FACULTY OF EDUCATION AND LANGUAGES
HBMT 3403 TEACHING MATHEMATICS IN FORM TWO
OCTOBER
2011
LETAKKAN
NAMA ANDA DISINI (EEISH COPY JE KEE)
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Introduction
Mathematics is knowledge that come from observation on nature. Mathematics
is a logic system that shows the formula, that creates the language of
Mathematics such as symbols, rules and operations.
Mathematics is a
way of thinking that is used to expand on reasoning and reach to a conclusion
from the existence of the universe.Mathematics is also interpreted as an art
form, because it contains its language and patterns in an interesting shapes.
A
character in Mathematics is Mathematics contains language, symbols and
operations. The basic of the language system is the grammar. In the Mahematics
language, the grammar consists of rules, theorems and formula that connects the
symbols.
According to Piaget, the mathematics language is abstract and only
secondary school students can understand it. Students in primary schools cannot
understand the mathematics symbols that are abstract without relating it to
their concrete experience. Thus,
teaching the Mathematics language must be related to their experience , relate
it with teaching aids and concrete items. For example, when we want to teach on
Mathematics, we relate to the spupil’s daily experience, for example, I am
purchasing three items at the store, at these prices, RM 19.95, RM9.98 and RM
29.97. About how much money am I spending ? The fastest way to solve this
problem is to round off and approximate. The first item costs about $20, the
second about $40, and the third about $30; therefore, I am spending about $90
on your shopping spree. Rounding is often an excellent heuristic for arriving
quickly at approximate answers to mathematical problems.
A problem is a situation that an individual wants to resolve arises but
no solution is readily apparent. It is also a situation which the individual is
confronted by something he/she does not recognise that he/she cannot apply any
model to solve for the unknown. In order to solve the unknown the individual
undergoes the problem solving. It is the process by which the unfamiliar
situation is resolved.
A
problem to an individual may not be a problem to another. For example,
determining the number of people in three cars in which each car consists of
five people may be a problem to some elementary school learners. They might
solve the problem by placing chips in boxes or making a drawing to represent each car and each person, and
then counting to determine the total number of people.
Hayes
(1981) described problem as finding an appropriate way to cross gap.”two major
parts of problem solving are representing the problem and searching for a means
to solve the problem. For example, in solving an algebra word problem, you must
translate the problem into an internal representation such as equation, and you
must be able to apply the rules of algebra and arithmetic to solve the equation
(Mayer, 1983) Mayer focuses on techniques aimed at improving the representing
phase of problem solving such as translation training and schema training, and
techniques aimed at improving the searching phase of problem solving such as
strategy training and algorithm automaticity.
Another
strand of thinking that appears to support the ‘conjecture-first’ approach to
thinking may be drawn from the words of that classic problem solving text by
Polya (1945); ‘How To Solve It’. This booklet is aimed at mathematicians. It
provides a suggested methodology to tackle the creative process of solving
mathematical and geometric problems. His ‘steps’ are:
“First. You have to
understand the problem”
Second. Find the
connection between the data and the unknown. You may be obliged to consider
auxiliary problems if an immediate connection cannot be found. You should
obtain eventually a plan of the solution.
Third. Carry out
your plan
Fourth. Examine the
solution obtained” [1945, p. xvi]
Polya then provides
further elaboration of what he meant under the second step in terms of needing
to seek analogous problems that have been solved. He suggests using existing
solutions to old problems. This is further understood by his elaboration of the
third step which he states as testing to see if the old analogous solution or
at least similar concepts work for the new problem. Polya, who is a well
respected author on problem solving, is therefore not suggesting that solutions
come after reasoning about alternatives. Rather, he suggests that you search
for a conjecture solution (from an analogous problem) and then think about its
usefulness for the current problem. This again seems very much like Dewey’s
advice and the advice of argumentative inquiry: conjecture something that might
work, and then think about the feasibility of this solution carefully.
HEURISTICS FOR PROBLEM SOLVING
This is a set of
heuristics that has been proven to be successful with students and teachers at
all levels of instructions
Read the problem
- Note key words
- Describe the problem
- Visualise the action
- Restate the problem in your own words
- What is being asked for ?
