Mastering Mathematics Smartly
by Wee Wen Shih

Dear students,

  After your midyear exams, it is important to pick yourself up from unsatisfactory performance by reviewing your mistakes and revisiting the exam questions. By revisiting, I mean doing the questions from scratch with the aid of worked solutions provided by the school and your teacher's explanations. Try to internalise the problem-solving approaches and appreciate the style and difficulty of your school exam paper. These proactive steps will certainly contribute to a better performance in your next exam.

  H2 Mathematics may look numerous in topics covering a wide array of concepts and skills at the beginning. After some time of struggling with the subject, you will surely notice some patterns about the types of question asked. What are these? I'd like to invite you to read some articles I have posted on SgForums which cover both Pure Mathematics (print version) and Probability/Statistics (print version) topics.

  On top of knowing patterns, it is crucial that one applies problem-solving steps effectively. Do read articles I have posted on Channel NewsAsia about Polya's problem-solving steps.

  Generalising patterns is yet another way to deepen your understanding of mathematics. Be sure to read my article "On generalisations".

  Continue to work hard for the rest of the year. Your best is yet to be!

Cheers,
Wen Shih

Dear precious friend of MMS (Mastering Maths Smartly),

  MMS was conceived close to two years ago and it has grown from strength to strength, thanks to your support and encouragement!

  In the last 30 days (i.e. 29 May - 27 Jun), MMS has welcomed more than 800 international visitors from 10 countries.

  Encouraged by this overwhelming response, MMS will continue to develop more student-friendly learning materials to help students in mathematics. Feel free to drop me an email (weews@yahoo.com) to provide ideas or suggestions about learning resources. Thanks in advance!

  Take good care :)

Cheers,
Wen Shih
28 Jun 2009

This site aims to provide smart ways for H2 Mathematics students in Singapore to master the subject. Undergraduate students can make good use of the content within to familiarise themselves of A-level Mathematics concepts. Mathematics educators are very welcomed to share the site's resources with their students.

Note: Students doing O-level mathematics may visit another of my site Mastering Mathematics II. Please be patient while more content is being written. Thanks for your support in advance!

Come 2009, I will write some articles to address learning difficulties of students taking the IB (International Baccalaureate) HL (Higher Level) mathematics. For a start, I found a set of IB mathematics problems for you :)

Afternote: I reckon I do not have to reinvent the wheel as I have just found a nice site written by a HK teacher on IB Maths, which I hope it will serve your needs :) Here is yet another great site that is full of resources :D

Many students of mathematics face much difficulties understanding the deep subject, including myself. Keep trying and don't give up without a good fight! It is a matter of time before you finally succeed. Perhaps this recent success story from the Beijing Olympics will inspire you, as it has for me.

We witness yet another struggle for success from the just-concluded inaugural night-race F1 Singapore which saw Alonso's miraculous climb from 15th position to become the eventual race champion. All is not lost, unless one thinks it is so.

Please drop me an email, if you'd like me to write an article for a topic.

The content is organised by topics in the H2 Mathematics syllabus. Key points are highlighted for each topic, accompanied by illustrated example(s) and/or practice question(s) and/or useful Internet link(s).

You will need Adobe Reader to access some content.

The webmaster (Wee Wen Shih) is currently pursuing his postgraduate degree by research in Mathematics at the National Institute of Education (NIE). His research interest is "Convergence Structures and Domains" and he is mentored by A/P Zhao Dongsheng. 

Wen Shih is a trained and experienced Mathematics teacher. Currently, he teaches part-time at NIE in 3 content courses (i.e. DSM100 Number Topics, DSM101 Geometry Topics, ASM401 Further Mathematics Topics) that are offered to trainee teachers.

Wen Shih has published A-level Mathematics solution books and ten-year series books with Dyna Publisher. In his free time, he enjoys reading books of all kinds and engaging in endurance sports. In September 2007, he completed the inaugural Singapore Aviva Ironman 70.3 in 6h 31m (view a race picture).

Please direct burning questions, if any, to him via email.

Both learners and instructors may find this learning model practical. Learning typically progresses through these stages:

• Unconscious incompetence - this is the stage where you are not even aware that you do not have a particular competence.
  
  
• Conscious incompetence - this is when you know that you want to learn how to do something but you are incompetent at doing it.
  
  
• Conscious competence - this is when you can achieve this particular task but you are very conscious about everything you do.
  
  
• Unconscious competence - this is when you finally master it and you do not even think about what you have, such as, when you have learned to ride a bike very successfully.

Have a good break!

Students often wonder what goes on after their exam scripts have been submitted.

In any case, scripts will be marked objectively according to a mark scheme prepared by the examining body. In general, several types of mark are awarded in the subject of mathematics. These are called M (for method), A (for accuracy which depends on M), B (for method and accuracy independent of M) and E (for explanation) marks. For finer details, you may wish to refer to the 2nd & 3rd pages of a real mark scheme.

For school assessments, I would encourage students to consult their teachers to find out where marks have been lost, with the objective to improve their grades.

How does an examining body set standards that are maintained consistently and objectively over time? Here is a detailed explanation (from a UK examination authority) for both teachers and curious students to read about.

