Number talk today:

Many kids found the precise answer within a minute, and there weren’t major misconceptions from the students who shared. That said, the big idea I’m looking for in most multiplication number talks — effective use of the distributive property — didn’t come from as many kids as I would’ve hoped. There were basically three camps:

- Students who found creative ways to multiply — for instance 51 x 9, then double the answer, or rounding 18 or 51, or 51 x 6, then triple the answer. All great strategies, but not quite what I was hoping would come out.
- Students who made use of the distributive property (usually by finding some way to calculate 18 x 50, then adding one more 18), but didn’t make that clear from their explanation.
- Students who used the distributive property, but were clear about breaking apart (for instance) the 51 into 50 and 1, and multiplying the two parts separately.

I would estimate students were about 30% using #1, 60% using #2, and 10% using #3.

This isn’t a big misconception, but students will be more powerful mathematicians if they can name and work flexibly with the distributive property.

Today made me think more about the way I scribe answers. Starting in my second class, I made a significant effort to probe students to be more explicit about their use of distribution, and to scribe it in a way to help other students make sense of it. I’ve had a desire to give more student ownership to number talks — for instance student scribes, or more partner-based interactions. But today reminded me of the value I have as the teacher — of taking a mathematical concept that one student is using effectively, and making it clear and accessible to the rest of the class. That’s not the student’s job; it’s mine. The best way to make that happen, however, is something I’m still working on.

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howardat58Hello DK

I don’t see the difference between methods 2 and 3. If I can see 51 as 50 and one more, then multiplying by 51 becomes multiply thing by 50 and add one more thing. To understand and appreciate aspects of numbers it is surely not essential to have a formal name for every, or any, process or rule or law. Of course if the CCSS requires that the laws of arithmetic are known by formal name then that is a very sad state of affairs. I didn’t encounter the distributive law until we got to groups, rings and fields in modern algebra. I would love to read your comments on this, as a rote learned law of arithmetic is no different from a rote learned rule for adding fractions.

dkane47Post authorHey Howard,

Thanks for your comment — I’m thinking again now about my goal with this number talk. A few things:

1. I think you’re right that pushing students to use language like “distributive property” before they’re ready is unhelpful to their mathematical development. A cursory glance at CCSS shows the distributive property introduced informally in 3rd grade, and explicit in the standards through 7th grade. It’s unclear the role the vocabulary plays.

2. That said, I think there’s value in students articulating the breaking-apart of numbers in mathematically correct ways. The commutative and distributive properties in particular are implicit through much of elementary mathematics, and the principles lay the foundations for algebra and abstract reasoning. I think you’re right not to force students to use specific vocabulary, but I think they should also be able to articulate more precisely how they are breaking numbers apart.

3. While students may not need to be able to explain a concept clearly in order to understand or apply it, most of my goal in number talks is to expose students to other students’ strategies and reasoning. Therefore I need to push students to articulate the concepts behind their reasoning for other students to learn from, and this is where my desire to scribe in mathematically provocative ways comes from. I agree that naming the distributive property is irrelevant here — but what is important to me is that students can see the mathematical structure in a strategy and consider applying that strategy independently.

Hope this clarifies my thoughts! Thanks for reading,

Dylan

howardat58and many thanks for writing back. I like what you are doing!

banderson02For me, I really wish my Elem Ts would use vocab, and make it a point to keep using it. It lifts the mystery of Algebra, Distributive Property, or anything if you give it a name when you first introduce it. Young Ss have a thirst and enthusiasm for learning, set the hook early and set it deep. When I tell Ss they have done Algebra sine K, they laugh at me. Then I throw up a K worksheet they have done or seen. I look at their faces, and when the light turns on- minds blown.

howardat58ps. I love your blog