Every time that a human being succeeds in making an effort of attention with the sole idea of increasing his grasp of truth, he acquires a greater aptitude for grasping it, even if his effort bears no visible fruit. Simone Weil
In mathematics the art of asking questions is more valuable than solving problems. Georg Cantor
For homework each week I ask tutees to bring questions to the tutorial. Overwhelmingly tutees will bring exercises to practice with, and that is absolutely fine because a part of tuition is to practice routine things with your tutor in order to gain confidence that you are doing those things correctly. And while it is true that the numbered parts of an exercise in a textbook or worksheet are called “questions” these are not questions in the sense which I mean.
What I am looking for are questions that come from you. Questions that arise because you are engaged with the maths that you are studying. In order to ask those types of questions you need to be able to articulate what you understand and don’t understand and that means you need to think about your maths.
How/where/when — often the easiest to answer — look it up in your notes? How difficult can it be?
Why — often applies as there is nearly always a reason for the steps in any mathematical process. Life is not like this!
What — these can be difficult to answer
It is difficult to be definitive about the questions you should ask, or how exactly you should frame them: a why question might, for instance, be framed as a what question, etc. But I feel confident in saying that the way to ask better questions in maths is just to start asking more questions ie you should aim to acquire a questioning attitude in respect of maths.
And notice that more general questions are often better to ask, as the answer will cover a greater number of instances.
Then again specific questions can be very penetrating and enlightening, and are crucial to solving problems.
Example: How do I do this ‘complete the square’ question? — look in your notes — look at some worked examples — what is the difficulty? Often the difficulty is in understanding why certain things are done in the process of completing the square.
Why questions can sometimes be more general — why are we being made to learn this? What is the point of it? See! Why is “completing the square” a thing? There is a reason — find out!
Asking either type of why question (specific or general) will usually be more productive than asking the how question. Reasons/motivations can be obscure; the methods are in your notes.
So perhaps it’s fair to say that most how questions contain why elements and your effort should be made in uncovering the buried why question.
How can be a very relevant question though!
Getting to the why question often involves sorting out what you do understand from the bit that stump you. You are stumped because you do not understand the reason for doing a certain step in a certain mathematical process. If you can have the reason explained to you all is good and you should be able to carry on and do infinitely many problems like the one that just stumped you.
So what do I understand in this question? is a good place to start as it may lead to why do I have to do this next step?
Asking questions is the way to learn most things, and maths is no different. Its content is more abstract, or less concrete than other subjects but you do not have to be a maths genius to learn it, just take your time to understand the reason for doing things, and proceed with a sure footing. You do have to be patient and pay attention in order to achieve this though.
Becoming proficient in numeracy and acquiring a basic understanding of maths is an important lifeskill (as well as being helpful in many professions). Unlike learning to play the violin or learning to speak French, learning maths can have a very broad and profound effect on our confidence in dealing with many life situations. The reason that I say this is because throughout our lives we need to make judgements and decisions where we are dependent on our ability to enquire (ask questions), think, and reason in order to get to the truth of the matter. One of the more important places where you are given a chance to acquire these skills is in your maths classroom.
What is a number? What is an equation? What is 2^3?
The first is difficult to answer, the middle question is a good one at school level, and the last is easy: 2^3 = 2 x 2 x 2 = 8.
What kind of equation is this? How do I solve this equation? Why do I need to solve this equation?
All 3 of these questions could be very relevant to ask (but they are just examples of specific types of questions).
What is an equation? How do you solve an equation? Why do you learn to solve equations?
Notice that these questions are more general questions and therefore excellent to ask!
So whether specific or general, who, what, when, how or why, ask away, and think!
Ad hoc operations
One more thing: learn to distinguish ad hoc operations. Ad hoc means ‘for this’ and algebra, at AH level in particular, is full of algebraic operations done for a specific purpose, say to help simplify an expression. These are things to attend to in class when your teacher carries them out, so please ensure that you understand what is going on. Rules and methods will be in your textbook and will be learned by repetition but ad hoc operations may not be so clear. If you want to raise your profile in class ask about these operations, your teacher will be dead impressed.
“The way Russians teach is that they make sure that every student, when they perform a mathematical operation, they understand why it is performed this way, not just learn how to do it,” Gerovitch says.
For now, at the individual level, Rifkin says her teachers see a difference when their students face a hard problem. They go from a knee-jerk “I don’t get it,” she says, to “Hmmm, let me think…”
A more balanced view of the Russian School: The New New Math: inside the Russian School of Mathematics