There are many key words you often encounter in mathematics exam questions. Please familiarise yourself with these words, and share them with your friends. Thanks!

Some examples of process words (or key words) and their implied meanings are:

1. 'Write down', 'State', 'Give', 'Express/Express briefly', 'List', 'Specify' mean write down without justification, i.e. no working need be shown (although you may include appropriate working if it helps you).

2. 'Find', 'Calculate', 'Determine', 'Simplify', 'Derive', 'Solve', 'Evaluate' 'Transform', 'Expand', 'Factorise', 'Differentiate', 'Integrate' mean work out and show your working, using standard results and techniques.

3. 'Prove', 'Show', 'Deduce', 'Explain', 'Indicate', 'Justify', 'Demonstrate', 'Determine', 'Decide (if, whether ...)', 'Verify', 'Confirm', 'Test', 'Predict', 'Illustrate', 'Identify' mean you must justify each step and provide a convincing argument.

4. 'Assume', 'Consider', 'Suppose', 'Apply', 'Use', 'Define (in terms of ...)', 'Sketch', 'Draw', 'Plot', 'Graph', 'Compile (a table ...)', 'Make (a table, list, ...)' indicate you must answer the question in a particular way indicated by whatever words immediately follow each of these cueing terms.

5. 'Explore', 'Investigate', 'Devise', 'Design', 'Obtain', 'Find', 'Adapt', 'Construct', 'Produce (an algorithm, argument, diagram, ...)', 'Translate (from, into, ...)' indicate doing a mathematical activity and then reporting on the process and result.

Source: Success with Mathematics by Heather Cooke, pp. 49

Here are relevant questions from the June '08 exam, to boost your mathematical confidence further :) Keep going!

Download the set of questions and share them with your friends and classmates.

These are the numerical answers I have worked out, for your reference.
NOTE: The conclusion to the question on hypothesis testing should be "insufficient evidence...that people...will lose an average of more than 3kg in a month". Sorry for my careless mistake! Thanks to Nee Huai and Steven for pointing out the error :)

Casio GC users may need to carry out linear interpolation to find inverse t. Find out how.

You may refer to Jan '08 AQA (i.e. an exam board in UK) exam questions and mark schemes from this site. Look at relevant questions (by now, you should be able to identify them) from Pure Core 2 - 4 and Further Pure 1 - 4 papers.

Exam fever is on. You are understandably anxious and raring (or not, perhaps) to score that elusive pass or impossible distinction grade.

Read this article for a better understanding of math anxiety and strategies to overcome it here and now. Think Nike: Just Do It!

It is important for a student to apply these steps (by George Polya) actively when he/she encounters a problem in Mathematics:

• Understanding the problem.
• Devising a plan.
• Carrying the plan.
• Looking back.

For details of each step, you may like to refer to this link.

I'd like to quote some interesting points that Polya made in a speech in the late 1960s, which I feel, are still relevant today:

• The most important action of learning is to discover it by yourself.
  
  
• Learning begins with action and perception, proceeds hence to words and concepts, and should end in good mental habits.
  
  
• The solution to a problem naturally starts always with a guess -- not always with a good guess. On the contrary, usually the guess is never completely good. It is just a little out of center and the art of problem solving consists in great part in correcting your guesses.

This website discusses ten practical ways. I would like to add some personal comments.

• Figure out the big picture: Can you summarise key concepts and approaches? Do you know what skills tend to be assessed often? Can you see interconnections between topics?
  
  
• Don't join the blame game: Don't blame your lack of ability or your past failures/bad experiences too. Drop the negative thinking, pick yourself up and just do this -- continue to work towards improvement with confidence, faith and optimism. Remember that your best is yet to be!
  
  
• Practice makes perfect: You will go very far if you practise questions with awareness. Be in the know of your lack of understanding of certain concepts and approaches, of your misconceptions, of your carelessness in working and of the types of question that appear frequently. Also, you should plan to have some practices carried out under time-constraint and without you referring to notes. If you have received feedback from teachers about bad handwriting, work diligently to improve on it as well.
  
  
• Time management: How much time should you allocate for one mark during a test or an examination? For example, it will be 1.8 minutes per mark for a 3-hour, 100-mark paper. If you encounter difficulty in any question, do not mull over it for too long. Just skip it and go on with another question, keeping in mind your goal of completing the paper within the time limit and scoring the maximum possible score.
  
  
• Don't fall into the trap of copying from a friend to survive: It will be better to have a really good discussion with him/her about the work he/she has shared with you. This may help you to gain initial understanding for you to begin your own work. Find out from your friend, too, what works or does not work for him/her in the learning of mathematics.
  
  
• Think "I can", then you will go far in the learning of Mathematics!

Pick up useful pointers to learn mathematics from this article. They include:

• Make connections between facts.
  
  
• Never memorise whole sections of notes.
  
  
• Follow the rules initially. Then strive to do the mathematical task automatically.
  
  
• Do lots of examples.
  
  
• Identify your difficulty.
  
  
• Learn the shape of any proof.
  
  
• Focus on useful activities during revision.