This is the title of a book I have recently purchased through the AAMT website. It pre-dates my other all-time favourite resource – Thinkers. Both are authored, in part, by John Mason and Anne Watson, of the UK.
It sets out some questions which can be asked about mathematical topics. These questions deliberately reflect and encourage noticing of the internal structures of mathematics and promote mathematical thinking (in both students AND their teachers!). It offers practical ways in which the culture of transmission that operates in many mathematics classrooms can become closer to a culture of thinking. It is firmly bound, and re-inforced, by educational theory.
For example, questions about geometry reflect the 5 sequential levels of awareness that take students from concrete thinking to more generalised thinking:
- descriptive analytic (distinction making…how is this the same as…how is this different to..)
- informal deduction (properties)
- formal deduction (relationships)
- rigour, meta-mathematical (structure)
Interestingly, in Hatties’ book, Visible Learning, he notes that the relationship between purposeful Piagetian programs (that deliberately plan instruction to move from the concrete to the formal thinking) and achievement is very high, particularly for mathematics. (p 43). I’ve always liked the Socratic nature of teaching through planned questions that ‘ratchet up’ the thinking as the lesson progresses. You can start with questions that determine ‘entry level’ and those that address misconceptions then prompt students to interconnect and interweave their constructed meanings with new knowledge.
Yes, the learning can be ‘scaffolded’ by purposeful use of questions by teachers but there must also be ‘fading’ or the engagement in the learning never moves from the teacher’s brain to the student’s. Teachers need to know when to intervene and when to ‘fade out’. If students don’t learn to ask themselves the ‘right’ questions to get them going on an unfamiliar and challenging learning task then we have failed as educators, in my view. As John Mason says in this book, ‘students are likely to remain dependent on the teacher asking certain questions. Thus they are only being trained in dependency’.
The questions in the book have been constructed to illustrate the 6 overarching aspects of mathematical thinking:
- Exemplifying and Specialising
- Completing, Deleting and Correcting
- Comparing, Sorting an Organising
- Changing, Varying, Reversing and Altering
- Generalising and Conjecturing
- Explaining, Justifying, Verifying, Convincing and Refuting
(1) For Completing, Deleting and Correcting
- To Use a Definition
Which statements can be deleted? A rectangle has four right angles, at least three right angles, at least two right angles, at least one right angle, two pairs of parallel sides, at least one pair of parallel sides, two pairs of opposite sides equal, diagonals that bisect, interior angle sum of 360 degrees
- To Use an Example/Counter-Example:
What can be done to 4/5 = ? in order to show that ‘division makes smaller’ is not always true?
(2) For Changing, Varying, Reversing,Altering
- Change one aspect of the example so that..
the equation y = 2x + 1 so that its graph decreases as the x values increase
A valuable resource for all teachers of mathematics. Highly recommended.