Category Archives: Talking math with your kids

The latest “Common Core” worksheet

You have seen this on Facebook.

Original (Click to enlarge)

Ugh what a mess.

Please share the annotated version widely.

I’ll say what I have to say (comments closed) and move on. If you wish to discuss further, hit me up on Twitter or pingback to the blog. Want to talk in private? Click the About/Contact link up top.

Also, Justin Aion—middle school teacher extraordinaire—wrote up his views on the matter. You can read them over in his house.

Here goes…

The intended answer

Dear Jack,

You only subtracted 306 from 427, not 316. You need to subtract another 10 to get the correct answer of 111.

Sincerely,

Helpful student

The purpose of this task

I cannot say whether this was the right task for this child at this time because I do not know the child, the teacher or the classroom.

I can say the following:

  • Analyzing errors is a useful way to encourage metacognition, which means thinking about your thinking. This is an important part of training our minds.
  • The number line here is a representation of a certain kind of thinking—counting back. The number line is not the algorithm. The number line records Jack’s thinking. He counted back from 427 by hundreds. Then he counted back by ones. He skipped the tens. We can see this error because he recorded his thinking with a number line.
  • Coincidentally, the calculation in question requires no regrouping (borrowing) in the standard algorithm, so the problem appears deceptively simple in its simplified version.
  • This task is intended to help students connect the steps of the standard (simplified) algorithm with reasoning that is based on the values of the numbers involved. Why count back by three big jumps? Because you are subtracting 300-something. Why count back by six small jumps? Because you are subtracting something-something-6. Wait! What happened to the 1 in the tens place? Oops. Jack forgot it. That’s his mistake.

So what?

The Common Core State Standards do require students to use number lines more than is common practice in many present elementary curricula. When well executed, these number lines provide support for kids to express their mental math strategies.

No one is advocating that children need to draw a number line to compute multi-digit subtraction problems that they can quickly execute in other ways.

The Common Core State Standards dictate teaching the standard algorithms for all four arithmetic operations.

But the “Frustrated Parent” who signed that letter, and the many people with whom that letter resonated, seem not to understand that they themselves think the way Jack is trying to in this task.

Here is the test of that.

A task

What is 1001 minus 2?

You had better not be getting out paper and pencil for this. As an adult “with extensive study in differential equations,” you had better be able to do it as quickly as my 9-year old.

He knows with certainty that 1001 minus 2 is 999. But he does not know how to get the algorithm to make that happen.

If I have to choose one of those two—(1) Know the correct answer with certainty based on the values of the numbers involved, and (2) Get the correct answer using a particular algorithm, but needing paper and pencil to solve this and similar problems—I choose (1) every time.

But we don’t have to choose. We need to work on both.

That’s not Common Core.

That’s common sense.

[Comments closed]

An interesting story about research and assumptions

Nature v. nurture. Age-old debate on relative importance. Not gonna settle it here. Not even in the limited context of factors influencing mathematics success.

There is lots of interesting research going on, of course. I want to tell you a quick story about a very small subset of that research.

A few years back, a group of educational psychology researchers published a study that phys.org headlined, “Math ability is inborn“.

The study investigated the ability of 4-year olds to choose the larger of two sets of dots when these sets were viewed briefly (too briefly to allow for counting).

They found that children who were better at this task also knew more about numeration and counting.

A quote from one of the researchers, Melissa Libertus:

“Previous studies testing older children left open the possibility that differences in instructional experience is what caused the difference in their number sense; in other words, that some children tested in middle or high school looked like they had better number sense simply because they had had better math instruction. Unlike those studies, this one shows that the link between ‘number sense’ and math ability is already present before the beginning of formal math instruction.”

Read more at: http://phys.org/news/2011-08-math-ability-inborn.html#jCp

Let’s pause for a moment to think, shall we?

If a child has not had formal instruction in mathematics, is the only remaining possibility that her mathematical performance is due to innate skill?

Of course it isn’t.

There is also the possibility that the child has absorbed some mathematical knowledge from her environment, and that different environments might provide differential input.

Maybe the child who is better at discerning the larger set has more practice doing just that. Maybe that child’s parents have been asking her how many? how much? and which is more? for the last two or three years.

Maybe that child’s parents have been Talking Math with Their Kids.

Mindsets, research and talking math with kids [#NYTEdTech]

This conversation happened in New York yesterday.

A view of New York City from the Times Center on Tuesday.

A view of New York City from the Times Center on Tuesday.

During a coffee break, I sat down on a white pleather sofa next to an older man.

Me: How has your day been?

Him: Good. You?

Me: Pretty good. Interesting.

What do you do?

Him: Retired.

Me: From what?

