Posted in A. Math, A. Math Tips, Metacognition, Pri Math, Sec Math

Metacognition and mathematical problem solving

Metacognition is a crucial element in problem-solving, which is itself a key component in mathematics learning. To monitor and regulate one’s cognitive processes in problem-solving, Polya (1945) describes a four-step method: first, one has to comprehend the problem by sub-dividing it into more manageable parts and recognize any given data, conditions and variables to be found; second, one devises or selects a strategy to find the connections between the known data and the unknowns to be found; third, one executes the plan, scanning, regulating and examining each step; and, finally, after obtaining the solution, one evaluates the results which may involve re-visiting the previously taken steps.

Building on Polya’s work, Schoenfeld (1987) describes effective mathematical problem-solving as being contingent on how one uses four types of knowledge/skills: (1) resource knowledge, which is knowledge about one’s abilities and cognitive processes including knowledge of how to perform tasks or procedures; (2) heuristics, which are specific problem-solving methods or strategies; (3) regulatory processes, which includes the organisation and selection of resources and strategies; and (4) beliefs, which includes perceptions of and assumptions about mathematics (Gama, 2004).

Ilyasa

 

Note: The above paragraphs are adapted from my minor research paper, Examining Supports for Metacognition in Singaporean Lower Secondary Mathematics Textbooks, NIE, 2011. All rights reserved.

Related links:

(1) Metacognition – The secret to learning and problem-solving;

(2) Metacognition enhances learning;

(3) Is Metacognition part of the Singapore Math curriculum?

 

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Posted in H1 Physics, JC Math (H2/H1), Pri Math, Pure Physics, Sec Math

Metacognition – The ‘secret’ to learning and problem-solving …

You don’t have to be ‘good in English but bad in Math’, or ‘good in Science but lousy in History or Literature’, or ‘good in Chinese but lousy in English’. It doesn’t have to be a one-or-the-other kind of thing. Every subject has its own way of learning and thinking about, but you can only discover it if you bring yourself one level up – think about the thinking itself, and think about how you learn. You can only learn how to learn if you think about learning. You can only learn how to think if you think about thinking.

Thinking about thinking has another name: metacognition, which formed the basis of my minor research work completed in 2011 at the NIE. A lot of research into metacognition centers around mathematical problem-solving, although metacognition can be applied equally well to other subjects. So for the purpose of illustration here, I will use math as the basis for our discussion about metacognition.

What is Metacognition?

In its simplest form, metacognition can be described as thinking about thinking (Wellman, 1985). Flavell (1976) has stated some examples of what constitutes metacognition:

I am engaging in metacognition if I notice that I am having more trouble learning A than B; if it strikes me that I should double-check C before accepting it as a fact; (…)if I become aware that I am not sure what the experimenter really wants me to do; if I sense I had better make a note of D because I may forget I; if I think to ask someone about E to see if I have it right. (p. 232).

The above sounds familiar, doesn’t it? Yes, you may have been practising metacognition sub-consciously, all your life, without realising it has a formal name, or without realising how important it is to learning and thinking.

Research in metacognition has examined various forms of cognitive processes by building on Flavell’s initial notion of metacognition as the monitoring, regulation and arrangement of thinking processes to achieve specified goals (Gama, 2004).

For example, Brown (1987) argues that metacognition consists of two components: knowledge of cognition, and regulation of cognition. The former involves being aware of one’s cognitive abilities through self-reflection while the latter pertains to mental activities such as monitoring and controlling one’s thinking processes in the course of learning or problem-solving. According to Brown, although these two types of metacognition are distinct from one another, they are usually used together in the same cognitive process as one type often utilises the service of the other (Gama, 2004).

Wishing you all the best in your studies,

Ilyasa

(hp: 97860411)

 

Note: The above paragraphs are adapted from my minor research paper, Examining Supports for Metacognition in Singaporean Lower Secondary Mathematics Textbooks, NIE, 2011. All rights reserved.

Related links:

(1) Metacognition and problem-solving;

(2) Metacognition enhances learning;

(3) Is Metacognition part of the Singapore Math curriculum?

 

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Posted in A. Math, Pure Physics, Sec Math

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Posted in A. Math, Educational Advice, JC Math (H2/H1), Pri Math, Sec Math

Beware the critical years in math education …

Be aware that there are Mathematics gaps that need to be handled with care:

The following is derived from my eight years of teaching mathematics from
Primary 1 level to JC 2.

