I still remember struggling with fractions when I was young. The funny thing is that I ended up majoring in math in college (at a highly ranked one at that), and I did very well, even publishing research in differential topology as an undergrad. How can someone who struggled to even grok fractions end up being successful in pure mathematics?
I think it's because all my teachers operated off the assumption that kids can't possibly understand abstract definitions, so everything must be explained via analogies. So nobody could tell me that fractions are equivalence classes. Everything was instead explained in terms of pies, which certainly seems like it would be intuitive, and there's really nothing technically wrong with it, but it encouraged me to think of, say, 1/2 and 2/4 as different entities instead of two representations for the same entity (since, after all, cutting a pizza into two slices instead of four is just not the same thing). I didn't "get" fractions until on my own I came to the conception that these are actually kind of abstract concepts that go by many different names.
For me, I was taught too much to identify numbers with concrete objects in the real world. This sort of works for integers (as long as you ignore negative integers and even zero to some extent), but it basically breaks down for all other types of numbers, including rational numbers. So it's a really bad expectation to set in my opinion. But you can hardly find an elementary school teacher who will dare approach this topic.
I honestly think that if someone had just explained that "1/2" is a symbol we've devised to represent the solution to the equation 2x = 1 -- an equation which otherwise has no solutions at all in the integers -- I would've grokked it a lot faster. It would've been clear to me that we're introducing an entire new set of numbers distinct from the integers. These new numbers do not have a direct correspondence with physical objects in reality. And it would've been easier to see that the same solution must go by many other names as well. For instance, multiplying that equation by 2 on both sides gives us a new equation which must have the same solution: 4x = 2. So "2/4" must represent the same entity as "1/2". And so on.
But because US educational culture has decreed that basic algebra is more advanced than fractions, the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom. It is assumed as a basic fact of human life that kids cannot understand the concept of solving an equation before learning about fractions. We'll teach kids to perform mechanical manipulations of decimal expansions before we ever begin explaining what any of those digits actually mean.
I am all for introducing things the correct way, but this example of yours just does not work for me.
Fractions as equivalence classes? or as a solution to an equation. No. That might work for you. In fact I might even argue that it only works for you - the child idealization created by "current you". There are many things wrong with US education. But I do not think introducing fractions using concrete examples is one of the problems. Of course for each his own and in all fairness it is also how the analogy is worded exactly. If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
It's fine if this wouldn't work for you. Nothing works for everyone, and that's sort of the point. Kids are funneled through an extremely rigid sequence of topics focusing entirely on rote application of manipulative techniques. Anyone who doesn't conform to this sequence is just left to painfully deal with it on their own.
I remember a bit later in school I discovered algebra on my own and became really interested in it. I asked if I could take a class on algebra ahead of schedule and was virulently refused by the school counselor, who scoffed at the very notion and declared it to be impossible. Her response was so indignant that it offended me very deeply and caused me more or less to rebel against school in general. I ended up turning away from math completely for many years and didn't come back to it with any real interest or passion until I was about to graduate high school.
> If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces. Combined they might have the same mass or the same number of calories or the same area, but the numbers you just gave me don't have units and can't be measures of area since they're independent of the radius of the pizza.
> > If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.
> Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces.
Why are we using pizza pies in the first place? I vaguely remember being taught something about pie as well. Instead how about this:
You have a piece of string 1 m in length. You cut it into 2 equal pieces. We can denote the length of one piece as 1/2 m.
You have a piece of string 2 m in length. You cut it into 4 equal pieces. We can denote the length of one of these pieces as 2/4 m.
These pieces of string are the same length, therefore 1/2 m is the same as 2/4 m.
Although in my head this feels more like a nice way of demonstrating they are the same, I'm not so sure if it helps a lot with reasoning about fractions. But that's fine, like the article said it's good to be able to explain something in multiple ways.
Personally that's one of the things that absolutely fascinated me about arithmetic when I was young; That you can play around and have multiple ways of arriving at the same answer, and that if you follow the rules of math right, they always end up as the same answer, sometimes even unexpectedly so (for young me).
I like your analogy a lot. It's hard to me to say definitively but I think myself as a toddler would have found it more compelling than pizza and pie analogies. Of course, the same sort of manipulation can be done with pizza/pie slices, but it's a much more natural and straightforward manipulation with strings.
[I would edit this onto my comment above, but it's too late so I'm just adding it as a reply.]
I realize this is going to go far beyond the thoughts of a toddler learning about fractions for the first time via the pizza analogy, but I think it demonstrates that even the seemingly obvious pizza analogy is more nuanced than it seems.
I've never bothered thinking about this before, but it seems obvious to me now. Using just a straight-edge and compass, it is not always possible to cut a pizza into N standard-shaped slices for all positive integers N. Doing so is equivalent to constructing a regular N-gon, and this is only possible when N takes on the special form
N = 2^k * p_1 * ... * p_M
for some nonnegative integer k and some sequence of distinct Fermat primes (that is, primes of the form 2^(2^j) + 1).