- What information is given ?
Explore
- Organise the information
- Is there enough information ?
- Is there too much information ?
- Draw a diagram or construct a model
- Make a chart or table
Select a
strategy
- Pattern recognition
- Working backwards
- Guess and test
- Simulation or experimentation
- Simplify the problem
- Organised listing or make a table
- Logical deduction
- Using formula
Solve
- Carry out the strategy
- Use computational skill
- Use geometric skills
- Use algebraic skills
- Use elementary logic
Look back
- Check your answer
- Find another way
- What if... ?
- Extend
- Generalize
Another
strand of thinking that appears to support the ‘conjecture-first’ approach to
thinking may be drawn from the words of that classic problem solving text by
Polya, 1945) ‘How To Solve It’. This
booklet is aimed at mathematicians. It provides a suggested methodology to
tackle the creative process of solving mathematical and geometric problems.
His ‘steps’ are:
“First. You have to
understand the problem”
Second. Find the
connection between the data and the unknown.
Third. Carry out
your plan
Fourth. Examine the
solution obtained” [1945, p. xvi]
Polya
then provides further elaboration of what he meant under the second step in
terms of needing to seek analogous problems that have been solved. He suggests
using existing solutions to old problems. This is further understood by his
elaboration of the third step which he states as testing to see if the old
analogous solution or at least similar concepts work for the new problem. Polya,
who is a well respected author on problem solving, is therefore not suggesting
that solutions come after reasoning about alternatives. Rather, he suggests
that you search for a conjecture solution (from an analogous problem) and then
think about its usefulness for the current problem. This again seems very much
like Dewey’s advice and the advice of argumentative inquiry: conjecture
something that might work, and then think about the feasibility of this
solution carefully.
In the past, ratio
and proportions were used in architecture, paintings and sculptures. Today, we
use them in many daily situations, they are used not only in preparing
chemicals, but also in shopping and cooking.
For
this task, I have taken Chapter 5, ratio, Rates and Proportions The Concept of
ratio of Two Quantities (page 86), the Concept of Proportion to solve Problems
(page 9). The textbook that I am using is Mathematics Textbook Form 2 Volume 1
2003, Arus Intelek Sdn Bhd. Cheang Chooi Yoong, Khau Phoay Eng, Yong Kien Cheng
ISBN 983-150-241-8.
Ratio
For the first
question, I have decided to use the strategy of simulation, that is, I use
pictures to show the syrups and the water used.For the second question , I use
the Maths formula.
Pak Salleh has 10
cups of syrup and he can make 40 cups of drinks. Unfortunately, he spilled 6 ½
cups of the syrup. How many cups of drinks can he prepare with the balance ?
1) Understand the problem
10 cups of syrup=
40 cups of drinks
How much can we make with 6 1/2 of syrup ?
2) Decide on the
plan
I use simulation
and I draw the syrup and water.
3) Carry out the plan
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1 cup of
syrup
= 4 cups of
drinks
½ cup of
syrup = 4/2
½ cup of syrup
makes 2 cups of drinks.
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  3 cups of syrup
X 4 |
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  =12 cups of drinks |
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 6 ½
= 12 cups + 2 cups
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So, 6 ½ of syrup
makes 14 cups of drinks
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Ratio of syrup to
water
 =1 : 4
Solve this: how
much is ratio of ½ syrup ?
So, the formula
is 4/2=2
 Therefore, ½ syrup is 2 to water
= ½: 2
The amount of
syrup is proportional to the amount of water.
 1 cup of syrup = 4 cups of drinks
½ cup of
syrup = 4/2
  ½ cup of syrup makes 2 cups of drinks.  3 cups of syrup
X 4 water =12 cups of drinks So, 6 ½ of syrup makes = 12 cups + 2
cups
So, 6 ½ of syrup
makes 14 cups of drinks
The ratio of
syrup to water
 =1:4
Solve this ratio:
½ : ?
4/2=2
 Therefore, ½ :2
The amount of
syrup is proportional to the amount of water.
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1) Check the answer.
