Tag Archives: time

Units, attributes and four-year olds

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!

D: Here?

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?

LD: Yes.

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?

LD: No.

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.

How many weeks until Christmas?

I don’t want to give the impression that Talking Math with Your Kids always goes well. It does not. There are bumps in the road for sure.

I was having breakfast with Griffin (8) and Tabitha (5) over Thanksgiving weekend.

Griffin: It’s exactly one month until Christmas.

Tabitha: I knew that!

Me: That’s right 30 days.

T: So it’s zero weeks.

Me: How many weeks?

T: Zero

Me: In 30 days?

T: Oh! I thought you said 3 days.

G: So how many weeks is it?

Me: What do you think?

G: Well…five sevens is…er…six fives is 30 so seven fives is 35…Five weeks and five days! Is that right?

Me: Tell me how you got that.

G: [long pause]

Me: [long pause]

G: [frustrated] Just tell me if it’s right!

Tears followed shortly afterwards. I suggested we move on and talk about something else. Griffin cursed Tabitha’s name for making him wonder how many weeks this was. Emotions ran high and I told him we needed to discuss it later.

An hour later…

G: [on the couch, addressing me as I came down the stairs] Is it four weeks and two days?

Me: Did you do two sevens are 14, then double? Or did you do five sevens are 35, so five weeks are five days too much?

G: I did the thing with 14.

The most important message I can send my kids is that they can make sense using what they know. Their minds and the world are the arbiters of right and wrong, not me. In not telling Griffin whether he was right the first time, I was reinforcing my insistence on this principle.

I knew what his mistake was and I knew that he would find it if he stopped to think. Five sevens are 35 because seven fives are 35. And 35 is 5 bigger than 30. But that means five weeks is five days too big, not five days too small. I knew he could figure that out himself. Further, I knew that if he could not, we could talk about it and that he would learn from this conversation.

I also know that it is important to compromise. In the second part of the conversation, I knew that he knew he was right. But I knew that prodding too hard on the emotional wound recently inflicted could quickly lead to trouble.

So I compromised. Rather than have him tell me what he did, I offered him choices. I essentially asked him whether he had started from scratch (finding four sevens), or adjusted his previous answer (five sevens are 35).

How long until dinner? Adventures in Kindergarten fractions

Regular readers of this blog may feel that Griffin has been featured more often in Talking Math with Your Kids than Tabitha has. But in fact it is tied 10 posts to 10 so far. Today, Tabitha takes the lead.

Tabitha got home from her third day of Kindergarten. School gets out at 4:00, it’s a long bus ride home and the bus has been running late these first few days.

Point being, she’s hungry when she gets home.

Tabitha (five): When can we eat dinner?

Me: I’ll go get it on the table. We’ll eat in about 15 minutes.

T: Awww! That’s like a half hour!

Me: No, a half hour is thirty minutes.

T: [long pause as she leaves the room, wandering thoughtfully into her bedroom] Awww! That’s a half-half hour!


Griffy counts

Dan Meyer developed a lovely challenge for math teachers and curriculum writers last week:

Give yourself one photo or one minute of video to tell a mathematical story so perplexing that all of your students will want to know the ending, without you saying a word or lifting a finger.

He used it to start a Twitter discussion [#anyqs] that has yielded some really interesting discussion both on Twitter and on Dan’s blog.

I want to follow up on one of my entries in this fun and challenging game.

Griffy counts

The following question have all come via Twitter.

  1. How many times will Griffy make it around?
  2. How many times would he make it around if he counting at the correct speed?
  3. How close will his counting be to 2 minutes?
  4. Which will occur first; 2 minutes or his counting to 120?
  5. What number will he be on when the 2 minutes is up?
  6. How many numbers/minute is Griffy counting?

Mission accomplished. Each of these is relevant to my intention; they overlap and interact in ways that would make exploration of any one of them relevant to any of the others.

These questions get at mathematical modeling. What assumptions should we build into the model? Should we assume that his counting continues at the same pace up to 120? Should we assume that his running continues at the same pace? If we answer no to either of these, do we think he’ll speed up or slow down? Etc.

Answers in video below. Note that he starts counting 10 seconds into the video.