"ARITHMETIC is generally repulsive," America's most widely read children's author of the 1840s and '50s, Samuel F. Goodrich, observed, "not, as I believe, from any native antipathy, but from a disgust engendered by the disagreeable form in which it is usually presented to children."
The idea that basic elements of math can and should be taught entertainingly, with a dash of wit and storytelling fancy, actually predates Goodrich by a hundred years or more. But how few authors, as nearly anyone can testify from harrowing grade-school experience, have ever realized this elusive goal. An exception working today is formidable, playful, artist-designer- teacher Mitsumasa Anno.
In this, his latest picture book experiment, Anno, in collaboration with his son Masaichiro, has crafted an illustrated puzzle-fable, suited to readers with a knowledge of multiplication, that leads step-by-step to an understanding of the mathematical concept of "factorials."
In a certain china jar, we're told, a small quantity of water mysteriously expands into a sea in which one island lies. On the island are two countries; within each country are three mountains; on each mountain four walled kingdoms stand. As the story telescopes further inward on the Annos' imaginary world-within-a-jar, the particulars of that world continue to multiply according to pattern, up to ten: in each of nine boxes found in each of eight cupboards are . . . ten china jars. Worlds within worlds! Now the question is: how many jars are there in all on the island?
The answer, as the Annos explain at the end of their little fable, is unexpectedly large: 3,628,800, or, in mathematician's language, "ten factorial."
Let us, as the authors also do, quickly define this rather terrifying term.
A factorial is the number arrived at by multiplying any positive whole number--four, let's say--by every positive whole number smaller than itself; hence, "four factorial" equals one times two times three times four, or 24. The Annos' fable turns out to be an ingeniously simple demonstration, carried to ten, of the patterned relationship of numbers known as factorials.
Factorials are worth knowing about partly because they provide the answers to many questions having to do with the number of choices available to us. Three July Fourth orators, for example, planning to speak one at a time, have three factorial or six different ways of ordering their turns.
Factorials rapidly build from relatively small, easily imagined numbers to numbers of forbidding proportions. Ten July Fourth orators can stir their countrymen in more than three and a half million different possible orders. One gains from Anno's Mysterious Multiplying Jar a more palpable sense of the sheer bigness of large numbers and of the relatedness of large numbers to small; and a heightened curiosity about patterned relationships of other kinds within the number realm.
The unpretentious, exquisite craftedness of Anno's illustrations--painterly yet precise, whimsical and lyrical--has always played a more than merely decorative role in his work. Underlying the displays of artist's skill in picture books as various as Anno's Alphabet, Anno's Counting Book and Anno's Journey is the suggestion that pens and brushes, like alphabet letters and numbers strings, are tools by means of which our experience of the world can be enlarged. From this standpoint, Anno's Jar is topheavy, with the fable-portion of the book painted with festive high imagination, but with the latter pages, devoted to definitions and schematic diagrams, uncharacteristically textbook-pallid.
Some readers among the mathematically disinclined will also find the authors' afterword, in which a few practical uses of factorials are touched upon, all too cursory. But other readers, whether at home or school, with or without the aid of elders, will doubtless gather material enough from this Multiplying Jar to keep pocket calculators and personal computers whirring for days or weeks.