1st Gap – From Lower Primary to Upper Primary:

Somewhere in Primary Three problem sums that require the drawing of simple
models begin to appear and in some schools, this happens in P2 and even P1.
However, these problems tend to be simple enough so as not to cause problems for
students who don’t draw models. Generally, parents report their children doing
badly and losing interest in math in P4. This is because in P4, complex problem
sums begin to appear. It also coincides with the appearance of Decimals. Thus
students who have not mastered Fractions as well as simple models by the end of
P3 will find P4 a tough and demoralising year, with some probably staying away
from Math for the rest of their lives. However, in P4, Section C (problem sums)
still only take up about 20% of the marks, so pupils will still survive and
scoring above 75% is not a problem for the hardworking student who is not
careless.

However, this ecstasy is short-lived. In P5, Ratio, Average and
Percentages start to appear, on top of decimals and fractions, and only the
well-taught and discerning student will understand that they are all roughly the
same thing in different forms. To add to the agony, Section C in P5 takes up
about 45% of the total marks! It is a very big jump from P4; students can no
longer afford to just concentrate on their short questions in order to score a
Band 1. P5 is the year that separates the men from the boys (or the women from
the girls). In P6 or PSLE, Section C’s weightage is increased to about 55%,
wiping out all remaining students who have not mastered complex problem sums and
non-routine questions. That is NOT the bad news yet. The worse news is, the
joy of quite a number of students who scored A-star in math at PSLE is also
short-lived (I have encountered quite a number of students doing badly in
secondary math even though they scored A-stars or A’s at PSLE).

2nd Gap – From P6 to Sec 1:

Why is it that some students can score A-stars or A’s at the PSLE yet become average or even failures in math at the secondary level? The answer lies in two words – Algebra and presentation. It’s unfortunate that even at the upper primary level, students are not taught to form and solve equations using algebra, and they are also not taught how to present their answers in logical and coherent mathematical statements. Thus I find that many Sec 1 students provide math workings that will not earn full marks by ‘O’ level standards, and these habits are hard to change. Inability to use algebra properly also means inability to master important fundamentals such as algebraic expansion, factorisation and manipulation, resulting in poor performance at the upper secondary and JC levels.

Whenever I ask an upper secondary or JC student to state the main reason why he thinks he’s doing badly in math, the reason given is almost always that he had difficulty handling algebraic concepts and formulae while in Sec 1 and Sec 2. Thus parents and students need to comprehend fully the importance of mastering algebra in the lower secondary years.

3rd Gap – From Sec 2 to Sec 3:

Even students who perform well in Sec 1 and Sec 2 may suddenly suffer a drop in their math performance by the middle of Sec 3. This is largely due to the full impact of Additional Math and the pure sciences taking place and finally being felt by students around that time. A. Math can be a shock to some students who are not used to algebra-intensive questions with solutions that are  one-page long. Trigonometry in A. Math is also substantially more difficult to grasp than it’s counterpart in elementary Math.

4th Gap – From Sec 4 to JC 1:

H2 Math is more shocking to new JC students than A. Math is to new Sec 3 students. H2 Math is significantly more difficult than A. Math and from my experience, students who do not get an A1 for A. Math will have a hard time even in completing their JC tutorial worksheets. This is because on top of having to write out solutions that are often more than one page long, students have to familiarise themselves with a new graphical calculator. Many topics in H2 Math are also completely new to students, such as Complex Numbers, Series and Sequences and Probability Distributions, just to name a few. H2 Math is also difficult for most students because some parts of its topics are taken from the former subject Further Math, which was meant for only top students in Math. Thus it is not surprising to find many students failing in Math tests in their first year in junior college. From my experience, two topics in H2 Math that most JC students complain about are Complex Numbers and Vectors. This is largely because these topics speak their own language.

My main point is – Concerned parents must monitor their children’s
mathematical development extra closely when the kids go through the above
stages.

Good luck.

Best Regards,

Ilyasa

Related Links:

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Posted in Educational Advice, Sec Math

It’s a pity that negative numbers are not taught in primary math …

Recently I tested two Sec 2 students, who are both very weak in math, on their knowledge of arithmetic involving negative numbers, and the result was what I suspected about lower sec students who are very weak in math. It reminded me of a stint I had giving math enrichment to very weak students in a secondary school. Their stories are the same – inability to handle negative numbers and negative signs in algebraic equations.

If you are in Sec 1, take the following test to see if you have acquired basic mastery of negative numbers. You PASS the test if you can answer every question correctly in not more than 8 SECONDS (yes, ONE SECOND per question). (Note: some qns may not have any negative numbers.)

