FACULTY OF EDUCATION AND LANGUAGES
HBMT 3403 TEACHING MATHEMATICS IN
FORM TWO
OCTOBER
2011

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
shape.
A
character in Mathematics is Mathematics contains language, symbols and
operations. The basic of the language system is the grammar. In the Mathematics
language, the grammar consists of rules, theorems and formula that connects the
symbols.
George
Polya (1944) emerged as a reference, emphasizing the meaning of discovery and
of encouraging students to think by means of solving a problem. In his book How
to Solve It, he states “A great discovery solves a great problem, but there is
always a bit of discovery in the solution of any problem”. In 1949, he wrote
that solving problems is the specific realization of intelligence, and that if
education does not contribute to the development of intelligence, it is
obviously incomplete. In 1948, the work developed by Herbert F. Spitzer in fundamental
arithmetic, in the U.S.A., was based on learning with comprehension, always
using problemssituations; and in 1964, in Brazil, the teacher Luis Alberto S.
Brasil stressed on teaching mathematics using problems that generated new
concepts and contents.
At
the beginning of the 1970s, systematic investigation of problem solving and its
implications for curricula was initiated. Thus, the importance attributed to problem
solving is relatively recent, and only in this decade did mathematics educators
come to accept the idea that the development of problemsolving abilities
deserved more thought. At the end of the 1970s, problem solving emerged,
gaining greater reception around the world.
In
my school, teaching becomes more sophisticated. In “Unfinished Business:
Challenges for Mathematics Educators in the Next Decades”, Kilpatrick & Silver (2000) outline what they
believe to be the chief challenges: guarantee mathematics for all; promote
students’ understanding; maintain balance in the syllabus; use evaluation as an
opportunity for learning; and develop professional practice. Cai (2003) stresses, however, that although very
little information is known regarding how students attribute meaning and learn
mathematics through problem solving, many ideas associated with this approach –
the change in the teacher’s role, the selection and elaboration of problems,
collaborative learning, among others – have been researched intensively,
offering answers to various frequently asked questions about this way of
teaching.
Some problems can be successfully solved by following specific,
stepbystep instructions—that is, by using an algorithm. We can correctly
assemble the pieces of a new bookcase by following the directions for assembly
that come with the package. We can calculate the length of a slanted roof if we
use the Pythagorean theorem. When we follow an algorithm faithfully, we
invariably arrive at a correct solution.
However, the world gives many problems for which no algorithms
exist. There are no rules we can follow to identify a substitute metal ship, no
list of instructions to help us address the destruction of rain forests. In the
absence of an algorithm, learners must instead use a heuristic, a general
problemsolving strategy that may or may not yield a successful outcome. For
example, one heuristic that we might use in solving the deforestation problem
is this: Identify a new behaviour that adequately replaces the problem behaviour
(i.e., identify another way that peasant farmers can meet their survival
needs). For another example of a heuristic, consider the addition problem in
the exercise that follows.
Dewey’s
defined thinking, which he wants to distinguish from ‘daydreaming ’. He
specifically means focused, purposeful, rational and intelligent thinking. He
uses the label ‘reflective thinking’ in a way we might talk of ‘critical’,
‘careful’, ‘considered’, or ‘deep’ thinking today, but importantly he inserts
the word ‘reflective’ apparently in order to emphasise critically reflecting on
a prior belief, ‘first thought’, conjecture or some other ‘supposed form of
knowledge’.
Active, persistent, and
careful consideration of any belief or supposed form of knowledge in the light
of the grounds that support it, and further conclusions to which it tends,
constitutes reflective thought…
…it is a conscious and
voluntary effort to establish belief upon a firm basis of reasons.
In
the second part of the book (which starts at chapter 6) Dewey explains how this
“established belief upon a firm basis of reasons” might be analysed (where
analysed is distinguished from synthesised). The analysis or “picking apart”
(zooming in) provides the ‘steps’ (constituents, parts) and is spelt out under the
chapter heading, “The Analysis of a Complete Act of Thought”.
As
a flowchart, his steps of thinking might look like figure 1. The feature I am
trying to highlight is that a quickly guessed or ‘first’ solution seems to come
before the collection of supporting evidence (as reasoning or as empirics).
This is why Dewey calls it ‘reflective thinking ’, that is, thinking back on a
possible solution. Dewey provides synonyms for ‘possible solution ’, first
thought’ or ‘quick idea.’ He suggests, ‘conclusion’, ‘supposition’,
‘conjecture’, ‘guess’, and ‘hypothesis’.
This
idea of placing the solution prior to the thinking will only be a novel
suggestion to those unfamiliar with the ideas behind argumentative inquiry, or
of the pattern recognition literature from psychology. Of course, the example
provided are very simple, as are the other two examples that Dewey discusses.
Both of which follow the same pattern. Seeing them as simple examples of
thinking might however mistakenly open up the possibility of distinguishing
reflective thinking as all right for quick everyday decisions, but not for
important thinking or project management. This would allow a return to the line
of argument that ‘jumped to conclusions’ need to be avoided until after some
careful consideration of the facts. I would suggest any split between everyday
thinking and ‘big project’ thinking is a error, as the big project thinking
only includes what is happening in the heads of the individuals involved in a
project.
Dewey,
described by some as the most influential philosopher on thinking and education
in the 20th Century, spent the second half of his life in the department of
philosophy at Columbia University. Like Schon, he is thought of as a writer on
educational philosophy yet both of their works have been seminal to the
management literature. Newell and Simon (1972) cite Dewey in their own oft
cited book ‘Human Problem Solving,’ as does Churchman (1971) in his [oft cited]
book, ‘The Design of Inquiry Systems”. Simon won a Nobel prize and Churchman was
shortlisted. In more modern times Mintzberg (2001), a seminal figure in the
management research literature, directly attributes the rational steps of
decision making to Dewey (1910). Dewey spells out the steps (or as he says,
constituents) of reflective thinking, in his small book, “How We Think”.
Figure
1: Dewey’s Steps to Thinking
YES OR NO
Suggest
Solution
Collect Evidence
(reasoning and information)
Conclude?
Identify
or Define Problem
Another
strand of thinking that appears to support the ‘conjecturefirst’ 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”
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.
For
this task, I used Chapter 4, Linear Equations topic equality (page 70) , linear
equations in one unknown (72) , solutions of linear equations in one
unknown(75). 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 9831502418.

