How did you solve this problem? “Assuming everyone selects a number at random,” you think, “the average would be 50. Two-thirds of 50 is about 33. I should select that!” But your other mental voice interjects, “Ah! But everyone else is thinking the same! Therefore, they all select 33. Two-thirds of 33 is 22. That’s the number...” “Wait!” Back and forth your mind lobs. “I can see this continuing...I’ll select 0.” Phrased in mathematical jargon, the above problem is asking you to find the Nash Equilibrium of the given situation. If we had written the problem in such terms, some of you would not have made any progress while others would perhaps have selected 33 or one of the numbers along the sequence. Still others would have seen that the answer is 0 and engaged in the type of iterative thinking caricatured above. The present article expands on this last type of experience, of solving a mathematical problem by first seeing a resolution and a path leading to that resolution unfold into the near, problem-solving future. The above game is seemingly a small, one-off problem. We claim, however, that this kind of thinking—seeing a resolution and way to that resolution—is a general phenomenon in certain mathematical work, that in which no solution schema is readily available. There is a sense in which mathematicians (broadly construed as users of mathematics) engaged in this type of work are imagining future activity and this “future thinking” drags their present actions forward, motivates them, and helps them to persist. Where do these abilities come from? Experience? What aspect of experience? Can these abilities be taught? Perhaps the question that first needs asking is: how can we best characterize these future-thinking abilities? There are many ideas in the mathematics education literature that may aid in a description—intuition, strategic knowledge, an aesthetic sense, experience, heuristics, problem solving, meta-cognition, and many more. We argue here that none of these existing constructs quite captures a mathematician’s future-thinking processes. Though each of these constructs can weigh in on features of future thinking, they each have come to encompass so much and in such diverse mathematical settings that we find we desire a precise, restricted description of future-thinking processes in mathematics. In this article, we therefore introduce and elaborate the construct of mathematical foresight, drawing on a series of interviews with mathematicians to do so.

This study examines the language factors in senior secondary mathematics for students who do not have English as their first language. It involved 40 Years 12 and 13 Chinese students at Auckland Girls’ Grammar School. Parallel tests in English and Chinese were given to the students some weeks apart. The study indicates that these students experienced, on average, a 15 percent disadvantage in overall performance in the English test compared to the Chinese test. Some specific language features causing difficulty emerged from the analysis. Background Auckland Girls’ Grammar School is a decile 4 state school situated within a 2 kilometre radius of the Auckland CBD. Our focus was the difficulties our Chinese Mandarin speakers encounter when learning mathematics in English. The girls all spoke Mandarin as a first language. Their length of time living and studying in New Zealand varied between less than 1 week and up to 5 years. A large majority of these Mandarin speakers had already studied up to Year 11 mathematics in China. There was a big variation in the students’ mathematical abilities in their first language. We believe their abilities could be modelled by a normal distribution. This needs to be considered, as a common perception is that “Asian” students are typically well drilled in mathematics problems. The students’ life experience is also a factor when considering their responses and ability to solve mathematical problems based in a New Zealand English context. The teacher/researchers, Jushi Hu and Anne Blundell, both qualified as secondary mathematics teachers in 2002. Jushi is a native Mandarin speaker with 12 years’ experience lecturing mathematics in China. She has been teaching for 4 years at Auckland Girls’ Grammar School. Anne is a native New Zealand English speaker with 3 years’ teaching experience in New Zealand and 1 year in England. She has been teaching at Auckland Girls’ Grammar School since February 2004.

