at least

explain how far they have got in their investigation to finding out. The statement or question

effectively becomes the anchor for the lesson.

Statements can be always true, sometimes true or never true. The statements which tend to enrich

learning most are those that are sometimes true; the focus can be on establishing when and why the

statement is true or false.

Investigating statements lends itself well to extending higher achieving students. Investigations can

be extended by asking further questions linked to the ideas in the initial investigation. Questions

that begin with the words, ‘What if….’ are a useful way to do this. For example, to extend the

symmetry investigation described previously students could be asked: What if we cut part of the

shape off – could we make a cut and still retain the lines of symmetry? Once students have been

exposed to these sorts of questions, they will be able to generate their own questions to extend

their thinking.

Examples of statements and questions are integrated in the outcome elaborations for each strand of

the curriculum. Appendix E provides an example of how to adapt the structure of a mathematics

lesson to use statements or questions to begin the lesson.

Teachers need to make sure that they create a range of problem solving situations for the four

operations that challenge students to think. Appendix F provides examples of three different

structures on which to base problem solving situations for the four operations.

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Appendices

Mathematics Scope & Sequence

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Appendix A

Strand: Thinking Mathematically

Level 1 Level 2 Level 3 Level 4 Level 5

By the end of level 1, students should be able to:





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1TM 01 Describe mathematical situations and solutions to practical problems using everyday language, actions, concrete materials and informal recordings

2TM 01 Describe mathematical situations and methods using everyday language and some mathematical language, actions, concrete materials, diagrams and symbols; present solutions in an organised way

3TM 01 Describe and explain mathematical thinking and solutions orally and in writing using mathematical vocabulary, symbols, diagrams appropriately for an audience

4TM 01 Describe and explain mathematical thinking and solutions using mathematical vocabulary, diagrams and symbols; refine ways of recording using images and symbols

5TM 01 Communicate mathematical ideas and solutions in a variety of ways using appropriate mathematical vocabulary, diagrams and symbols

Pro

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ing

1TM 02 Use concrete materials, actions, technology to model and explore authentic problems using trial and error

2TM 02 Use concrete materials, actions, diagrams and technology to model and explore mathematical problems involving one and two steps; begin to check the reasonableness of solutions

3TM 02 Select and apply appropriate calculation and problem solving strategies to solve problems involving one and two steps, using technology as appropriate; estimate before calculating and check the reasonableness of solutions; begin to formulate problems and suggest extensions to investigations

4TM 02 Select and apply appropriate calculation and problem solving strategies to solve multi-step problems, using technology as appropriate; estimate, check and evaluate methods; formulate problems and extend mathematical investigations

5TM 02 Select, apply and evaluate strategies to solve a range of multi-step problems, using technology as appropriate; formulate problems and suggest and test hypotheses for mathematical investigations

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1TM 03 Describe simple patterns and classifications orally; use concrete materials and/or pictorial representations to demonstrate how a solution was reached

2TM 03 Recognise and describe patterns and relationships involving numbers or shapes; make and test predictions (conjectures)

3TM 03 Investigate and check the accuracy of a statement; explain the reasoning used and give an example to justify the conclusion; make and check conjectures and begin to form generalisations based on results of investigations

4TM 03 Investigate and check the accuracy of a statement or solution to a problem; give a valid reason and/or examples to justify supporting one solution over another; make and check conjectures, and form valid generalisations based on mathematical investigations

5TM 03 Justify conclusions or solutions by explaining mathematical relationships; use deductive reasoning to prove a hypothesis

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Appendix B

Processes for Solving Problems in Mathematics

These processes can be used to support students in solving word problems. Example 1 is recommended for use

with students working at level 3 and above. Example 2 is a simple process to use with students progressing

through levels 1 and 2 of the curriculum. If displayed in the classroom, visual clues and examples would need

to be displayed with the processes. Example 3 is suitable for levels 2 – 3.

