HBMT 3403 TEACHING MATHEMATICS IN FORM TWO(4)

FACULTY OF EDUCATION AND LANGUAGES

HBMT 3403 TEACHING MATHEMATICS IN FORM TWO

OCTOBER 2011

	Tutor MR.

 

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 problems-situations; 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 problem-solving 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, step-by-step 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 problem-solving 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 short-listed. 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 ‘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”

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 983-150-241-8.

1

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 ?

1. 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

2. 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

4. 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

1. Understand the problem

Brand X : RM 25

Brand Y :RM 4

Both costs her RM 600.

2. 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.

3. 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.

4. 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  ?

1. 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	=8.35-0.33 y          0.2
x	=8.35-0.33 y                0. 2
X+y	= 32
Y	=32-x

this is the equation number 2

Substitute equation number 2 into equation number 1

0.2x+0.33 y	=8.35
x	= 8.35-0.33y                  0.2
32-y	=8.35-0.33y          0.2
6.4-0.2y	=8.35-0.33y
0.13y	=1.95
y	=15

3. Check the answer

The second method is by guess and checking

1. 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 32-10=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 32-16=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 32-17=15. If this is true, then the total bill will be (17 X 20 cents) +(15 X 33 cents)=RM 8.35

4. 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

1. Understand the problem
2. Planning a device
3. Make the plan
4. 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

1. http://www.cliffsnotes.com/study_guide/Linear-Equations-Solutions-Using Eliminations.topicArticleId-38949,articleId-38875.html
2. http://www.cliffsnotes.com/study_guide/Linear-Equations-Solutions-Using-Substitution.topicArticleId-38949,articleId-38874.html

3. http://www.archive.org/details/howwethink000838mbp

4. J. R. Anderson, 1990; J. E. Davidson & Sternberg, 1998, 2003; H. C. Ellis & Hunt, 1983; Halpern, 1997a)

5. Kilpatrick & Silver, 2000, “Unfinished Business: Challenges for Mathematics Educators in the Next Decades”
6. Krulik, S.Redneck,JA (1989) “Problem Solving A Handbook for Senior high School Teacher”.

7. Mohd Nasir Mahmud.(2010). HMBT3403 Teaching Mathematics in Form Two Kuala Lumpur:OUM
8. Polya G., 1945, How To Solve It, Princeton University Press, New Jersey.