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Thinking Mathematically

The excerpts below make up one of the most powerful lessons you will find here at this blog.  The reason they are being posted is in connection to my current series, Your Daily Recruiting Journal.  It is one of the greatest failures of education today that most who read the following excerpts will have NEVER thought of mathematics this way before.

The word “sad” is simply too small, too weak, not adequate to express the loss we all suffer as a result.  Mathematical thinking, as you will read about it here, is perhaps the single most powerful thing a human brain can be used to do.

Let me express it this way.  Why do you want to build your journal writing skills?  In order to solve problems in your life and work.  You want to solve all problems that hold you back from filling your potential, hitting your goals, and rising to become greater than you have ever been, continuously, between now and when you die.  Your daily journal is the place to invest into your performance, you future and the powers of your mind itself.  So, any guide that empowers you to use your mind better is worth…what?…how much?  What is it worth to you to be able to strengthen your mind’s abilities?

Consider that question the other way round.  How much cost would be too much; what price would be too great to pay in order to improve your mind’s powers, in order to set up the foundation for perpetual performance growth for the rest of your life?  Whatever that price is, the one too high to pay…you won’t be required to pay that price here.  Here, it is free of dollar cost.  It is not free though, because you’ll have to invest time to read, focus and effort to learn and execute.  So, those prices you will have to pay.  But, as you can tell, my promise is that you will reap such infinite rewards you’ll find the price meaningless in comparison.

There is, however, a dollar price I do recommend you invest, too. That is simply the dollar cost of buying the book itself.  Here’s a link for that, if you’re sold:

Last point.  The series you read here, Your Daily Recruiting Journal, is a living conversation at our LinkedIn discussion group.  You should consider going over there and joining us live!



Excerpts From Thinking Mathematically

By John Mason with Leone Burton & Kaye Stacey



Thinking Mathematically is about mathematical processes, and not about any particular branch of mathematics.  Our aim is to show how to make a start on any question, how to attack it effectively and how to learn from the experience.  Time and effort spent studying these processes of enquiry are wisely invested because doing so can bring you closer to realizing your full potential for mathematical thinking.

Experience in working with students of all ages has convinced us that mathematical thinking can be improved by

tackling questions conscientiously;

reflecting on this experience;

linking feelings with action;

studying the process of resolving problems;


noticing how what you learn fits in with your own experience.

Consequently while encouraging you to tackle questions, we show you how to reflect on that experience by drawing your attention to important features of the process of thinking mathematically.


How to use this book effectively!

Thinking Mathematically is a book to be used rather than read, so its value depends on how energetically the reader works through the questions posed throughout the text.  Their purpose is to provide recent, vivid experience which will connect with the comments that are made.  Failure to tackle the questions seriously will render the comments meaningless and empty, and it will be hard to use our advice when it is needed.  Three kinds of involvement are required: physical, emotional and intellectual.

Probably the single most important lesson to be learned is that being stuck is an honourable state and an essential part of improving thinking.  However, to get the most out of being stuck, it is not enough to think for a few minutes and then read on.  Take time to ponder the question, and continue reading only when you are convinced that you have tried all possible alleys.  Time taken to ponder the question and to try several approaches is time well spent.  Each question is followed by suggestions under the heading STUCK? to provide signposts when progress seems blocked.  Because different resolutions follow different paths, some of the suggestions might be mutually contradictory, or irrelevant to your approach, so do not expect every suggestion to provide immediate insight!

Recalcitrant questions which resist resolution should not be permitted to produce disappointment.  A great deal more can be learned from an unsuccessful attempt than from a question which is quickly resolved, provided you think about it earnestly, make use of techniques suggested in the book, and reflect on what you have done.  Answers are irrelevant to the main purpose of this book.  The important thing is to experience the processes being discussed.

(Pages ix and x)


Writing yourself notes

Before we go on….I wish to introduce a technique for recording mathematical experience.  The reason for introducing it now is that you should begin now to record your experiences so that they are not lost, but can be analyzed and studied later.  Recording your experiences will also help you to notice them and this contributes to developing your mathematical thinking.  Aim to record three things:

all the significant ideas that occur to you as you search for a resolution to a question;

together with

what you are trying to do;

your feelings about it.

Obviously this is a tall order, but it is well worth attempting.  In particular, it gives you something to do when you get stuck – write down STUCK!  Recognizing that you are stuck is the first step towards getting out of it.

Writing down the feelings you have and the mathematical ideas that occur to you will destroy the stark whiteness of the piece of paper that confronts you as you begin a question.

Once a start has been made, ideas often begin to flow more freely.  Then it is important to write down what you are trying to do as it is easy to lose track of your approach, or the reasons for embarking on some long calculation.

(Pages 10 & 11)


I suggested before {while working on a specific example on pages 1 – 16}… that you write down your thoughts, feelings and ideas.  You may have decided that it was not necessary for one reason or another, and so not bothered.  If you did not write things down, then you missed an opportunity to learn something about yourself and about the nature of thinking.  I recommend that you take the time to work through most questions conscientiously.  If you did try it, I suspect that you did not find it easy.  It does at first seem awkward and unnecessary, but a little self-discipline at this stage will reap rewards later.  To make it easier to write helpful notes, I am going to be more specific about useful things to write down.  This added detail or structure can then become a framework to assist your mathematical thinking.  The framework will give you continuing assistance if you get it inside you and make it your own.  Otherwise it can provide only vague, temporary help.

