# Down the Rabbit Hole

“The rabbit-hole went straight on like a tunnel for some way, and then dipped suddenly down, so suddenly that Alice had not a moment to think about stopping herself before she found herself falling down a very deep well.”― Lewis Carroll, Alice’s Adventures in Wonderland

Alice had it easy.

In a recent colleague’s blog, I was characterized as “the guy with all the questions.” In my defense, everywhere I look there’s another rabbit hole I need to explore, and then, down I go. Recently, I began to question the development of number as ordinal and cardinal during a child’s development of number. Steffe’s (1994) makes the statement, “In my work with children, I have found an ‘ordinal approach’ to number to be more appropriate” (p.132).

I began to think of our work with children and considering times when their responses indicated an ordinal approach with number. One such instance occurred while asking a first-grader “What happens when you add “one” to a number?”. Her response was “It equals the number after it.” Another time a child was presented with the problem of 21 – 6. She counted backward by ones, while extending her fingers one at a time, and saying 20, 19, 18, 17, 16, 15. She was then presented 21 – 7. She sat quietly thinking for a moment before saying 14. When asked to explain she said, “Well, seven is next to six, so it’s 14 because it’s next to 15.” In these children’s responses, their attention is focused on the relative position of the numbers. Does this imply an ordinal or cardinal understanding of number in their thinking?

The difficulty arises because a child’s mathematical thinking is never simple nor static. Fuson (1988), in her definition of cardinality, states that it refers to the quantity or a total number of items in a set and can be determined by subitizing or counting. A child’s initial understanding of cardinality is when they understand that the last number name said tells the number of objects counted. Fuson (1991) labels this as an initial understanding of cardinality and that “Children’s understanding of cardinality continues to grow throughout the primary grades” (p. 33).

Why is this important? Many teachers work from an assumption that children have the same depth of understanding of quantities that adults do. A child’s ability to provide correct answers to addition or subtraction problems is not a guarantee they have an understanding of the quantity it represents or the parts a number (e.g.that seven can be composed of five and two). As teachers, we need to understand the thinking of young children which is different than that of an adult. A model of the mathematics of children can be developed by interpreting a child’s language and actions. Teachers who understand the mathematics of children can facilitate a higher quality instruction.

In asking questions, rich conversations occur which then send us tumbling into deep investigations of the topics, but in the end improve our understanding of how children come to learn mathematics. That is well worth the journey.

“I almost wish I hadn’t gone down that rabbit-hole—and yet—and yet—it’s rather curious, you know, this sort of life!”― Lewis Carroll, Alice’s Adventures in Wonderland

**Sources**

Fuson, K. C. (1988). *Children’s Counting and Concept of Number.* New York, NY: Springer-Verlag.

Fuson, K. C. (1991). *Children’s early counting: Saying the number-word sequence, counting objects, and understanding cardinality.* In K. Durkin & B. Shire (Eds.), Language and mathematical education (pp. 27-39). Milton Keynes, GB: Open University Press.

Steffe, L.P. (1994). Children’s construction of meaning for arithmetical words: A curriculum problem. In D. Tirosh (Ed.), *Implicit and explicit knowledge: An educational approach*. (pp. 131-168). Norwood, NJ: Ablex.