From mrdardy in the comments recently:
Slightly off topic, but I wanted to share a conversation with my soon to be four year old daughter from this past weekend, We were on a long car drive and she was asking how far we were from our hotel. I replied that we were twenty minutes away. Later in the pool she was jumping to me from the pool steps and commanding me to back up some. I asked her how far I should go and she told me to be five minutes away. I said “Do you mean five feet away?” and she replied, firmly, that she meant for me to be five minutes away. I am wrestling with whether I think this is just charming and (semi) clever on her part or whether I need to start answering her pleas in the car with distances. Curious to hear some ideas on this.
I am happy to weigh in here.
Anna Sfard describes knowledge as participation in a discourse and learning as changes in that participation. That is, we can measure whether someone knows something only to the extent that they can talk in ways that adhere to the norms of other knowledgeable people. And when these behaviors change to conform more closely to these norms, we can say that they are learning.
Nowhere is this more clearly demonstrated than in the learning of young children.
The four-year old in question here (let’s call her “Little Dardy”) is trying very hard to participate in conversations about measurement. Measurement, though, is a challenging and rich domain.
mrdardy outlines two scenarios in which the concept of how far comes up for Little Dardy. It shouldn’t be at all surprising—considering Sfard’s model—that she answers a distance question in the same way her father had earlier on. She has taken his example in using units of time to discuss how far something is.
My approach would not be to avoid using units of time to answer the question how far? After all, people do this frequently; it is part of the discourse of measurement.
No, I would use this tension to encourage Little Dardy to think about the two attributes in question here: time and distance. It might go something like this…
Little Dardy: (four years old) Back up, Daddy!
Daddy: This far?
LD: More! You need to be five minutes away!
D: Do you mean five feet away?
LD: No! Five minutes!
D: OK. Tell me when I’m there. But then don’t jump right away; I want to ask you a question before you do. [Daddy backs up slowly...]
LD: OK! There!
D: Right. Here’s my question: Do you think it will take you five minutes to get to me from where you are?
D: Do you know how long five minutes is?
LD: That far.
D: No, no. Can you think of something we do together that takes five minutes?
D: It takes us about five minutes to read [INSERT TITLE OF FAVORITE PICTURE BOOK HERE] together. Do you think it will take that much time for you to get to me?
At this point, I have no idea how Little Dardy will respond (which is what fascinates me so much about talking math with kids). I do know that pretty soon, she is going to want to jump, and that whether that’s right away or after a few more exchanges doesn’t really matter.
What matters is that she’s been asked to think.
This line of discussion lays the foundation for thinking about distances, times and their relationships to each other. It supports Little Dardy’s attempts to participate in the discourse of measurement.
My recent conversation with Tabitha about the height of our hill was in a similar spirit; we worked on the meaning of height when she asked me to lie down on the hill.