%0 Journal Article %1 Falk89 %A Falk, Ruma %A Lavy, Shlomit B. %D 1989 %J Thirteenth annual conference of the international group for the psychology of mathematics education %K bibtex-import cognition grainsofsand ijtme2006 infinity mythesis %P 252--259 %T How big is an infinite set? Exploration of children's ideas %V 1 %X Children were required to compare the size of different pairs of sets. In particular, the set of all numbers was compared to a finite set considered very large by the child. They also located symbols of different sets on a long strip in an attempt both to order them according to size and to represent the gaps between them by the distances. Although children (from about 8 on) generally knew that the set of al. numbers was the largest, the infinite gap between this set and huge finite sets was not conceived until much later.

@article{Falk89, abstract = {Children were required to compare the size of different pairs of sets. In particular, the set of all numbers was compared to a finite set considered very large by the child. They also located symbols of different sets on a long strip in an attempt both to order them according to size and to represent the gaps between them by the distances. Although children (from about 8 on) generally knew that the set of al. numbers was the largest, the infinite gap between this set and huge finite sets was not conceived until much later.}, added-at = {2006-03-07T17:56:12.000+0100}, author = {Falk, Ruma and Lavy, Shlomit B.}, biburl = {https://www.bibsonomy.org/bibtex/2b6e0c85d43e220a77120e502b1425e90/yish}, citeulike-article-id = {532436}, comment = {Comparison of Sets We presented 38 children, aged 5 to 13, with simple questions requiring a judgment of "what are there more of, X or V?", where X and V were replaced, in turn, by members or different sets, like leaves on all the trees in the forests, hairs on all people's heads, grains of sand on earth, etc. The set that was judged larger in each step was next compared to a new one. The "winning" set in that sequence of binary comparisons (assuming transitivity) was then compared, in our target question, to the set of all numbers. The children were also asked to justify their judgments. The percent of children who knew that there are more numbers than members in their largest finite set, and knew why, in each of three age groups, was as follows: 5-7: 29%; 8-10: 72%; 11-12: 83% Location of Sets In the course of our research on potential infinity in children (Falk et al., 1986), it became evident that sometimes children were able to say that "there is no end to numbers," but they were unable to apply the implications of that statement beneficially in a competitive game. Children had often heard from parents or siblings that there are infinitely many numbers, without internalizing the meaning of this contention. We therefore wished to test the understanding that not only are there more numbers than grains of sand, but that there are infinitely more numbers than objects in any finite set. Consequently, we asked the children also to represent the gaps between different sets. This exploratory task was run with 91 children (48 boys; 43 girls) aged 6-12. Method Small cards, each bearing the name of one of the compared sets, were prepared. They were presented after being through with Comparison of Sets. We read together the names of the sets, then we asked the child to arrange the cards from left to right according to their estimated set size. A long strip (ca. 90cm) was now presented, and the child was instructed to carefully paste the cards on the strip so that the spaces between the cards would represent the gaps in the set sizes. Before doing the job, the procedure was explicitly demonstrated and explained with familiar sets like "fingers on one hand" "fingers on two hands" and "all the people of Jerusalem." The children were guided to keep the two first cards very close to each other so as to leave space for the much bigger difference between the second and third set. We tried to explain the task of representing the gaps proportionally by the distances, without using technical terms. Children were also asked to explain their placements. We offered this task halfheartedly, since the correct response to the -task of placing the set of all numbers should be to refuse to do it. No finite strip, whatever its length, can suffice for placement of that card, no matter what scale is applied. Presenting an impossible task to children, who are eager to cooperate and comply with the experimenter's requests, is not a desirable procedure. We felt, however, that if we do only with Comparison of Sets, we run the risk of missing important information. Consequently, one of the most important features of the procedure was our last instruction: "If you feel the strip is not long enough for you, please let us know, and we'll see what we can do about it." If a child did respond that way, we tried to find out why did the strip not suffice. We asked whether an extension would solve the problem, how long an extension, and if not, why.}, interhash = {1a99326e44a1f16d3db2e00c9334606f}, intrahash = {b6e0c85d43e220a77120e502b1425e90}, journal = {Thirteenth annual conference of the international group for the psychology of mathematics education}, keywords = {bibtex-import cognition grainsofsand ijtme2006 infinity mythesis}, pages = {252--259}, priority = {2}, timestamp = {2008-04-27T13:33:18.000+0200}, title = {How big is an infinite set? Exploration of children's ideas}, volume = 1, year = 1989 }