Example 1: Example 2:

Read the problem, underline the key

words

Decide which operation/strategy

to use

Approximate your answer

Calculate

Check 1 – is your calculation

correct?

Check 2 – does it answer the problem?

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Example 3:

STOP!!!

Read the task

What do you have to do, find out or solve?

Underline important information

READY TO GO!!!

Think

Which operation?

Which strategy?

GO!!!

Solve the problem

Work hard to find answer

Show your working

Check your answer

Don’t give up

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Appendix C

Mathematical Investigations

This work sample shows the processes for students to follow when investigating a mathematical idea:

generate results using a systematic approach, look for patterns in their results, make generalisations and

justify their statements.

SOURCE: The Mathematics Teachers’ Development Series, Pack 1: Investigative and Problem Solving

Approaches to Mathematics and their Assessment, Alice Onion, 1990.

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Appendix D

Problem Solving Strategies

Name of Strategy Explanation Sample Activity Levels Most Suitable For

1 - 2 3 4 5

Construct a table

A table helps to organise the information so that it can be easily understood and so that relationships between one set of numbers and another become clear. A table makes it easy to see what information is there, and what information is missing. When a table is drawn up, the information often shows a pattern, or part of a solution, which can then be completed.

Research shows three out of ten people are blond. How many blonds will be found in 1000 people?

Further example:

Your teacher agrees to let you have 1 minute of recess on the first day of school, 2 minutes on the second day, 4 minutes on the third day, and so on. How long will your recess be at the end of two weeks?

Day 1 2 3 4….

Time 1 2 4

X X X X

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1 - 2 3 4 5

Look for a pattern A pattern is a regular, systematic repetition. It can be numerical, visual or behavioural.

Many patterns are similar and so can be predicted.

Talking about patterns will help students identify them and use them when solving problems.

Students may be asked to complete gaps in a simple number sequence, the colour of the next shape or calculate how many items will be in the nth term.

The figure shows a series of rectangles where each rectangle is bounded by 10 dots

a) How many dots are required for 7 rectangles?

b) If the figure has 73 dots, how many rectangles would there be?

Solution:

Rectangles Pattern Total dots

1 10 10

2 10 + 7 17

3 10 + 14 24

4 10 + 21 31

5 10 + 28 38

6 10 + 35 45

7 10 + 42 52

8 10 + 49 59

9 10 + 56 66

10 10 + 63 73

X X X X

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1 - 2 3 4 5

Guess and check

Guess and Check involves taking a guess, checking the answer and then making another, more informed guess.

Guessing often produces the wrong answer. But the strategy called "Guess and Check" often produces the right answer. Students need support to evaluate their original guess and think about where to go to make their next guess more accurate.

Guess, Check and Adjust might be a better name for the strategy.

I have chickens and goats on my farm. The animals have 26 heads and 68 legs. How many of each animals are there?

Guess

Chickens Goats

Number of

heads

Number of legs

1 13 13 26 78

The number of legs is too high, so I need to reduce the number of goats and increase the number of chickens. I already have the correct number of heads.

Guess

Chickens Goats Number of heads

Number of legs

1 13 13 26 78

2 20 6 26 64

3 18 8 26 68

Further example:

A café advertises twin flavor milkshakes – that’s two flavours of milkshake poured into a glass. They claim to offer 28 different twin flavour combinations. How many different single flavours of milkshake must be available?

X X X

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1 - 2 3 4 5

Act it out

(Use manipulatives to model the problem)

Using real materials to model the information may help solve the problem. Blocks, counters, rulers, protractors, compasses, dice, play money and squared materials are some examples of materials that students may use to help understand the information given in a problem.

It can also be very helpful for students to act out the roles of the different participants in the problem.

