1. Teaching and Learning Mathematics Laurinda Brown
Gattegno
Teaching and Learning Mathematics
Laurinda Brown

2. A farewell address
People often say: ‘I teach them but they don’t learn’. Well, if you know that, stop teaching. Not resign from your job: stop teaching in the way that doesn’t reach people, and try to understand what there is to do…

3. There is only one instrument in research … and that is to raise questions, to ask questions. To question is the instrument. And if you don’t question, then don’t be astonished that you don’t find anything.

4. This is the state of mind of the learners, that they don’t rest after reaching mastery. They aim at mastery; when they have it, they use it to conquer the next thing…they apply it, they transfer it…

5. The awareness of mathematization
Rather than showing how every part of elementary mathematics may be treated, we prefer to concentrate on the central themes:
educating mathematical awareness and
making students act as mathematicians.

6. How is mathematical awareness educated?
First, as awareness of language and a corresponding notation, then as an opportunity to ask questions about both the language and the notation, to see how far they can be extended. Articulating the awareness provides new mathematical openings for both students and teachers.

7. Mathematization
With awareness comes certainty; for students now recognize that there are real criteria for deciding things, not just a number of seemingly arbitrary activities.

8. Opportunities with geoboards
Encountering difficult challenges that no one may have yet solved - some may turn out to be insoluble in this context, opening up awareness that other lattices, or other material, may be required;
Making definitions that are clearly grasped and verbalised;

9. Extending basic, recurring propositions to become theorems which may be able to be proved valid for extended lattices, with varying degrees of certainty;
Working on the verbal statements offered by students, to neighbours or to the class, in order to chastise the language used and create habits of careful, unambiguous expressions;

10. Developing a sense of constant growth in knowledge of what is being done, can be done, remains to be done, and reclassifying knowledge.
What is important is that we can keep learners very close to everything that is being done, involving them affectively, perceptively, actively, verbally, in fact totally in their contemplation of a challenge.

11. Students start from something perceptible and label it with the help of the teacher; further developments continue based on what can be perceived.
In any consideration of mathematical activity we examine the triad action-image-thought. If we temporarily distinguish one or other of these it is salutary to remember that we do experience them simultaneously.

12. What helps them to learn mathematics? What can they do?
Which numbers give you the most possibilities of making rectangles? Least?
The language - ‘ings’; rotating, accelerating, being