Help me come up with a snappy term/metaphor for this math concept
September 27, 2021 8:57 AM   Subscribe

In math, there are two basic ways to describe a set: (1) Directly specify what is in the set, either by simply listing the contents or by giving a rule to directly generate them. (2) Start with a larger set, and describe which items in that set are to be retained. I'm looking for a memorable pair of contrasting terms to describe these two approaches broadly to students. These terms do not need to be in existing use; be as creative as you like.

One way this comes up is in parametric vs. implicit equations for a graph. For example, the same circle could be described as the set of points (cos(t),sin(t)) (approach (1)), or as the set of points (x,y) satisfying x2+y2=1 (approach (2)). But "parametric vs. implicit" is too narrow and technical for my purposes. Extension vs. intension has the broad scope I want, but those terms are way too dry and unfamiliar.

What I want is something more like this. A cake decorator making a circle would want to know where to apply the frosting directly (approach (1)). But an ice sculptor would want a rule for which points to chip away and which ones to keep (approach (2)). So, we could call the two approaches the "cake-decorator way" and the "ice-sculptor way". What other phrases can you come up with to capture the broad idea of direct vs. subtractive description?
posted by aws17576 to Education (26 answers total) 1 user marked this as a favorite
Pyramids vs Mount Rushmore.

If this is for young enough students, I bet there's a Minecraft meme or metaphor that would work.
posted by mrgoldenbrown at 9:08 AM on September 27, 2021 [1 favorite]

Classic vs diet?
posted by olopua at 9:20 AM on September 27, 2021

Maybe tap into the middle school experience by doing something like clubs vs. cliques?
posted by El_Marto at 9:22 AM on September 27, 2021

include / exclude
put in / take out
add / remove
uh huh / nuh uh
posted by signal at 9:24 AM on September 27, 2021 [2 favorites]

Method 1 = List all the marshmallow shapes in a box of Lucky Charms
Method 2 = Dump the contents of different types of cereal out on the counter, and ask them to pick out only the marshmallows.
posted by ericbop at 9:30 AM on September 27, 2021 [5 favorites]

By name/By not



Nice list/Naughty list
posted by wenestvedt at 9:30 AM on September 27, 2021 [1 favorite]

Either kitchen chores you are responsible for, or kitchen chores you are not responsible for.

And hopefully there won't be a wise guy/gal in the class that knows about Russel's paradox..
posted by forthright at 9:41 AM on September 27, 2021

Build a house for the Smurfs by:

a) ordering all different types of Legos, but in blue only; or
b) take a huge bag of Legos, and pick out only blue ones.

cost/benefits are interesting here, it must be expensive to order blue-only Legos, but time spent picking them out from a huge bag is valuable too.
posted by Dashy at 9:44 AM on September 27, 2021

Every sandwich in the bag versus "these" sandwiches (meaning sandwiches that satisfy certain requirements).
posted by history is a weapon at 9:46 AM on September 27, 2021

I think your sculpting metaphor is pretty good, but I'd pair it with clay sculpting vs ice/wood/stone instead of cake decoration. The only thing it doesn't include is the programmatic based method. I think it might be hard to make a two term trick because of this.

This is maybe a bit far afield, but it seems like a checkers board could be useful here. An image can be "drawn" in checkers, or created via rules ala game of life. Or the board can be filled, and pieces removed.

On the goofy side: "Family" vs "Leave the gun, take the cannoli"
posted by Jack Karaoke at 9:49 AM on September 27, 2021

Recipe book vs. salad bar?
Constructed vs. revealed?
Forged vs. knapped?
Engineer vs. detective?

(Personally, teaching students the words extension and intension doesn't seem like a bad thing to me. But, you know your students and the context lot better than I do.)
posted by eotvos at 9:51 AM on September 27, 2021 [2 favorites]

Taxonomy might offer some relatable examples, but not good terminology. A taxonomist could 1) list all of the breeds of dogs that there are or 2) list the qualities that define what makes a critter a "dog"
posted by OHenryPacey at 10:12 AM on September 27, 2021 [1 favorite]

Burrito vs hamburger. I'll have chorizo, lettuce, rice, beans, cheese, pico de gallo, guac ... vs one hamburger hold the pickles please.
posted by february at 10:25 AM on September 27, 2021 [2 favorites]

I've taught this as construction vs reduction. (It's not perfect but it's good enough and readily accessible)
posted by SaltySalticid at 10:26 AM on September 27, 2021 [1 favorite]

I think whatever metaphor you choose please ensure it connects down to what we use the items in the set for, and what's the purpose for having different methods so that the metaphor doesn't break as soon as you understand it. In other words if there is a reason for why we use implicit vs explicit *in math* (which, I don't know because this chapter of math broke my brain in 11th grade) then that has to be preserved in the metaphor for it to be useful.
posted by bleep at 10:27 AM on September 27, 2021 [1 favorite]

In software whitelist/blacklist - or the less loaded but less common allow-list/deny-list - are pretty close. Similarly you could think of a curated wedding guest list vs a bar that allows everyone over 21 except for these specific jerks.
posted by february at 10:27 AM on September 27, 2021 [1 favorite]

The burrito take makes me also think many people have familiarity with the idea of ordering a "#6" from a menu, which kinda satisfies the programmatic part. So you have: #6, make your own, and "hold the..."
posted by Jack Karaoke at 11:03 AM on September 27, 2021 [1 favorite]

Oh my how has no one mentioned a filter yet?

So you either make direct selections or you put everything through a filter and some things are retained.

Like some sort of Metaphor Filter, I don’t know.
posted by iamkimiam at 2:13 PM on September 27, 2021 [2 favorites]

My initial though was Bucket vs. Net - with a bucket you'll scoop things up indiscriminately, or keep everything you throw into the bucket, where a net will catch and keep only things bigger than a certain size. You could even have a visual demonstration where you scoop something up with a bucket, then tip it into the net.
posted by Jon Mitchell at 5:23 PM on September 27, 2021

I like iamkimian: filter versus select?
posted by Valancy Rachel at 6:56 PM on September 27, 2021

Build-it-up vs carve-it-out?
posted by flabdablet at 9:47 PM on September 27, 2021

Assembly vs whittling?
posted by flabdablet at 9:48 PM on September 27, 2021

Defined by inclusion vs defined by exclusion. Enumerated would be a subset of defined by inclusion.

The distinction could get murky. For example if you say "the set of integers which are not prime", the phrase "set of integers" could be taken as either including the integers or excluding everything which is not an integer.

You might consider the words used for different kinds of proofs: direct, by contradiction, etc.
posted by SemiSalt at 6:16 AM on September 28, 2021

Back in high-school programming class, we had to write programs to generate a list of prime numbers using the "sieve" and "div" methods.
posted by adamrice at 8:12 AM on September 28, 2021

In manufacturing it’s additive vs. subtractive.

You could probably make things memorable on the subtractive side with: "Carving an elephant is easy. All you do is start with a block of marble and remove everything that doesn’t look like an elephant."
posted by Tell Me No Lies at 11:33 PM on September 28, 2021 [1 favorite]

Little late, but I'd go with "explicit" and "reductive".
posted by Zudz at 8:26 AM on September 29, 2021

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