John is queuing at the canteen. There are 50 people in front of him, but he is very impatient. Each time a student in the front is served, John slips past two students. How many people will be served before John reaches the front of the line? It will help students to visualise the movement in the problem by actually moving the counters or blocks to each new position. Further examples: Six students are standing at the teacher’s desk. Five students join them. How many students are now at the teacher’s desk? There are 24 students in a class. Each student gives a Christmas card to each of the other students in the class. How many Christmas cards are exchanged? A man buys a horse for $60, sells it for $70, buys it back for $80 and sells it for $90. How much money does the man make or lose?

X X X X

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1 - 2 3 4 5

Draw diagram or make a model

Drawing a diagram is a common problem solving strategy. Very often, a problem solver needs to draw a diagram just to understand the meaning of the problem. The diagram represents the problem in a way they can “see” it, understand it, and think about it while they look for the next step.

First they need some practice to learn how to interpret a problem and draw a useful diagram. Give them an easy problem to solve, i.e. a problem for which it is easy to draw a diagram, and then, once the diagram is drawn, to see the next step.

A piece of wood has to be cut into eight pieces. Each cut takes you 30 seconds. How long will it take to cut the wood into pieces? In order to make the eight pieces cut the dowel in seven places.

Seven cuts multiplied by 30 seconds per cut equals, 7 x 30 = 210 seconds. It will take 210 seconds or 3 minutes and 30 seconds to make the cuts. Further example:

A snail is at the bottom of a jar that is 15 cm high. Each day the snail crawls up 5 cm, but each night it slides back down 3 cm. How many days will it take the snail to reach the top of the jar? Note: The answer is not 7 ½ days – draw a diagram to see why not!

X X X X

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1 - 2 3 4 5

Draw a diagram

Continued.

Refer to previous cell. There are eight netball teams representing schools in the local area. The teams are playing in a netball competition to decide on the champion. It is a knock-out competition, which means that once a team is beaten they leave the competition. How many games will be played during the entire competition?

Seven games of netball will be played during the competition.

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1 - 2 3 4 5

Work backwards

It is sometimes necessary for to start with the information given at the end of a problem and compute your data working toward the information presented at the beginning of a problem.

Jack is 35 years younger than Karen.

Frank is half of Jack’s age.

Jennifer is 17 years older than Frank.

If Jennifer is 35 years old, how old is Karen? Jennifer is 35. She is 17 years older than Frank. So using the opposite operation plus becomes minus. So Frank is 35 – 17 = 18 Frank is half Jack’s age so the opposite operation is 18 x 2 = 36 Jack is 35 years younger than Karen so 36 + 35 = 71 Therefore Karen is 71 years old.

Further examples:

Sue baked some cookies. She put half of them away for the next day. Then she divided the remaining cookies fairly among her two sisters and herself, so each got 4 cookies. How many cookies did she bake? Complete the addition table:

X X

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1 - 2 3 4 5

Solve a simpler problem

Using simple numbers to make an easier problem can sometimes help solve problems. Taking simple numbers and using them for the given numbers can give a clearer picture. The same idea can then be used for the larger numbers.

How many times will the digit seven appear as part of a number, when counting all the numbers from 1 to 374?

Work out how many times seven appears from 0 to 10, and then how many times from 10 to 20. Once the pattern for the lower numbers has been worked out, it is easier to calculate how many sevens appear in the higher numbers and so reach a solution. Further example:

64 students play in pairs in a tennis competition. Losers are out of the competition. Winners play until only one winner is left. How many games must be played before there is one winner left? First try and solve how many games would be played if there were only 2 players… 3 players…4 players

X X

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Appendix E

Thinking Mathematically: Open Ended Investigations

Scaffolding a Lesson: Using Statements or Questions to Start a Lesson

It is a recommendation that every mathematics lesson should begin with a mental starter linked to

the content of the main part of the lesson.

When planning and teaching a lesson which begins with a statement or question to investigate, it is

useful to structure the main part of the lesson on four stages:

1. Recruitment stage

2. Getting started

3. Maintaining participation

4. Plenary

Stage 1: Recruitment

How do we get students to want to investigate the question or statement?

