Howard Stern Loves to Show How He Does the Math

My Next Guest Howard Stern

About six and a half minutes into the latest episode of My Next Guest Needs No Introduction, host David Letterman asks guest Howard Stern how long they’ve known one another. Viewers are treated to a number talk. The transcript:

David: You know how long you and I have known one another?

Howard: How long?

David: Well, it’s pretty much to the month since 1984.

Howard: Wow. Now I’m gonna do some quick math and figure out how long that is, if you don’t mind. Now math happens to be… I’m good at it. This is how I do it. This is 2018. Right?

David: It’s 34.

Howard: Oh, you gave it away.

David: It’s 34 years.

Howard: Let me check your math.

David: Yeah.

Howard: The way I get to it is, you say 1984 and I add ten immediately.

David: Yeah.

Howard: That brings us to 1994.

David: That’s right.

Howard: That’s ten.

David: Yeah.

Howard: 1994, then 2004 is 20.

David: Yeah.

Howard: Now here’s tricky ’cause I get confused. 2004 to 2014 is another 10. That’s 30. You’re absolutely right. That’s 34 years. Good for you.

David: Now…

Howard: I love to show how I do the math.

David: Speaking of which, you realize that all of that will be subtracted from the show?

Howard: Wow. But really for you… I guess the premise of this show, although who knows what this show is… you know, I don’t even know what I’m doing here, but I thought the premise was that… you’re choosing six people… and I’m way more fun than Obama already, I’m sure. I mean, this is fun.

David: Really?

Howard: Oh, for God’s sake, yeah.

Lucky for us, Letterman didn’t subtract all of this from the show. Some observations…

Despite David giving away the solution, Howard continues to share his strategy. David is not the ultimate authority; Howard is eager to prove this solution. Howard, at least, is interested in Howard’s reasoning. He’s focused on sense-making, not answer-getting; how?, not what? All of this is typical of a classroom number talk.

Howard uses an adding up (or add instead) strategy for 2018 − 1984. He moves forward from 1984 to reach 2018. The context implies distance–not removal–which lends itself to this strategy. Stern’s jumping by tens gives us an opportunity to discuss efficiency, e.g., one jump of thirty rather than three jumps of ten. For what it’s worth, I used an adding up strategy too. First I added 16 to 1984 to get to 2000 (or six and ten to get to 1990 and 2000), then I added 18 to get to 2018.

David, of course, does not record Howard’s thinking. I might use this video clip to have teachers anticipate possible strategies for 2018 − 1984 and consider how they would record them. I chose an open number line to model Howard’s adding up strategy:

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Howard is confident: “Now math happens to be… I’m good at it.” He is enthusiastic: “I love to show how I do the math.” He is joyful: “I mean, this is fun.” Over the last two years, it has been my privilege to work alongside Surrey teachers Alex Sabell and Jonathan Vervaet (and others) as they’ve incorporated number talks in their classrooms. These same positive attitudes towards mathematics come through in their students’ interviews (see Alex & Jonathan).

What did you notice in this clip? What did I miss?

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Paint Splatter Arrays

This isn’t Splat!

Paint Splatter Arrays (.key)
Paint Splatter Arrays (.pdf)

In Steve Wyborney’s Splat!, the total number of dots is given and the number of dots under each splat is unknown. In my Paint Splatter Arrays, the total number of dots is unknown. My paint splatters do cover some dots but how many is beside the point. Also, Steve’s dots are scattered; mine are arranged in arrays. (More on that below.) Steve’s splats splat. My splatters are there from the get-go. See? Not the same.

h/t Andrew Stadel

Here’s why I created this activity…

Number Talks -- Dot Cards

T: “How many do you see?”

S: “Twenty-five.”

T: “How do you see them?”

S: “Two, four, six, …”

Every. Single. Time.

Not all students. Most students do see and use groups or arrays to figure out how many. Those strategies are described in this post. But some students don’t seem to make sense of others’ ideas. That’s a greater challenge than I’ll tackle here. (Recommended: Intentional Talk by Elham Kazemi and Allison Hintz.) Instead, I designed the activity above to (gently) shove students towards looking for and making use of arrays.

The first three are softballs. For example, the second:

Paint Splatter Arrays.004

Students can still see each dot and count all by ones or twos. But a more efficient strategy is to see 3 × 5 (3 rows, 5 columns).

The next several slides completely cover at least one dot, so students can’t count all by counting what they can see. In each, at least one complete row and one complete column is visible. For example:

Paint Splatter Arrays.010
4 × 6
Paint Splatter Arrays.028
6 × 6

I had some fun with the last two. In the next-to-last one, the middle column is completely concealed.

Paint Splatter Arrays.030
3 × 7

In the last one, most of the dots are hidden. A bit of estimation. How many?

Paint Splatter Arrays.032

How confident are you?

Paint Splatter Arrays.033
<shrinks> 6 × 8

What about now?

I test-drove these on my daughters. (Keira likes Booger Math! over Paint Splatter Arrays, by the way. It is catchier.) I’m looking forward to trying this out in Surrey classrooms. Feedback welcome!

And mine goes ding ding ding di di ding ding DING ding ding ding di di ding ding.

Halving & Doubling: Very Fun to Play With

On last week’s Last Week Tonight with John Oliver, John Oliver used the mental math/computation strategy of halving and doubling as a punchline to a news story on nuclear waste.

The graphics nicely–and quickly!–illustrate why this strategy works. Starting with 1 × 20 (one football field twenty feet tall), if we double the first factor (area in football fields) and halve the second factor (height in feet), the product (volume in piles of nuclear waste), expressed as 2 × 10, remains the same. Similarly, we can halve and double to visualize that 1 × 20 is equivalent to ½ × 40. (Oliver also throws in the commutative property at the end–twenty football fields one foot tall.)

This reminded me of a video clip from Sherry Parrish’s Number Talks. In it, the teacher poses the problem 16 × 35. The fifth graders share several strategies: partial products (10 × 30 + 10 × 5 + 6 × 30 + 6 × 5); making friendly numbers (20 × 35 − 4 × 35); halving and doubling (8 × 70); and prime factors (ultimately unhelpful here).

I’ve probably shared this video in about a dozen workshops. There are some predictable responses from attendees. Often “not my kids” is the first reaction. I remind teachers that the teacher in this video has implemented this routine three to five times a week in her classroom. This isn’t her kids’ first number talk. Pose 16 × 35 in your fifth–or ninth!–grade classroom tomorrow and, yeah, the conversation will probably fall flat. Also, this teacher is part of a schoolwide effort (seen in other videos shared at these workshops).

Teachers are always amazed by Molly’s halving and doubling strategy. Every. Single. Time. I ask attendees to anticipate strategies but they don’t see this one coming. I note that doubling and halving wasn’t introduced through 16 × 35. I would introduce this through a string of computation problems (e.g., 1 × 12,  2 × 6, 4 × 3). “What do you notice? What patterns do you see? Does it always work? Why?” We can answer this by calling on the associative property: 16 × 35 = (8 × 2) × 35 = 8 × (2 × 35) = 8 × 70 above. Better yet, having students play with cutting and rearranging arrays provides another (connected) explanation.

Rather than playing with virtual piles of nuclear waste, I had fun with arrays of candy buttons:

Number Talks (pdf)