# Mathematics

## Mathematics Department

jau_hsiao_chien@moe.edu.sg

tan_kian_tiong@moe.edu.sg

### Vision

*Innovative Problem-solvers, Creative Thinkers*

MISSION**Developing lifelong learning with a passion for Mathematics.**

#### Desired Outcomes

Developing lifelong learners with a passion for Mathematics abf

#### Signature Programmes

STAR approach to problem-solving (P1 – P6)

S – Study the Problem

T – Think of a plan

A – Act on the plan

R – Reflect on your own answer

Rosyth Thinking Programme

In this Rosyth Thinking Programme (RTP), P3 students are challenged with Thinking Tasks that are set in interesting and authentic contexts, and constructed with an open question or a twist. These tasks are content-based, and are usually open and multi-dimensional. Thus, they arouse students’ curiosity and naturally lead them to search for solutions. As they search for out-of-the-box answers, they will need to think critically and creatively about the problems and their possible solutions. They use skills such as making conjectures, analysing, synthesising, and evaluating information gathered from observations, experiences, reasoning and communication.

The school integrates a differentiated **‘Challenge-Enrich-Support’** approach into its tasks and activities. Higher-ability students are challenged their thinking, while support structures are in place to encourage the less able.

#### Problem Solving

Mathematical problem solving is central to mathematics learning. It involves the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended and real-world problems.

One of the aims of mathematics education is to develop the mathematical thinking and problem solving skills and apply these skills to formulate and solve problems.

Problem solving can be made easier when children are mentally equipped with a ready set of thinking skills and heuristics.

The essential thinking skills are:

- Classifying – arranging pieces of information into meaningful groups
- Comparing – making comparisons among groups or pieces of information
- Sequencing – arranging information into a meaningful/logical order
- Analysing parts and wholes – comparing, visualizing and synthesizing various bits of information and making sense of them as a whole
- Identifying patterns and relationships
- Induction – making generalizations using specific examples
- Deduction – infer various specific examples from given generalizations
- Spatial visualization – mentally manipulate (“logical imagination”) an object/a problem without concrete materials
- The heuristics are the tools which children use based on the plan that they have created from their thinking skills.
- Listed below are the heuristics, categorized into 4 groups, that children can employ in helping them solve problems.

Source: Primary Mathematics Syllabus 2007 ( click )

- To give a representation – e.g. draw a diagram, make a list, use equations
- To make a calculated guess – e.g. guess and check, look for patterns, make suppositions
- To go through the process – e.g. act it out, work backwards, before & after concept
- To change the problem – e.g. restate the problem, simplify the problem, solve part of the problem

**Steps for problem-solving**

Knowing how to solve problems is an important skill and an essential part of our lives. George Polya, a mathematician, spent considerable effort on trying to characterize the methods that people use to solve problems, and to describe how problem-solving should be taught and learned.

Polya devised a general approach that one can take to solve a problem.

**4 Steps for Problem-Solving**

__Step 1: UNDERSTAND the problem__

- Read the problem carefully to understand what is required in the problem.
- Break up the problem into smaller sections and understand each section thoroughly before moving on to understand the next section.
- Draw or write down the information given in the problem in a simpler form to help you understand better.

__Step 2: PLAN what to do/Devise a plan__

- Choose a heuristic to use to solve the problem

__Step 3: DO it/Carry out the plan__

- Use computational skills, geometrical skills and logical reasoning to carry out your plan to solve the problem.

__Step 4: CHECK the solution/Review__

- Check the reasonableness of your solution
- Improve on the method used
- Seek alternative solutions
- Extend the method to other problems

__In short, the 4-steps for problem-solving is
__

- Understand
- Plan
- Do
- Check

#### For Parents: Helping your child with homework

**Solving word problems**

Go through the steps for problem solving together.

__Step 1: UNDERSTAND the problem__

Help your child understand the problem by getting him to read aloud one sentence at a time. Ask your child to explain his understanding of the sentence read in his own words. Once your child understands the sentence, move on to the next sentence.

__Step 2: PLAN what to do__

After understanding the problem, prompt your child to think of how to solve the problem. Give your child time to explore different methods to solving the problem. Encourage him to talk about what he is thinking. Challenge your child to find alternative ways to solving the problem.