Let students know that they will be able to investigate the question or statement; although the

question or statement might initially seem a bit tricky it is important to let the students know

they have, or will be given the tools to investigate it.

Make it clear that the process of investigating – the grappling with the problem – is as important

as finding the solution.

Allow student generation of the question or statement, as appropriate. This might come about:

as the result of a previous investigation – answering one question may have generated other

questions for students to investigate

by giving students a statement and getting them to generate related ones

Unpack the question or statement so students are clear about the investigation. Focus on

explaining key vocabulary, explaining key terms and going through examples.

Stage 2: Getting Started

Often, just faced with a statement or question, students do not know what to do or how to get

started. Students should not just be sent off to investigate a statement or question unless they have

previously investigated a very similar statement.

Depending on the question or statement there are a variety of ways of helping students get started:

Ask students to discuss how they might get started, with a partner or a group. Students need to

be open to completely rethinking their strategy at this stage.

Start the investigation off together, then once you have given them a way of working encourage

them to continue in pairs (with differentiated levels of support).

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Investigate a statement together, then give them a similar statement to investigate in the same

way with a partner.

Brainstorm variables that may affect whether the statement is true or not before starting e.g.

suggest they try decimal numbers, negative numbers, fractions, big numbers, small numbers,

multiples of 3, prime numbers, square numbers, consecutive numbers etc.

Give different groups different variables to investigate.

Narrow the field of investigation by giving students a starting place e.g. just look at the numbers

to 20 initially

Refer them to other investigations they have done to draw out generic problem solving

strategies they are familiar with e.g. being systematic

Simplify the problem – start smaller.

Stage 3: Maintaining the Participation

Often students start off enthusiastically but can soon lose interest and/or focus. One way to prevent

this happening is to stop the class or groups and have mini-plenaries throughout the lesson.

During the mini-plenaries you might:

Take feedback. The aim is to highlight ways of working that other students may benefit from.

Refocus. Remind the students of what they are doing and why. Group feedback can sometimes

serve this purpose.

Extend the investigation. One way of doing this is to ask ‘What if….?’ questions

Stage 4: Plenary

In the plenary it is important to bring students back to the original question or statement and for

students to articulate their findings related to this. The statement or question should be related to a

content description so achievement against this will need to be assessed. One way is to ask: Is the

statement true or false? How do you know?

Once establishing if the statement is true or false, discuss if there is any way they could add or take

away a word to make the statement always true or always false e.g. ‘You can make all the numbers

from 1 to 30 by adding together consecutive numbers’ becomes true if you add the word ‘odd’

before ‘numbers’ i.e. ‘You can make all the odd numbers..’.

Returning to the original statement is a crucial part of the plenary but there are other additional

things you could do in the plenary:

How might you use this? We want the students to identify how they might use what they have

learned. We want them to see connections with the bigger picture of mathematics.

What could we investigate next? Ask students to generate the ‘what ifs’ even if they do not go

on to investigate these.

Consider the application of generic problem solving strategies. Identify and evaluate strategies

used in the investigation.

Acknowledgement

Adapted from Using and Applying In Every Maths Lesson: Ideas for the Primary Classroom, Devon County

Council, UK.

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Appendix F

Problem Solving: Addition & Subtraction Situations

Result Unknown Change Unknown Start Unknown

Add to Two rabbits sat on the grass. Three more rabbits hopped there.

How many rabbits are on the grass now?

Two rabbits were sitting on the grass. Some more rabbits hopped there. Then there were five rabbits.

How many rabbits hopped over to the first two?

Some rabbits were sitting on the grass. Three more rabbits hopped there. Then there were five rabbits.

How many rabbits were on the grass before?

2 + 3 = □ 2 + □ = 5 □ + 3 = 5

Take from Five bananas were on the table. I ate two bananas. How many bananas are on the table now?