Question 1
Mr Salleh bought 45 pencil cases for his students. He bought 2
types of pencil cases, under brand x and y. Brand X pencil cases cost RM 25
each and brand Y pencil cases cost RM 4 each. He spent RM 600 to buy both
brands, the questions is how many of each type of pencil cases did he buy ?
 Understand the problem
Mr Salleh needed 45 pencil cases. There are two types of pencil
cases in which one is RM 25.00 each and the other is RM 4.00 each. We need to
find the total number of each type that she bought for RM 600.00
 Devise a plan
Interpret the information in a table format. The scope of this
problems is:
·
The total number of both
types is 45
·
The total cost spent is RM
600.00
3.Carry out the plan
Brand X

Brand y

Total cost(RM)

decision


Amount
40
30
20

Cost (RM)
1000
75
500

Amount
5
15
25

Cost (RM)
1020
810
600

1020
810
600

>RM
600.00 Too large
>RM
600.00 Too large
>RM
600.00 Correct

 Check the answer
Verify the answers. Could there be other sets of guesses? Why or
why not
The second method I use is Guess and test
 Understand the problem
Brand X :
RM 25
Brand Y :RM
4
Both costs
her RM 600.
 Devise a plan
Use
suitable sets such as pictures. Use addition, subtraction, multiplication
X
represents brand x pencil cases, and y represents brand y pencil cases.
 Carry out the plan
Based on
the picture and guessing that is done to Brand x
If he buys 15 brand x pencil cases, he will pay RM 375 and if he
buys 35 brand y pencil cases, he will use RM 140, RM 375 + RM 140 will only
cost RM 715, that answer is wrong.
If he buys 20 Brand X pencil cases, then, the price will be RM
500, and he would have bought 25 brand y pencil cases and that will cost him RM
100.
When doing addition, RM 500 + RM 100 will cost him RM 600.
 Check the answer
Total
calculators bought: 45
Brand X:20
X RM 250=RM 500
Brand y:25
X RM 4=RM 100
Total
amount RM 500+ RM 100= RM 600
Second Question
Miss Saleha posted
altogether 32 hari raya cards and letters. She has to pay RM 8.35 for the
postal fees at the post office. The postcards cost 20 cents each and the
letters cost 33 cents each. How many postcards and letters had she sent ?
 Understand the problem
Total cards and
letters=32
32= RM 8.35
One postcard costs 20
cents, one letter costs 33 cents.
2 Devise a plan
3 Carry out the plan
Method number one : using formula
0.2
x + 0.33 y