The framework consists of a number of key words.  As you use these words they become endowed with associations with past thinking experience, and through these associations they can remind you of strategies that worked in the past.  In this chapter four key words are suggested, and in Chapter 2 they will be considerably augmented.  The whole framework of key words is called a RUBRIC, following the medieval custom of writing key words in red in the margins of important books.  The activity of writing yourself notes I call RUBRIC writing.

The four key words that I suggest you begin using in your notes and in your thinking are




Whenever you realize that you are stuck, write down STUCK!  This will help you proceed, by encouraging you to write down why you are stuck.  For example:

I do not understand . . .

I do not know what to do about . . .

I cannot see how to . . .

I cannot see why . . .



Whenever an idea comes to you or you think you see something, write it down.  That way you will know later what the idea was.  Very often people have a good idea, but lose it subsequently and cannot recall it.  In any case, it feels good to write down AHA!  Follow it with

Try . . .

Maybe . . .

But why . . .



Check any calculations or reasoning immediately.

Check any insight on some examples.

Check that your resolution does in fact resolve the original question.



When you have done all that you can or wish to, take time to reflect on what happened.  Even if you do not feel that you got very far, it helps to write up what you have done so that you can return to it freshly and efficiently at some later date.  It is also the case that the act of summarizing often releases the blockage.  There are several things worth noting particularly:

Write down the key ideas.

Write down the key moments that stand out in your memory.

Consider positively what you can learn from this experience.

I strongly recommend getting into the habit of RUBRIC writing when working on any question.  You may wish to change the key words to suit yourself, but what is important is to develop rich associations with the words you use so that they conjure up the more detailed advice being offered in this and subsequent chapters.  You cannot possibly memorize all the helpful advice that could be given.  Instead of relying on someone else to get you unstuck with some timely advice, you can draw on your own experience.  A RUBRIC is the means for drawing on that experience, and Chapter 7 discusses how the links between detailed advice and RUBRIC words are forged.

The RUBRIC should not be followed slavishly or dogmatically.  Rather, with a little practice, the RUBRIC phrases will arise naturally, identifying what is to be done and suggesting what might be done.  Sometimes you can be almost afraid to write down an idea in case you lose hold of it as it is forming itself in your head, so do not rush into writing.  It is true though that having these standard key words automatically available helps in pinning ideas down.  Conversely, try to avoid writing down scraps of ideas haphazardly in random places on a page.  Everyone finds it hard at first to write RUBRIC notes, but those who persevere find it a real benefit.

(Pages 17 – 19)


Chapter 3 Responses to being STUCK

Everyone gets stuck.  It cannot be avoided, and it should not be hidden.  It is an honourable and positive state, from which much can be learned.  The best preparation for being stuck in the future is to recognize and accept being stuck now, and to reflect on the key ideas and key moments which begin new and useful activity.




There are several different feelings connected with being stuck.  For instance I can find myself

staring at a blank page, at the question, or into space;

resisting a computation or some other action;

growing tense or even panicking because I cannot make progress;

feeling frustrated because nothing seems to be working;

to mention only a few.  In my experience, I have usually been stuck for some time before I become aware of it.  At first the awareness is hazy and indistinct.  It slowly grows until I am definitely both stuck, and aware of being stuck.  Only when I feel stuck and I am aware of my feelings can I take action.  That is why RUBRIC writing is so useful, and particularly the writing of STUCK! or something similar.  The act of expressing my feelings helps to distance me from my state of being stuck.  It frees me from incapacitating emotions and reminds me of actions that I can take.

So what can be done about being stuck?  Having recognized and accepted that you are STUCK, you can either stop working altogether, take a short break, or keep going.  Although it is often inviting to give up when you are stuck, it is not always the best idea.  Good ideas often come just when it seems most hopeless.  If you are going to take a break, remember first to record as clearly as possible what it is that you think is blocking you.  There will be more on this topic in Chapter 6.

(Pages 49 & 50)


Chapter 7 Developing an internal monitor

Each of the previous six chapters has offered advice on how to think mathematically, but having advice printed in a book is not very useful when you find yourself stuck in the middle of an investigation.  Trying to select from the book which bit of advice is most appropriate is tantamount to resolving the original question!  You may have noticed that whenever possible my suggestions have been in the form of questions or exhortations.  The reason is that specific ‘hints’ remove the opportunity for you to do the thinking, and mask the important points which are the impulses which produced those ‘hints’.  Furthermore, the whole nature of ‘hints’ suggests a view of mathematics as a bag of tricks that have to be discovered or revealed.  I find this attitude inappropriate and unacceptable.  When you are stuck, and even when you are not actually stuck, what you need is a tutor to ask you a helpful question which gets you going again.  This chapter describes the process whereby your own internal tutor can grow in strength and effectiveness.

The main thrust of the book so far has been to try to relate my advice to your experience.  The purpose of the many questions embedded in the text has been to provide specific, direct experience of thinking so that my advice becomes associated with your experience.  In that sense, REFLECTING is probably the most important activity to carry out.  It is sometimes said that

the only way to learn is from experience

but experience alone is not enough.  The experience must leave its mark.  REFLECTING on the key ideas and key moments intensifies the critical moments of an investigation and helps to integrate their resolution into your thinking repertoire.

Chapter 5 introduced the idea of an internal enemy or skeptic who seeks holes in your arguments.  This chapter develops the notion of an internal skeptic into an internal monitor who is always present, and who has a number of roles to play.  After describing these roles, the chapter concentrates on the mechanism by which such a monitor can be encouraged to grow.

(Page 115)


List of section headings for the rest of Chapter 7:

Roles of a monitor

Emotional snapshots

Getting started

Getting involved


Keeping going


Being skeptical