This child made it clear that

• She knew that an array was a rectangle
• That this was technically a rectangle
• These super-long folks were not arrays, or at least she didn’t think they were, because they didn’t look like a rectangle
• The 2 x 17 was an array

To what do you attribute this perception? (You can check your answers in the back of the book.)

• So NONE of these are arrays? Or are the 2 x 17 (which she mislabeled as 2×34) arrays?

I’m tempted, if the 2-by’s are arrays, to draw a connection between the mislabeling and the definition of an array that she’s currently working with.

I won’t check my answer ’til I hear back which are arrays.

• mpershan

Ooh, I should clarify: it’s the 1 x stuff that aren’t arrays. The 2 x 17 is, by all accounts, an array.

• Okay, then I believe that the student things an array is something with rowS — like if you have 34 things, they could be in a line, or they could be in an array of 2 rows.

It makes sense to talk of the 17-across by 2-down rectangle as a 34×2 array because it’s 34 things arranged by 2’s, or into 2 rows. Its 2-ness is what makes it an array, the 17-ness is incidental… and its 34-ness is what tells you what it is or how much. 34 and 2 are its salient qualities.

I’m very curious about what she might call 17×17 arrays, or for that matter 34×34 arrays. What if you arrange 17 objects into 17 rows, or 34 objects into 34 rows. Still an array? Or must an array have rowS AND columnS?

• When arrays are introduced to students, is it typical to use them in the 1 by n cases? Or do teachers usually move to the 2 by n type cases first?

It also seems like we should dive deeper into the child’s assertion that the super-long strips are not rectangles. How does she define a rectangle?

• I think it is natural (that is, it arises from early interaction, pre-school) to consider the special case as different from the general case, as in ellipses and circles. I am quite sure that kids have a hard time accepting that a square is a rectangle, and that the term “straight angle” (see CCSS doc) is not an attempt to confuse. Of course, in mathematics, eventually, it is really convenient to treat a square as a special case rectangle. Later on with matrices it is so convenient to treat a vector as a 1xn or an nx1 matrix, and a linear function as a special case of a polynomial, and a pair of straight, intersecting, lines as a special case of a conic section. A one dimensional array is a list ! We are trying to push deep mathematical notions at the kids far too early.

• Alison Hansel

This is really interesting and is something I will definitely keep in mind when I work with young students on arrays. Better give them plenty examples of the “1x” arrays!

I’m wondering about her difficulty in seeing those “1x” arrays as two-dimensional. I know that when we start working with arrays, we are often using tiles, the same sorts of manipulatives students have, until that time, used for simple counting. Earlier they would line up those tiles to model a problem and count off the number. Only the length of that row of tiles is relevant. It only has one attribute or dimension: how many? Perhaps that is playing a part here. Clearly, two rows of something is different. That’s an array with two discernable dimensions. But one row is just that number.

Is the word array so terribly important? What if we used an intermediate word like “arrangement”? Would that give the student a meaningful way to describe these models? W have 17 tiles arranged in 1 row. The next has 34 tiles arranged in 2 rows of 17 each.