Five bananas were on the table. I ate some bananas. Then there were three bananas. How many bananas did I eat?

Some bananas were on the table. I ate two bananas. Then there were three bananas. How many bananas were on the table before?

5 – 2 = □ 5 - □ = 3 □ – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown1

Combine (join)

Partition (take apart)

Three red apples and two green apples are on the table. How many apples are on the table?

Five apples are on the table. Three are red and the rest are green. How many apples are green?

Serah has five flowers. How many can she put in her red vase and how many in her blue vase?

3 + 2 = □ 3 + □ = 5 5 – 3 = □

5 = 0 + 5 5 = 1 + 4 5 = 2 + 3

5 = 5 + 0 5 = 4 + 1 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

2 + □ = 5 2 + 3 = □ 5 – 3 = □

Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

5 – 2 = □ 2 + 3 = □ □ + 3 = 5

Addend: A number to be added. In 7 + 4 = 11, 7 and 4 are addends and 11 is the sum.

1 These partitioning or taking apart situations can be used to show all the ways a given number can be partitioned or split.

The number sentences help students understand that the = sign does not always mean makes or results but can sometimes mean is the same number as.

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Problem Solving: Multiplication & Division Situations

ACKNOWLEDGEMENT: Adapted from Common Core State Standards for Mathematics, 2010 p88-89

Unknown Product Group Size Unknown

How many in each group? (Division)

Number of Groups Unknown

How many groups? (Division)

6 x 3 = □ □ x 3 = 18

18 ÷ 3 = □

6 x □ = 18

18 ÷ 6 = □

Equal Groups

There are 3 bags with 6 mangoes in each bag. How many mangoes are there altogether? Measurement example: You need 3 lengths of string, each 6 centimetres long. How much string will you need altogether?

If 18 mangoes are shared equally into 3 bags, how many mangoes will be in each bag? Measurement example: You have 18 centimetres of string, which you will cut into three equal pieces. How long will each piece of string be?

If 18 mangoes are to be packed 6 to a bag, then how many bags are needed? Measurement example: You have 18 centimetres of string which you will cut into pieces that are 6 centimetres long. How many pieces of string will you have?

Arrays Area

There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example: What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example: A rectangle has an area of 18 square centimetres. If one side is 3 cm long how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples how many rows will there be? Area example: A rectangle has an area of 18 square centimetres. If one side is 6 cm long, how long is a side next to it?

Compare A blue hat costs K6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example: A rubber band is 6 cm long. How much will the rubber band be when it is stretched to be 3 times as long?

A red hat costs K18, it costs 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example: A rubber band is stretched to be 18 cm long, which is 3 times its original length. How long was the rubber band before it was stretched?

A red hat costs K18 and a blue hat costs K6. How many times as much does the red hat cost as the blue hat? Measurement example: A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

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References

The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6.

UK Department for Education & Employment, 1999.

Numeracy Development Project Books 1 – 8, Numeracy Professional Development Projects, Ministry

of Education, New Zealand, 2008.

The Australian Curriculum: Mathematics, Australian Curriculum Assessment & Reporting Authority,

2013

NSW Syllabus for the Australian Curriculum: Mathematics K-10, NSW Board of Studies, 2012.

Reys, Lindquist, Lambdin, Smith (2012) Helping Children Learn Mathematics 10th Edition, Wiley, USA.

Assessment Through Guided Maths, Devon Learning and Development Partnership, Devon County

Council, UK, 2009.

Using and Applying In Every Maths Lesson, Devon Learning and Development Partnership, Devon

County Council, UK, 2005.

The National Primary Strategies: Teaching Children to Calculate Mentally, UK, 2010.

The Mathematics Teachers’ Development Series, Pack 1: Investigative and Problem Solving

Approaches to Mathematics and their Assessment, Alice Onion, 1990.

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