The title of the text is “Modern Mathematics for Junior High School Book 2” the publisher was Silver Burkett. I believe that I understood the use of the projector as an attempt to model negative values of time in the d=rt formula, but there were the wake of the boat and the eyes of the pilot pointing in the real direction of time, and where were the models for the negative areas in the A=lw formula? This just did not seem like math. By the end of ninth grade, with the help of a very dedicated teacher and without ever hearing the terms group or integral domain I understood that the integers were a system which expanded the natural numbers in a manner that included an additive identity and inverses, was closed under addition, in which the non commutative operation of subtraction was not needed and that if the commutative properties of addition and multiplication along with the distributive property were to be preserved in the extension then a negative times a negative must be a positive.

]]>Thanks for your calling my attention to this book, and for your many years of serious engagement with elementary mathematics! I hope it’s apparent from the post that I’d like to see everyone (mathematicians, teachers, teacher educators, etc.) attend with loving care to the logical structure of elementary mathematics, and you’ve been doing that for a long time. Any bibliography of the kind I evaded in note [3] would include numerous of your publications, certainly including the book in question.

(One could be forgiven for the misimpression that you’re asserting that everything contained in the post is already in your book. It seems to me our treatments of theorem 2 of the post are essentially the same in their logical content [the equality of the quotative and partitive models of division is a consequence of commutativity], but our treatments of theorems 1 and 3 are very different. Your proof of theorem 3 is very nice!)

]]>Thanks for this. What was the text?! And maybe it’s a conversation for another place and time, but I’m now curious how you made sense of the idea that the boat analogy had anything to do with multiplication?

]]>Of course you’re right that it’s not a ÷ b that’s a binary operation, but just ÷. Perhaps I was cavalier in my “how could they be?” comment.

The magic you refer to, namely understanding the equivalence of interpreting the slash as a division vs. as a fraction, is the whole point of #3. That’s the theorem! If you like, you can forget that I brought the obelus into it at all. As you say, it’s possible to have the whole conversation without it. I included it both for the sake of the section 3 lede, and more importantly because without it, I’m forced to state the theorem as a/b = a/b, with one side denoting a fraction and the other a division. Written this way, the notation serves to hide the underlying theorem. The fact that the slash has both meanings packages the theorem into the notation, simultaneously encoding it and rendering it invisible.

(This is what I was referring to in the post when I said that using the slash to denote division is built on, but also elides, the theorem in question.)

There are other instances of this, where we package theorems into our notational conventions, and I think it often makes for “good notation” in the sense that the notation reflects the underlying mathematical situation. The first examples that come to mind for me are from calculus. Leibniz notation for the derivative, dy/dx, makes the chain rule look like an evident fact about fraction multiplication. It’s not actually a fraction multiplication, but we can get away with this because we can prove the chain rule in other ways. Meanwhile, definite and indefinite integrals are written with the same symbol even though modern math understands them as two completely different things: one is a limit of Riemann sums (or their Lebesgue analogues), while the other is an antiderivative. We can get away with this because of the Fundamental Theorem of Calculus.

At the same time that these notations are “good” for doing math with, I think they can also conceal interesting ideas from learners, precisely because they hide the theorem they encode. Reflecting on my own experience of learning calculus, I think it took me a long time to appreciate the details involved in proving the chain rule, precisely because the notation makes it look so obvious. It also took me a long time to appreciate the vastness of the conceptual difference between definite and indefinite integrals.

All this is to say: my purpose in writing #3 is to call attention to the theorem that reconciles the two interpretations of the slash (as fraction bar and as division). Because you acknowledge this as “magic”, I assume we agree that it’s a theorem. I don’t think this theorem gets its due, hence its inclusion here. Let me add a speculation: I suspect part of the reason it doesn’t get its due is that we use the same symbol for both meanings, a convention that encodes this theorem but in the same moment hides it from view.

The challenge of getting used to the fact that our symbols also have synonyms, such as the obelus vs. the slash (in its division meaning, not as a fraction bar), the times sign vs. juxtaposition for multiplication, and the fact that we sometimes write a calculation horizontally and at other times vertically, is also interesting and important, as you say. It’s not particularly the subject of this post, though — hopefully this digression has helped to isolate the point I’m trying to make.

]]>Ultimately I think this idea of synonymous expression is simultaneously the concept of biggest import and greatest difficulty in elementary math. Division vs. fractions is one of the more subtle examples, but just as important are examples of 4×3 vs 4(3) vs 4*3 vs etc … Or even 12+23 vs it’s vertical counterpart where place values are aligned (I remember that equivalence throwing my daughter into an absolute tempest of confusion at first). And none of that even touches on translation between spoken/written language and math language when trying to interpret expressions or model real world scenarios.

Just my 2 cents.

]]>Just wanted to say I enjoyed the article, thank you, and I believe anyone seriously concerned about how our children are encouraged to think is fighting the good fight. ]]>

The fact that a/b=a÷b presupposes that we KNOW what a/b is, what a÷b is, and we can prove that these two objects are EQUAL. This means we must provide a robust foundation for fractions and whole numbers so that students KNOW what a/b is and what a÷b is. More over, the foundation must make clear what “equality of two numbers” means. The above book provides such a foundation for all of elementary school mathematics, and this fact explains why it is 551 pages long.

]]>You ask what I think is the central question of mathematics education. So I don’t have any advice that would fit into this small margin. And in fact I find myself continually learning new ways to encourage reasoning, to uncover misconceptions, to stimulate the thinking process. I find it helpful to listen to students carefully, with an ear to how they are connecting statements, what assumptions they are making, what gaps are in their arguments. Then find ways to make them conscious of these gaps or assumptions, revisit them, and continue their exploration. This is pretty vague and general ‘advice’, but it’s really all I can give.

]]>I have recently taught a lesson on logic and problem-solving. In this lesson, I have noticed that logic is not something that can be taught. It is something that students have to come to an understanding by themselves and practice on. What advice would you give to us if we want to improve our student’s logical and reasoning skills? ]]>