(a)  5 + 3 = ____

(b)  5 – 3 = ____

(c)  – 5 + 3 = ____

(d)  – 5 – 3 = ____

(e)  3 + 5 = ____

(f)  3 – 5 = ____

(g)  – 3 + 5 = ____

(h)  – 3 – 5 = ____

Answers: (a) 8, (b) 2, (c) – 2, (d) – 8, (e) 8, (f) – 2, (g) 2, (h) – 8

If any of your answers are wrong, or you took more than 8 seconds to complete the test correctly, then you need to seriously revise and understand the concept of negative numbers, else your other topics such as algebraic simplification, expansion and factorisation will also be adversely affected. For eg, try the next one:

Simplify     – 3a – [- b – (- 2a)]

Ans :  b – 5a

Of course there are far harder questions. My point is, get your negative numbers in order, then you’ll see improvement in the way you solve algebraic equations. Remember, you’ll get ZERO marks for just one mistake made in the opening line of your half-page long math solution. So do not underestimate the significance of the negative sign.

The importance of Sec 2 Math and Science

Beware the critical years in math education

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Posted in A. Math, Educational Advice, Sec Math

What to do if you had done badly for Math in SA1 in sec school …

Actually, sec students will normally do badly in their SA1 exams, although this should not be used by the students themselves to justify their poor performance. Why does this “mass slaughter” of math students happen near the middle of every year?

Well, my educated guesses are as follows:

(1) If you are in Sec 1, you probably did badly because of your poor or zero exposure to algebra, negative numbers and proper presentations while in primary school. Please realise that your single arrows, your poorly labelled statements and your misuse of equation signs will really worsen your grades this time. So please listen carefully to your sec school teacher’s illustrations on how math solutions are presented in major exams. Many of you would also have made careless mistakes in opening brackets with minus signs on the inside and the outside, and in simplifying or solving algebraic expressions.

(2) If you are in Sec 2, in addition to not having overcome the above problems earlier, you were probably slapped with an exam paper that was designed to “kill”. The ‘logic’ is simple. Sec 2 is a crucial year; it is the year you are “streamed”, the year that decides whether you will be allowed to do pure sciences or  additional math in sec 3. So teachers feel that they need to “wake you up”, to humble you, so that you will work harder for SA2. So most exam papers for SA1 in Sec 2 have been set to “wipe out” students. It is not something I fully agree with, as such tactics can also demoralize students for life.

(3) If you are in Sec 3, it was probably because you underestimated the effort needed to master A. Math. Topics like Quadratic Functions and Logarithms need a lot of attention and deep thinking, and even ‘simple’ topics like Remainder Theorem and Partial Fractions have their difficult parts, the kinds that teachers like to select for exams. Make no mistake about this – teachers KNOW what concepts you find difficult to apply.

(4) If you are in Sec 4, the reason is the same as in (2) above, that your Math teachers want to “wake you up” before the prelims and the actual O-Level exam, and also because you wouldn’t have had enought time to revise both Sec 3 and Sec 4 work before the exam which started right after you were taught the last topic. That’s why some schools remove the mid-year exam in Sec 4, or postpone it to late June.

What you have to do now to avoid a total disaster for SA2?

(1) If you are in Sec 1, focus on mastering the solving of Algebraic Equations.

(2) If you are in Sec 2, focus on mastering the solving of equations involving Algebraic Fractions and mastering all the topics on Graphs (Linear and Quadratic Equations). If you have been taught Indices, master that too!

(3) If you are in Sec 3, for Math, focus on mastering Indices, Coordinate Geometry and Trigonometry. For A. Math, focus on mastering Trigonometry.

(4) If you are in Sec 4, for Math, focus on mastering all of Graphs and Vectors. For A. Math, focus on Applications of Differentiation and Integration. If you have no time to revise everything since Sec 1, focus on the topics mentioned in (1) to (3) above. If you have not mastered all the topics mentioned in (1) to (4) above, you will be in serious trouble if you do Math in JC.

Click on How to study for Math or Physics exams to know more about exam preparation.

Click on O-Level Math or Additional Math to know what books you can use for revision and practice.

Or consult Mr Ilyasa @ 97860411.

All the best in your exams!

 

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Posted in A. Math, Integrated Programme (IP), Sec Math

Sec 4 IP Math group tuition …

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Sec 4 Integrated Programme (IP) Math:

Sun 10.00 am – 12.00 pm;  Course Code:  s4amipsun10. (Investment amt: $240 per 4 sessions) (1 vacancy)

Conducted by our Principal Tutor, Mr Ilyasa (h/p: 97860411)

Current students in the class are from RI, RGS and ACS(I). Mr Ilyasa has been giving tuition in IP Math since 2005. His IP students are mainly from RI, RGS, HCI, Nanyang Girls’ and River Valley High. To view Mr Ilyasa’s educational qualifications and teaching experience, click here.

Posted in A. Math, Administrative, Sec Math

Sec Math Textbook Solutions

If you are looking for fully-worked solutions to problems found in commonly used mathematics (Additional or Elementary) textbooks in Singapore, you can check out this wonderful website created by Lee Chu Keong, a lecturer at the Nanyang Technological University: http://www.openlysolved.org

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