Title is a quote from Shakespeare’s Macbeth.
Whilst we may not be able to attain our ideals in the teaching and learning of mathematics, I believe it is important to have a vision of the way you want things to be; something to aim for.
My vision for the teaching & learning of mathematics:
We need to develop processes and approaches which ask “What are the powerful ideas and processes ..that we believe are important for young people to learn ?”within each topic we teach.
We need to start with our focus on desired understandings, the big ideas, and then ask “What information and what experiences do I engage the students in, in order for them to develop these understandings ?”
The focus of assessment needs to be in terms of the further question “If students understand this idea, how can they demonstrate it ?” 
The task of the teacher is not to put knowledge where it does not exist ,
but rather lead the mind’s eye so that it might see for itself. 
Human understanding requires processing in the individual’s mind. You can transfer information (rules, procedures, formulas etc.) but understanding has to be developed. Processing for understanding cannot be left to chance – we have to provide learning opportunities that develop the students’ capacity to construct their own understandings.
A “standard’ lesson before the 1980’s consisted of teacher led theory at the blackboard, examples so that students could attempt the homework then individual practice on problems that were frequently frustrating and had no perceived relevance to the world students occupied. A dull, tiresome subject full of drilled routines, rules and testing that served to confirm students’ beliefs that they couldn’t ” do Maths “. It had (has ?) an aura about it that only served the needs and interests of the chosen few that had been initiated into its secrets. It provided a vocational basis but was really more useful as an intellectual sieve ; sifting the academically capable from the rest. Has much changed?
We have seen a great change in the way mathematics is assessed in Year 12 and hence taught. The main thrusts of this change have been problem solving (the use of mathematical methods in unfamiliar situations) and modelling (the applications of mathematics in real life situations).
Mathematics, more than any other subject, can present students with the stark alternatives of success and failure. Many students have “learned”, as opposed to “learning”, difficulties with the subject. By being taught that there is an accepted way of doing problems to obtain the ‘ right ‘ answer without discovering the method for themselves, students can remain passive and blocked in a school situation. If the student passes the test he/she will obtain some reward but the student not knowing how to acquire the skills needed to pass the test ( and these are not necessarily mathematical skills ) has only the opportunity to be further convinced of his/her inadequacy and reinforce his/her idea of being a failure at mathematics.
We are the students’ role models and we should act in class the way we want them to think. If we are uncomfortable taking risks (such as admitting we don’t know the answer) then we cannot expect our students to do so – we must model the behaviours we want them to attain. We need to search for the answers with the students and encourage the development of effective thinking strategies rather than just present them with these answers and methods. We need to use problem solving as a teaching method rather than a marginalised activity and not be satisfied with translating a text book into more user friendly language…ie. the use of other resources apart from a text should be expressly encouraged. We need to give the students opportunities to explore, to discuss, to investigate, to reflect, to formulate, to criticise, to develop, to create, to go wrong and try a new attack – in short, to learn mathematics by acting like mathematicians….an activity we could discourage by being ” the expert ” at the front of the room ( “sage on stage” as Julia Atkin calls it ), creating a dependency and directing their learning experiences according to a strategy that we are aware of but they are not.
The former behaviours listed are the ones we should be trying to promote – examples include using questioning to elicit quality thinking, using “mind journeys” (a visualisation technique), drama, physical activities, technology etc. We should want our students to actively engage in the doing of mathematics. Not always providing the immediate “how to” answer that students of maths sometimes demand is threatening to them as they must fall back on their own resources and it is also threatening to teachers who fear retribution along the lines of..” You are paid to teach us – so do it !“ It is also far easier to ‘tell’ rather than ask another question and lead students to the realisation that they hold the key to a solution within themselves. We must teach for understanding – understanding that it is a language in which we can describe the world’s patterns and relationships; a language we can use to formulate mathematical models which we can then use to predict things ; a language that is actually understood and used appropriately by students rather than repeated parrot-fashion.
We should also want our students to become more autonomous learners; more in control and responsible for their own learning. Self-knowledge as learners through the use of various forms of self-assessment or reflection should be encouraged.
Julia Atkin suggests the following questions as a guide for students reviewing the work covered:
· What do you know and what do you want to know ?
· What’s the ‘big idea’ ?
· What are some of the little ideas ?
· What can you do now that you could not do before ?
· How did you learn that ?
· Compare how you learned it with how others learned it
· What do you understand now that you did not understand before ?
· What do you want to know now ?
 Taken from Julia Atkin’s Article : Enhancing Learning with Information Technology November 1997
 Plato ( quoted in above article)