**Summary.**The aim of this contribution is to present some innovative aspects of teaching mathematics at secondary level in order to face two key issues an innovative teaching: why and how.

*To Paolo Abrantes, in memorian*

## 1. Introduction

Our aim in this presentation is to face the interesting questions of “why?” and “how” we can approach innovative teaching of mathematics at secondary level. Following the main conclusions of the Topic Study Group 14 at ICME-10 that Anne Watson and myself coordinated we will describe different ways of innovative actions in the classroom and their pedagogical objectives.

Of course the first remark that we need to make is on the word “innovative”. When we talk about innovation we are thinking in ways of teaching and learning different from the traditional exposition-and-exercise practices, the dominant use of textbooks or programmed published materials. But, of course, innovation is always a relative issue that many depend of the country’s traditions, the social-educational paradigms, etc. (Steen, 1998, 2001).

## 2. Innovations in the curriculum

We need to teach well what people need to learn. But learning needs are related to people’s future. This claim justifies that curriculum must be sensitive to changes. First of all we may consider the innovation of goals and contexts, i.e., new concepts and topics to be learnt as well as new competencies to be achieved (Niss, 2001). To work the so-called “big ideas” in mathematics is a must.

In addition to content’s changes, the way we plan to develop the curriculum in the classroom may be also innovative. While the traditional way of teaching follows what we may call a linear ordering of topics and skills it is interesting to explore different strategies like the one of *mixing*the realization of projects, the experimentation with resources, the use of technology and the problem solving approach (Aubanell, 2006). We may consider also the interest of cooperation between teachers of different subjects, giving opportunities to students to learn and practice, at the same time, various topics (e.g., teaching mathematics in a foreign language, working statistics in social sciences, etc.).

Clearly in United States of America there are various projects following NCTM principles (NCTM, 2000) that have been developed in a very interesting way, i.e., Tom Romberg and Jan de Lange’s project “Mathematics in context” (Romberg-de Lange, 1998) and the Arise Project: Modeling Our World directed by Sol Garfunkel (COMAP 1998). At European level curricula from Denmark, the Netherlands, Germany and UK are more innovative than others.

## 3. Innovations in the use of technology and resources

To build the classroom of the XXIth Century we need to open the full integration of today’s technologies in the learning and teaching processes. The traditional scientific calculators (well prepared for secondary schools) have experimented a positive evolution towards calculators which integrate numerical calculus, symbolic manipulation, functional graphs, statistical packages and representation software. No need to say we have today’s computers with very useful software for all kinds of simulations, representations, geometrical constructions (2D and 3D), etc. But beyond the power of all these devices we have at our disposal Internet, i.e., we can integrate today e-mail communication, webs, interactive activities, applets,… and all kind of multimedia resources.

In the classroom we need to use technology for all kinds of mathematical activities. Personal computers and electronic boards will play a crucial role in the innovative uses of technology. But in the learning of mathematics we combine words, symbols, numbers, diagrams and we can use technologies as well as hands-on materials.

Hands-on materials, complementary of technologies, may play also an important role at secondary level. Their role in primary school is well recognized since the contributions of Maria Montessori or Caleb Gattegno. The role of manipulatives at secondary level was a key question in 1954’s European meeting in Madrid organized by Pedro Puig-Adam. Now it seems important to return to this need. As mentioned in (Alsina and Nelsen, 2006):

Experimentation is important in mathematics and it plays a significant role in learning. So by organizing a mathematics laboratory with hands-on materials, or by bringing them in to the classroom one can provide students with materials to help them develop visual thinking in three dimensions [Alsina, 2006]… But in many instances teachers are not confident in dealing with three-dimensional geometry, there is a lack of good three-dimensional models in the teaching resource catalogues, and what is worse, many children end compulsory schooling without spatial literacy.

Some people believe that making models and experimenting may have a role in the early grades, but that it is something to be replaced by more sophisticated linguistic and symbolic descriptions later on, i.e., “real mathematics comes after experimental work». This belief is incorrect, as research has shown that if we do not provide a stimulating reference for abstract concepts, then formal approaches degenerate into merely an intellectual game. Visual thinking is not just an appetizer for the main course of abstraction. Clearly, at certain levels one is restricted to a selection of spatial items, but there are opportunities to offer a broader spatial culture at all ages.

To sum up, the following are four important contributions that hands-on materials make in school classrooms [Alsina, 2005]:

- Hands-on materials may open windows to creative solutions that are impossible using traditional tools.
- Images and hands-on materials may be needed if the problem in question requires an explicit practical solution.
- Hands-on materials may facilitate visual thinking, and constitute a more important step than making plane representations or more formal calculations.
- Images and hands-on materials may be the only feasible way to exhibit examples of or solutions to planar or spatial problems.

## 4. Innovations in teaching and learning processes

The best curriculum plan and the richest use of all kinds of resources are not enough for making an innovative approach to mathematics. We need to promote the engagement of students in their own learning process and to teach integrating various stimulating strategies.

On one hand *research practices, cooperative work, development of projects*and *formative assessment*have shown to be key innovative ways of learning.

At the teaching level George Pólya and Hans Freudenthal pointed out, many years ago, how we may face mathematical modeling, applications and problem solving (Alsina, 2002). In a recent ICMI Study (Blum et al., 2007) one can find many research results in mathematics education showing the interesting use of context, of motivating applications and how the modeling approach give wonderful perspectives at secondary level (see e.g. (Galbraith, 1998), (Matos, 2001), (Pollak, 1997), (Tanton, 2001)).

## 5. Challenging creativity and positive emotional attitudes

Innovative teaching is an open attitude in front of the learning process which has in each generation people of reference. We are not dreamers starting a new adventure. We want to give a high quality education following the rich iniciatives coming from masters like Maria Montesori, Pedro Puig Adam, George Pólya, Hans Freudenthal, Caleb Gattegno, Emma Castelnuovo, Miguel de Guzmán, Paolo Abrantes, etc., i.e., innovation has its own story.

We make innovations because we want to educate creative people, with positive emotional attitudes towards mathematics. At this level we can pay attention to the emotional intelligence of students, i.e., to develop an *affective innovation*. Leonor Roosevelt said once:

* “future belongs to those who believe in the beauty of their dreams”*

We have the dream of sharing beautiful mathematics with all.

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