Ask leading questions such as...

- "What should you do next?"
- "Will your method work?"

__Step 3: DO it__

Advise your child to write proper mathematical sentences to show the process of solving the problem. Develop the habit of showing all working clearly as method marks will be awarded in the examinations.

Your child should read the question again and answer according to what is asked for. Reading the question and writing the final answer statement is a checking mechanism to ensure the correct answer is given. (eg. Giving the answer in the unit required)

__Step 4: CHECK the solution__

Ask your child to check his answer. Ask leading questions such as...

- "How did you get this answer?"
- "Is your answer reasonable?"
- How do you know that your answer is correct?"
- "Did you use another method to check if your answer is correct?"

Going through the steps for problem solving, will help your child to become an independent thinker and problem solver.

**Helping your child when his answer is wrong**

If your child gets a wrong answer, ask your child to explain how he solved the problem. His explanation may help you discover if he needs help with computational skills such as addition, subtraction, multiplication and division or with the concepts involved in solving the problem.

Remember:

Don't provide the answers immediately. Giving the answers will not help your child. Learning mathematics is more than finding the correct answer. It is a process of solving problems and applying mathematical knowledge to new problems.

Common reasons why some children do not do well for long structured questions

**Using a tedious method**

The key to doing well in Mathematics is learning when to apply the methods learnt. There is a basic set of methods which children have learnt that can be used for all questions e.g. unitary method, listing, working backwards.

Although children know how to use various different problem-solving methods, they have difficulty knowing when to apply them effectively. Many children often choose the wrong (and often more difficult) methods instead of the ones mentioned above. If children choose the wrong or more time-consuming methods (e.g. Guess and Check), they may not have enough time to sufficiently complete and check their solutions during an examination.

**Poor time management**

Sometimes children spend too much time on questions that they cannot solve easily. If they encounter difficulty solving a question, they should skip that question and continue to solve the remaining questions. They can come back to attempt the question again when all the other questions have been completed.

Spending too much time on a question may result in less time or insufficient time for other questions that could be solved easily.

__The PSLE format__

PSLE Standard Mathematics Examination Format

Paper |
Booklet |
Item Type |
Number of questions |
Number of marks per question |
Total marks |
Duration |
---|---|---|---|---|---|---|

1 |
A |
Multiple-choice |
10 |
1 |
10 |
1 h |

5 |
2 |
10 |
||||

B |
Short-answer |
5 |
1 |
5 |
||

10 |
2 |
20 |
||||

2 |
- |
Short-answer |
5 |
2 |
10 |
1 h 30 min |

Structured / Long-answer |
12 |
3, 4 or 5 |
45 |
|||

Total |
47 |
- |
100 |
2h 30 min |

PSLE Foundation Mathematics Examination Format

Paper |
Booklet |
Item Type |
Number of questions |
Number of marks per question |
Total marks |
Duration |
---|---|---|---|---|---|---|

1 |
A |
Multiple-choice |
10 |
1 |
10 |
1 h |

10 |
2 |
20 |
||||

B |
Short-answer |
10 |
2 |
20 |
||

2 |
- |
Short-answer |
10 |
2 |
20 |
1 h |

Structured |
6 |
3 or 4 |
20 |
|||

Total |
46 |
- |
90 |
2h |

The duration for Paper 1 is 50 minutes.

The duration for Paper 2 is 1 hour 40 minutes.

__The suggested time frame for completing each section is:__

Paper 1

- 20 minutes for Booklet A Multiple-choice questions
- 20 minutes for Booklet B Short questions
- 10 minutes for checking

Paper 2

- 15 minutes for Short questions
- 60-65 minutes for Long and structured questions
- 15-20 minutes for checking

Do advise your child wisely on his time management.

**Rosyth Learning Support Programme for Mathematics**

- The programme provides early intervention so that pupils will have a good foundation in Mathematics.
- The focus is on equipping pupils with basic mathematical skills and concepts.
- It aims to build pupils’ confidence and positive beliefs about their ability to do Mathematics
- The programme supports Primary 1 and 2 pupils.

**Important features in the Learning Support Programme for Mathematics**

- Small group teaching so pupils receive more attention
- Parallel teaching but at a slower pace
- Pupils learn through hands-on-experiences