=8.35

0.2x

=8.350.33 y
0.2

x

=8.350.33 y 0. 2

X+y

= 32

Y

=32x

this is the equation number 2
Substitute equation number 2 into equation number 1
0.2x+0.33
y

=8.35

x

= 8.350.33y 0.2

32y

=8.350.33y
0.2

6.40.2y

=8.350.33y

0.13y

=1.95

y

=15

 Check the answer
The second
method is by guess and checking
 Understand the problem
Total cards and
letters=32
32= RM 8.35
One postcard costs 20
cents, one letter costs 33 cents.
2.Devise a plan
Suppose we guess that
the number of postcards is 10. It means the number of letters is 3210=22. If
this is true, then the total bill will be (10 X 20 cents) +(22 X 33 cents)=RM
9.26.
This tells us that the
10 cards and 22 letters would cost more, so we need to guess one more time.
3 Carry
out the plan. Let us guess
Suppose we guess that
the number of postcards is 16. It means the number of letters is 3216=16. If
this is true, then the total bill will be (16 X 20 cents) +(16 X 33 cents)=RM
8.48
Suppose we guess that
the number of postcards is 17. It means the number of letters is 3217=15. If
this is true, then the total bill will be (17 X 20 cents) +(15 X 33 cents)=RM
8.35
 Check the answer
Reflections
When carrying out the process to solve the problem, there are many
obstacles that I have encountered. They are:
I do not understand the maths terms used. Generally students who
are competent in English will face difficulties to understand the Maths terms,
compared to students who are high achievers. They must remember the terms while
performing the Maths solution. The pupils cannot translate the Maths problem
into Bahasa Melayu.
I do not know what strategies to be used. The pupils are not
exposed the strategy to be used, they use the formula trial and error. The
pupils assume Maths is boring and not practical
Question situation given is not suitable. The problem must follow
the pupils existing knowledge. They can visualise the problem if they have
experienced it before, and this makes them able to solve the problem the
simpler way.
I do not understand the operation used. Low achiever students do
not understand the operations to be used for a problem, whether it is addition,
subtraction, multiplication or divide.
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 ;
The
problems given must be in the form of a story that relates to the pupil’s
lives. Second, I must 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.
Then, 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.
Drilling is another way too. 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.
Students 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
 Check
Conclusion
In the questions, I have used formula and guessing method. In
teaching linear equations, the best way to expose them is giving them daily
life situations. People are busy with live that they separate linear equation
formula they have learnt in class from their lives. It is shown that by
presenting the word problem, the students learn better. Not
only that, the Polya’s and Dewey’s methods which is proven will guide the
students in making better judgments in facing their daily problems, and they
can calculate using one of the three methods, how much money they have spent
for that week, even though they do not have the information organised neatly.
They also can create new statistics on the aspect of time management or how to
manage time better using the heuristics. They can solve their problem of not
having much time on studying at home. Time can be calculated better using the
maths formula they have learnt in class. There are two methods shown, the
formula way and other ways that we usually use to calculate, such as using the
guessing and testing method. It takes simple maths to perform the task.
References
 http://www.cliffsnotes.com/study_guide/LinearEquationsSolutionsUsing Eliminations.topicArticleId38949,articleId38875.html
 http://www.cliffsnotes.com/study_guide/LinearEquationsSolutionsUsingSubstitution.topicArticleId38949,articleId38874.html
 J. R. Anderson, 1990; J. E. Davidson & Sternberg, 1998, 2003; H. C. Ellis & Hunt, 1983; Halpern, 1997a)
 Kilpatrick & Silver, 2000, “Unfinished Business: Challenges for Mathematics Educators in the Next Decades”
 Krulik, S.Redneck,JA (1989) “Problem Solving A Handbook for Senior high School Teacher”.
 Mohd Nasir Mahmud.(2010). HMBT3403 Teaching Mathematics in Form Two Kuala Lumpur:OUM
 Polya G., 1945, How To Solve It, Princeton University Press, New Jersey.
No comments :
Post a Comment