½
cup of syrup equals to 2 cups of drinks, so
6
½ cups of syrup means
½
X 6 ½ =
= 14 cups of drinks
Maths
formula
Pak Salleh has 10
cups of syrup and he can make 40 cups of drinks. Unfortunately, he spilled 6 ½
cups of the syrup. How many cups of drinks can he prepare with the balance ?
1) Understand the problem
10 cups of syrup=
40 cups of drinks
How much can we make with 61/2 of syrup ?
2) Decide on the
plan
I use Maths formula
3)Carry out the
plan
10 cups of syrup=40
cups of drinks
40/10=4
So, 1 cup of syrup
makes 4 cups of drinks
How much can ½ cup
of syrup makes drinks ?
So, 4 / 2=2
It means ½ cup of
syrup makes 2 cups of drinks.
3 cups of syrup X 4
=12 cups of drinks.
6 1/2 cups of syrup
makes 14 cups of drinks.
The ratio of syrup
to water is 1so, what is the ratio of water if syrup is ½ ?
4/2=2
Therefore the ratio
of syrup ½ to water is 2= ½ :2
The amount of syurp
is proportion to the amount of water.
4)Check the answer
1 cup of syrup
makes 4 cups of drinks
and ½ cup of syrup
makes 2 cups of drinks.
6 ½ cups of syrup makes 14 cups of drinks
Second
question
The
ratio of cows to bulls in Pak Salleh’s farm in Ayer Pa’abas, Melaka is 5:7.
There are 48 more bulls than cows. After one year, 3 bulls are transferred to
Pak Samad’s farm, and Pak Salleh buys 5 new cows. Find the ratio of cows to
bulls in Pak Salleh’s farm.
1) Understand the problem
The ratio of cows
to bulls is 5:7
Cows is x and bulls
is y.
Cows =48 is more
than bulls
2) Decide on the plan
We use the formula
method.
We find X first and
we use linear equation.
3) Carry out the plan
x= cows
y=bulls
x:y=5:7
x/y =5/7
This is the first
equation
y=48+x
x
______= 5
48+x 7
7x + 240 + 5x
7x-5x=240
2x=240
x=240/2
x=120
Put the new x value
into the first equation
y=148+x
y=148+120
y= 168
After a year,
Bulls: 168-3=165
Cows:120+5=125
New ratio=125:165
25:33
4) Check the answer
Find the ratio of
cows to bulls in Pak Salleh’s farm.
Bulls: 168-3=165
Cows:120+5=125
New ratio=125:165
25:33
The
second method
The
ratio of cows to bulls in Pak Salleh’s farm in Ayer Pa’abas, Melaka is 5:7.
There are 48 more bulls than cows. After one year, 3 bulls are transferred to
Pak Samad’s farm, and Pak Salleh buys 5 new cows. Find the ratio of cows to
bulls in Pak Salleh’s farm
1) Understand the problem
Cows:bulls=5:7
The difference
between cows and bulls =48
3 bulls are
transferred and 5 new cows are added.
Find the new ratio of cows:bulls
4) Decide on the plan
I use simulation
and experimentation method.
5) Carry out the plan
2 parts represent
48 animals.
1 part represent
48/2=24 animals
5 parts represent 5
X 24=120 cows
7 parts represent 7
X 24=168 bulls
\
After a year,
Bulls=168-3
=165
Cows=120=5
=125
New ratio=125;165
(divide both by 5)
=25:33
5) Check the answer
25:33=125:165
125-5=120 cows,
165+3=168 bulls
Female:male=120:168
(divide both by 24)
=5:7
Reflection
I
have shown these examples to two classes, that is 2A and 2F. These two methods
have been shown to them and i have explained it differentlly for both classes.
The first class, ihave explained in detail, and the second class, I explain the
basics only, I don’t want them to get confused which method to use.