(a) Colette Laborde Le neuvieme congres international sur l'enseignement des mathematiques (ICME-9) s'est deroule dans un ensemble complexe de bâtiments destines a recevoir de grandes manifestations d'aspect futuriste, architecture que Jules Verne se serait complu a decrire avec talent. Makuhari, centre de convention de taille gigantesque, accueillait en effet le congres ICME-9. La geometrie des passerelles de niveau multiple et des edifices verticaux fournissait un objet d'etude inepuisable pour tous les amateurs de geometrie 3D ou de reperage dans l'espace. Les 2069 participants d'ICME 9 (venant de 79 pays differents) se sentaient un peu perdus dans ces locaux prevus pour des masses plus importantes. La plus faible participation (les congres de Seville et de Quebec avaient attire 4000 et 3000 participants) a confere au congres un caractere plus intime qu'a l'habituel et apres deux jours de congres, chacun etait assure d'avoir rencontre tous ses amis, collegues et connaissances, ne serait-ce qu'a l'occasion de la Happy Hour qui se tenait le soir de 19h a 21h, dehors dans le parvis au pied d'escaliers menant au grand hall. Un autre avantage de l'environnement tenait certainement a la qualite l'equipement audiovisuel mis a la disposition des conferenciers et groupes divers. Comme prevu, mes journees ont ete bien remplies entre 7h 30, heure a laquelle je quittais Tokyo pour prendre le metro puis le train, et 22h 00, heure de retour a Tokyo. Un peu plus de deux heures de transport, ou je surveillais avec anxiete les noms des arrets du metro et du train quand je croyais etre proche du lieu de changement ou d'arrivee. Les deplacements ont ete nombreux aussi pendant la journee et s'il reste un souvenir fort du congres c'est bien celui de marcher a vive allure pour rejoindre un groupe de travail ou une conference, tout en saluant au passage les nombreux amis et collegues. Des moments calmes ou je pouvais etre assise, je retiens quelques points forts. Une conference particulierement reussie sur tous les plans a ete la conference pleniere de T. Nunes qui cloturait de facon heureuse le congres: 'How mathematics teaching develops pupils' reasoning systems', dans laquelle elle a exploite l'approche vygotskienne de signe et d'outil dans l'apprentissage de la division. Clarte du propos s'appuyant sur un diaporama Power Point, insertion de videos montrant des enfants resolvant des problemes, ... Une autre conference brillante a ete la conference reguliere de C. Alsina sur la geometrie de Gaudi, architecte catalan, concepteur de la Sagrada Familia et aventurier audacieux dans l'etude des formes fondee sur une grande connaissance de la geometrie. Les aspects culturels ont aussi ete fortement presents dans la conference de T. Osamu ('Some characteristic features of Wasan, the Japanese traditional mathematics') sur les mathematiques Wasan, c'est-a-dire les mathematiques japonaises, developpees en dehors de toute influence occidentale avant l'ouverture du Japon a l'ere Meiji. Diversite et variete dans le Working Group 1 1 sur l'usage des technologies (groupe de taille geante, il a attire un grand nombre de participants, en particulier parce qu'il etait le seul sur ce theme) et le Topic Group sur la geometrie plus modeste en audience: presentation des usages varies de differentes technologies (calculatrices, tableau blanc electronique interactif, tableur, ...) dans le premier groupe, experimentations avec des eleves sur l'apprentissage de notions specifiques, exposes theoriques sur les processus d'apprentissage en geometrie ou les aspects epistemologiques pour le second groupe. Grande variete de propos aussi au groupe international Cabri-geometre qui a bien reflete la diversite culturelle de ses participants. Les exposes tenus ont porte sur les environnements de geometrie dynamique, leur usage dans differents pays du monde, leur integration dans le curriculum, aussi bien que les nouveaux problemes conceptuels rencontres par les eleves dans leur usage, ou les modelisations rendues possible de proprietes mathematiques (theoreme des residus, transformations conservant l'aire, 5eme probleme de Hilbert). En conclusion, un congres de contrastes, un lieu futuriste et des danses traditionnelles, des locaux immenses et un nombre plus restreint de participants qu'habituellement, de nombreux 'anciens' mais aussi beaucoup de nouveaux participants des pays d'Asie.

Anyone with an interest in both mathematics and linguistics must grin with rueful fascination as they watch the way that the term 'ethnomathematics' is thrown about both in the public media and in the mathematics education literature. It is a versatile toy, fulfilling several functions in an on-going game of catch. Since the 1984 ICME address by Ubiratan D'Ambrosio (1985) which is widely regarded as establishing the field in its contemporary form this word has been used in the political games of curriculum development in many countries. It has also:

The study of mathematical discourse in different languages offers insights into alternative mathematical conceptions. It also highlights the role of imagination in the development of mathematics, and raises questions about the learning of mathematics. It is concluded that there are pedagogical opportunities to be exploited in the relations between language and mathematics. In particular, there are language-derived alternative ways of approaching aspects of conventional mathematics. In this paper, recent research into the everyday discourse of quantity, relationships, space, and change in some non-Indo-European languages is described. In the second part three of these examples are transformed into practical classroom activities.