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Can 1/3 and 1/3 = 2/6? It seemed so

marilynburnsmathblog.com

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#301

Post by p4bl0 »

tsimionescu wrote: post #156
p4bl0 wrote: post #128

Earlier quoted context omitted.

abstract class Fruit: prop name: string prop rotten: bool class Apple (Fruit): prop name: string = "apple" prop rotten: bool = False class Orange (Fruit): prop name: string = "orange" prop rotten: bool = False func add (a: List , b: List ) -> List : return foldl(lambda l, f: l.append(f), a, b) func filter (p: Func , l: List ) -> List : return [f for f in l if p(f)] func is_rotten (f: Fruit) -> bool: return f.rotten a…

This does nothing to elucidate the problem. 1/3 apples + 1/3 oranges could be 2/6 fruits, or 2/781 fruits, or any other number you want. In normal mathematical notation, 1/3 is interpreted as 1/3 of 1, and in that case 1/3 apple + 1/3 orange is 2/3 fruit. And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types.…

You're being dishonest in your reasoning.

> In normal mathematical notation, 1/3 is interpreted as 1/3 of 1

Why would you deny the possibility of 1 being 1 list of fruits rather than 1 fruit?

In the original example, it was 1 table of students, not 1 student.

Everything you say after that is based on this broken assumption.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#302

Post by tsimionescu »

macspoofing wrote: post #295
simiones wrote: post #294

Earlier quoted context omitted.

You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics. Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when…

>The modern formal definition is a post-hoc formalization of an existing intuitive concept Sure, but we're aiming to teach those 'post-hoc' formalizations. What fourth-graders are learning is supposed to be a foundation for future abstract mathematics. And we are using the operators and syntax of the modern formalizations as well. Again we're not just teaching kids how to think about ratios and do practical arithmeti…

From my own experience as a pupil, teaching purely formal systems with no help in building the intuition of why the formal rules are what they are is a recipe for disaster. It invites thinking of mathematics as a game, and when you forget the rules, you tend to invent new ones.

My belief is that is exactly how you end up with students doing fraction addition as a/b + c/d = (a+c)/(b+d); or the infamous shepherd's age problem[0] - they forgot the actual rules, and picked a different rule that makes just as much sense to them.

And regarding analogies, I think that the best approach is to pick a real-world problem, and translate that into math, as a starting step for explaining the formal rules and building this intuition. Doing thing the other way around is much more likely to lead to contrived examples. But math rules have good intuitive reasons for existing, and explaining these as you introduce the rules is likely to help rather than hinder.

Of course, I wouldn't advocate for having students go back to the analogy while solving more advanced exercises with the rules that they have internalized. But having lots of exercises initially that try to drive home the intuition behind the rules is going to be very helpful in my opinion.

[0] "A shepherd has 25 goats and 53 sheep. How old is the shepherd?" A lot of kids will give you an answer: if they do, they will probably say that it's 25+53 or 53-25, since they may apply some common-sense reasoning after they "do the math", but at the wrong end of the problem.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#303

Post by imetatroll »

forbiddenvoid wrote: post #174
joshocar wrote: post #119

Earlier quoted context omitted.

This. What the student said was 'wrong' and they need to look at the whole. This "1/3 + 1/3 = 2/6" problem would be a great example to use for a lesson on units though.

What they said wasn't wrong. Their mental model was absolutely correct. "One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.' Their written representation of the mental model was incorrect because their instruction was fo…

Math is about learning how mathematical language is applied to problems. She is incorrect about how the math is applied. Ergo she is wrong. Her intuition is on the right track.

The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#304

Post by imetatroll »

Have the kids stand up and count out numbers.

Now they know how many kids are in the room.

Then have them go and stand in two different corners. Count one corner.

Then have them stand in three different corners. Count one corner.

Finally the four corners. Count one corner.

Then have the two tables distribute their students to the corners and figure out these elusive fractions.

Have each corner be a Pokemon or something. That should do the trick.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#305

Post by matt-attack »

romwell wrote: post #266
TeMPOraL wrote: post #259

Earlier quoted context omitted.

Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?

This hits the nail on the head: 1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.

Why? Each is a pure number without units. Since the units cancel in each term.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#306

Post by tome »

Denvercoder9 wrote: post #74
wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.

Are you sure it wasn't sin (30 pi / 180), because that actually makes sense.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#307

Post by ValorieO13 »

I solved this by then introducing the kids to slices in a pie. Visually, the kids can see that 1/3 of a pie plus 1/3 of a pie is not the same as 2/6. Fractions are often introduced as "parts of a set" which is great for the first day. It's easy for kids to see and understand, but kids are more familiar with trying to slice the birthday cake for all the kids at the party. After the kids realize there are two kinds of fractions, we spend time at each problem thinking through is this a set (which we'll return to in ratios) or a whole (which can't be expanded) and visually representing fractions both ways to check their answers. Adding units is absolutely important, but when I teach it, that's what I add in fourth or fifth grade when I'm teaching them to reduce.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#308

Post by kortilla »

TeMPOraL wrote: post #271
kortilla wrote: post #269

Earlier quoted context omitted.

> you can multiply your variables by these amounts, and now both expressions have the unit In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.

Since numbers are usually introduced to children in terms of quantities of things (e.g. 3 apples), I think it might suffice to stress the importance of keeping these things always attached to those numbers. In general, numbers very rarely go alone.

Sure. Might point is that you’re whole statement is way beyond the math comprehension of grade schoolers.
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