Him: I was president of [small New England college]. How about yourself?

Me: I teach math at a community college in Minnesota.

But I’m also working on a project. I work with future elementary teachers, so I have studied the mathematical development of children.

Him: Uh huh.

Me: And I want to use that knowledge for something else, which is this: I am trying to understand what knowledge parents need in order to support the mathematical development of their children.

Him: That’s important.

Me: Right.

[Short pause]

Me: Do you have grandchildren?

Him: Yes. They are 8 and 10.

Me: Oh nice! So their parents—your kids—are my target market.

Him: Yes. Their father is really into that. They use Khan Academy and all that.

—FIN—

If the end of that conversation makes no sense to you, I ask that you please, please, please spend the next 15 minutes over at my website, Talking Math with Your Kids. You might be especially interested in the research summaries, which demonstrate that young children need to talk about number and shape with their parents rather than (or at least in addition to) being sent to website, iPad apps and decks of flash cards.

Kids need mathematical conversation. And they enjoy it.

Talking Math with Your Kids for Kindle!

Someday there will be a full-sized paper version of a Talking Math with Your Kids book (Hear that publishers? Wanna talk? You can find me at the About/Contact page.)

Until that day, there is now a mini-version (15,000 words; roughly three chapters, $4.99) available on Kindle (and readable on other devices with the Kindle app).

Tabitha is delighted by the news!

Tabitha is delighted by the news!

Go have a look, won’t you? Share widely and let me know what you think.

Table of contents:

  1. Introduction
  2. Counting and other adventures in number language
  3. Adding and subtracting: Two peas in a pod
  4. Conclusion
  5. References and further reading

The book is structured around conversations I have had with Griffin and Tabitha. About 1/3 of the conversations in the book have been previously documented here and/or on the new Talking Math with Your Kids site. The rest are new to readers.

There is lots of new content summarizing research in parent-friendly ways.

The impetus for getting this out now is this: funding my New York Times Schools for Tomorrow trip. I got partial funding from my college, but it’s an expensive conference. So I hacked a couple of chapters out of a draft I have been working on for quite a while now, tidied and edited them and voilá!

Griffin and Tabitha are moving

omt.moving

For two years now, I have been documenting the mathematical conversations we have around the house, filed under the category, “Talking Math with Your Kids”.

I have long wanted to bring these conversations to the attention of non-mathy parents. But seriously, they want to wade through my musings on College Algebra, elementary teacher preparation and Khan Academy? I don’t think so.

So these conversations are moving.

To talkingmathwithkids.com

Come join us over there, won’t you?

More importantly, do us the favor of sharing the link with parents and caregivers of your favorite 0—10 year olds. Especially those who might be a little bit afraid of math.

In writing for parents, I’m fleshing out the conversations with information about kids’ mathematical development, and with ideas for starting similar conversations with their own kids.

We’ll review products that are targeted at kids and math, and share relevant research and news.

There’s even a contact page where parents can report conversations they have with their kids, and ask questions.

We’re just getting started, so give us a hand won’t you?

 

The oldest man in the world

Tabitha (six years old) and Griffin (on the cusp of nine) are attending a three-hour soccer camp in the neighborhood every afternoon this week. Furthermore, she has been begging to come to Tuesday night Ultimate Frisbee with me this summer. This week was the first opportunity for her to come along. It’s about a half hour ride up to Blaine so we had time to chat in the car.

Tabitha (6 years old): My group is 6 and under, but everyone is 6.

Me: Interesting.

T: Yeah. Six and under, so even a zero-year-old could play.

Me: I suppose so. But everyone is six, so there are no “unders”.

T: Daddy, everyone is under something.

Me: Huh?

T: Like you. You’re under 100.

Me: I suppose so. But then everyone you know is under 100.

T: Not the famous guy.

Me: What famous guy?

T: The oldest man in the world. He’s not under 100.

Me: No he’s not. But you don’t know him either.

T: Yes I do. I read about him in a book.

Feeling smug for having won this round, Tabitha sits in silence for a moment.

T: Are his mom and dad still alive?

Me: Whose mom and dad?

T: The oldest man in the world.

Me: Let’s see if you can work this out yourself.

T: Oh! They’re not alive.

Me: How do you know?

T: Well, his mom and dad are older than him. So if they were alive, they would be the oldest people in the world.

Pause.

T: Or, they could be alive, but younger than him.

Postscript

Quick plug: Tony Sanneh is evidently from Minnesota. He has a foundation that, among other things, offers free soccer camps in Minneapolis and St Paul recreation centers. They seem to be really positive, well run affairs drawing kids of diverse economic and cultural backgrounds. From what I can tell, they are doing lovely work that we should applaud.

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.