These
two classes have different capabilities from each other. 2A can understand the
two methods, whereas 2F needs more explanations from the teacher. It looked 2F
needed a double period for linear equation topic, and I wish to include this
into their end year final examination, and I want both classes to be able to
answer the linear equation problems and get A One for Maths
The students are encouraged
to apply both methods so they can tell which method is better in a given
problem.Usually they like to use the guess and test method because the method
is simpler and not confusing. If we compare with the formula or drawing a table
method, there are many students who are confused. Perhaps the low achievers
will use the guessing method, but they will not get full marks if they write it
in the exam paper.So, the teacher will have to stress to them, to use the
formula or table method to get full marks. They will get full marks if the
method written and the answer are both correct.
\
Suggestions
The
ability to solve problem is important for a pupils to master. It relates to our
daily lives. This ability must be stressed to them. My suggestions are the
problems given must be in the form of a story that relates to the pupil’s
lives. Expose students to a few strategies such as Drawing,Find the pattern,
Building a table, Try and error and Building word bank. The usage of a certain
strategy must be suitable with the problem, and it is easier for the students
to solve the problem. The students must know the maths language in problem
solving. There are many pupils face problems in solving word problems. This is
because they must read the problem, understand it and think what method to be
used. So, the sentences that are used must be suitable with the pupil’s
understanding. The pupils must be competent in translating the maths problem,
from Maths language to their mother tongue and the other way around. Generally,
students who are weak in maths make less practice compared to pupils who makes
more exercises. Therefore, daily practises, with schedule is required as
additional homework, to help them in mastering problem solving. Next, solve
problem according to problem solving process.It is important for teachers to
give their pupils good questions. In good questions, the facts that are given
are presented in the pupils daily lives. The pupils understand that all Maths
lessons learnt in classrooms can be applied in their 24/7 lives. They can be
used in all aspects of life, whether it relates to Maths or not. The process to
solve the problem must follow Polya’s problem solving method that is Understand
the problem, Planning a device, Make the plan and Check.
Conclusion
By
using the Dewey and Polya’s method, it simplifies the student’s work in finding
the answer. The students are taught to organise the information in a manner
that makes them undrerstand the information that is given, turn in into
mathematical language, decide and carry out the plan. Lastly, they check the
answers, look whether it tallies with the question’s requirements. By using
simulation and formula, they are able to find the answers, assisted by the
textbook. The questions cover the student’s daily experience, where they are
taught to apply the heuristics approach to tackle their daily tasks.
References
- Bahagian Pendidikan Guru (1998), Pengajaran Pembelajaran Matematik : Pecahan
- Untuk Sekolah Rendah, Dewan Bahasa dan Pustaka, Kuala Lumpur.
- Dr. Mahmood Othman (2010), Teaching Mathematics in Year Four, OUM, Kuala
- Lumpur.
- Frank J. Sweet & Liew Su Tim ( 1987), Pengajaran Matematik KBSR, Penerbit Fajar
- Bakti Sdn. Bhd. Petaling Jaya.
- Linda Lim, S.F Liew & Ros Dhania, Success Mathematics UPSR, Oxford Fajar,
- Selangor
- Ling Choi Hong & Aw lai Sin (2010), Penilaian Topik BEST Mayhematics KBSM
- Year 1, PEP Publications Sdn. Bhd. Petaling Jaya, Selangor.
- Maktab Perguruan Persekutuan Pulau Pinang ( Julai 1995), Penyelesaian Masalah
- Dalam Matematik, Bahagian Pendidikan Guru, Kuala Lumpur.
- Mohd Nasir Mahmud (2010), HBMT3403: Teaching Mathematics in Form Two,
- OUM, Kuala Lumpur.
- Mok Soon Sang (2000), Pengajian Matemtik Untuk Diploma Perguruan, Kumpulan
- Budiman Sdn Bhd, Selangor.
- Zarina Law bt Abdullah & Abdul Latih bin Ahmad (2007), HBMT 1103 Introduction
- to Mathematics Education, OUM, Kuala Lumpur
- Polya G., 1945, How To Solve It, Princeton University Press, New Jersey.
- Krulik, S.Redneck,JA (1989) “Problem Solving A Handbook for Senior high School Teacher”.
- (J. R. Anderson, 1990; J. E. Davidson & Sternberg, 1998, 2003; H. C. Ellis & Hunt, 1983; Halpern, 1997a)
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