Can 1/3 and 1/3 = 2/6? It seemed so
This kind of thing fascinates me. I&#x27;ve always felt that you don&#x27;t really understand a thing, even a simple thing like fractions, until you can find and resolve all the paradoxes that arise in using it naively.<p>My take when reading the 1&#x2F;3 + 1&#x2F;3 = 2&#x2F;6 equation is that Addison was right but the &quot;+&quot; here is a different operation than the &quot;+&quot; of 1&#x2F;3 + 1&#x2F;3 = 2&#x2F;3. In one case we are adding disjoint parts of the same whole and counting how much of the parts of the whole are now included, and get 2&#x2F;3, and in the other case we are adding two wholes and counting how much of the new whole is included.<p>People who pass primary education without internalizing this distinction will have to do so later if they learn statistics or probability theory.<p>The answer &quot;what would you do now?&quot; is the same one I would try to ask the students. First I would stop, confused, and write 1&#x2F;3 + 1&#x2F;3 = ? elsewhere on the board, and then wait for the students to see why I&#x27;m confused. They already understand that we want mathematics to be consistent, and getting different answers means we were wrong at least once, but they will have a hard time to find out or explain why 2&#x2F;6 and 2&#x2F;3 both are intuitively and obviously right. You have to compare the intuitions directly to each other to see that in one case you&#x27;re adding parts of the same whole (set or group or table of kids) and in another case adding the wholes.<p>From what I know of elementary school, we focus a lot on fractions and the formal manipulation of them as unitless values, and drill into kids that 2&#x2F;4, 1&#x2F;2, and 0.5 are all &quot;the same thing&quot;. Bullshit! It&#x27;s no wonder they have problems with word problems! Two-out-of-four apples and one-out-of-two apples and a half apple are very different things, as every elementary student knows well. How can we expect them to believe something that&#x27;s obviously false unless we&#x27;ve introduced the narrow context in which it&#x27;s true? When used as unitless &quot;coefficients&quot; or bare numbers that exist only to scale another value that represents a real quantity <i>of</i> something, <i>then</i> 1&#x2F;2 and 2&#x2F;4 are the same. <i>This</i> is what we should be teaching when we teach fractions, not some mindless rote symbol manipulation like cross-multiplying or what have you without any ability to intuitively derive those rules yourself.<p>I&#x27;d spend the rest of the class time letting them work out all the different ways of looking at it, and the connections to everything else fractions and ratios and addition are good for.<p>Of course we want to be able to have &quot;+&quot; be well-defined, especially in elementary school, and the students understand that! We&#x27;ve now found a perfectly intuitive meaning of addition for which our standard &quot;+&quot; definition does not fit! This gives an opportunity to explore the connections between mathematics and intuition and the symbols and the real world operations they stand for. It gives us an excellent chance to, as Polya puts it, &quot;introduce a suitable notation&quot; for the setwise addition of &quot;marked&quot; sets that we have just discovered. Then we can explore the properties of this operation and the ones between it and the familiar addition operation. We can show that the plus sign and the addition operation we use it to represent are distinct, and our use of one for the other is a choice, and one we could make differently.<p>I saw some mentions of units in the comments, and yet 1&#x2F;3 + 1&#x2F;3 = ? is still ill-defined if we know only that &quot;1&#x2F;3&quot; means &quot;one out of three students&quot;. What we need to know is which students they are! So if one out of Alice, Bob, and Charlie has red hair, and one out of Charlie, Doris, and Evan has red hair, how many of Alice, Bob, Charlie, Doris, and Evan have red hair? Well, we made the question challenging even for adults if we ask it that way! And now you can introduce uncertainty and degrees of belief. (What if three people tell you they have one red-headed friend, in a class of 23 students? Do they all have the same friend, or is there more than one redhead?)<p>Of course what most of us will actually do as the teacher in this situation is freeze and think &quot;oh no, I&#x27;ve made a mistake and the teacher (in your mind) is going to point it out and the class is going to laugh at me!&quot;. Because that&#x27;s what we&#x27;ve learned happens in these situations. If we interpret confusion and being wrong as painful and embarrassing, little wonder that mathematics seems a long and painful road that we avoid whenever we can.