Thus, you cannot slice a pizza into 7 slices using a straight-edge and compass. So in the pizza analogy the number 1/7 corresponds to something that is not physically achievable with standard implements.
Children take a long time to realise that one half of pizza is the same as two quarters. In general you can give a child less pizza and as long as it's the same number of pieces, they can't tell the difference. The well-known experiment is with presenting two sets of pieces of chocolate, one with less chocolate but more pieces and most children say the one with more pieces has more chocolate.
I'm not sure I believe this. My experience with children is that they're almost always smarter than I expected.
I think I've found the Wikipedia article about the effect, and the criticisms section [1] matches my bias, in that it suggests the children are confused because they are thinking not just about the material presented to them, but also why they're being asked.
Depends on what you mean by children. Most of these concepts appear as a child ages but I think they get out of the simple ones pretty early.
I know before a certain developmental state children will say a taller thin glass holds more then a short wide glass even if you pour them back and forth to show they are equivalent. Once they "get" the concept it is for life but before that it is impossible, logic be dammed.
A practical impact of this - tool users from metric countries don't think in fractions. The notion that the next larger wrench after 1/8 is 5/32 is alien to people brought up with the metric system. Metric bolt heads are integral numbers of millimeters.
The irrational part of this is that the 'wrench system' is actually sized in 32nds of an inch, and thus would be much easier to comprehend with 'improper fractions'.
Yep, there are many counter-intuitive aspects to the rational numbers. They're ordered just like the integers, and yet given any fraction there isn't a well-defined "next" fraction. But there are still "gaps" (irrational numbers). The same fraction has infinitely many representations. Despite that there are no more rational numbers than integers, yet the number of "gaps" far exceeds either of these. Not every fraction even has a finite decimal expansion. And fractions allow you to specify a level of precision for measurements that could never be achieved in the real world under any circumstances, which immediately raises questions about what these extraordinarily precise fractions actually refer to in the real world.
I honestly believe that lots of kids detect these issues on some level, but their curiosity and confusion remain unresolved, possibly for life.
I have some memories from first grade in a California public school (in Santa Cruz) of "finding a number we can put in place of x so that x + 2 = 4" and how "at some point you will learn how to deal with x + 4 = 2". In 3rd grade we explored prime factorizations, and I also remember an emphasis on clearing out greatest common divisors for fractions -- so, while it didn't encourage thinking of fractions as equivalence classes, it at least encouraged comparing things by canonical forms. I even remember around then an introduction to base four representations, probably to illustrate the "carry the ten" when adding.
My experience, in contrast to yours, was well-meaning teachers drilling the algorithms while hinting at some underlying structure. I suppose it was enough inspiration to figure out what math was about, and I eventually found myself in a math PhD program. (Despite the fact I had a hard time remembering the so-called "math facts." During competitions, I would try to calculate, slowly, something like 7 x 9 using something like (8 - 1) x (8 + 1) = 8 x 8 + 8 - 8 - 1, since the only "facts" I ever remembered were multiples of five, 6 x 8 and 8 x 8. Those competitions did make me think I was a bit bad at math!)
Though, let me share my story about fractions. In fifth grade, I got a day planner because the school pushed for everyone to get one, and in the front leaves there were various references, like a periodic table, lists of equations, etc., and one which caught my eye was "a/b + c/d = (ad+bc)/bd". For some reason I thought "oh, that is what fractions are" and I tried to explain to a teacher how this defines fraction addition, how finding a common denominator is just a way to calculate this, but instead I was told I was incorrect, and you just find common denominators. I can't say I didn't feel somewhat vindicated when I learned, much later, about the Grothendieck construction.
I also remember trying to find a formula for triangular numbers in sixth grade, and I will never forget the look of horror on my "home-room" teacher's face as she backed away while I explained what I was trying to do.
I've heard that the people in the US who have developed a math phobia or who think that they have an inability to understand math self-determine this fact after particularly tough math concepts are being taught. IIRC, learning fractions is one of the first of those, and creates the largest cohort.
I believe it. I've known a lot of adults who don't quite get fractions. They generally also have no trouble working with money and giving change, but no ability to understand interest, especially compound interest. Somehow, the educational system is failing here, not the kid.
You're pinning too much on supposed bad teachers, and not enough on your past-self's youthful incomplete understanding.
1/2 of a pie is 2/4 of a pie, 2 slices of four is the same fraction as 1 slice of 2. It's exactly the same idea as the equivalence classes, where the ratio matters but the description does not. You just didn't understand that at the time.
> the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom.
My kid is in first grade and his homework is to solve equations of the form "7+2=__" and "9-__=7"
With all due respect, that sounds like straight-up circular logic to me. You're arguing that 1/2 = 2/4 because "1/2 [...] is [...] 2/4". You're just restating the conclusion as an explanation of itself, which is completely unsatisfactory.
Another way of putting it:
> 2 slices of four is the same fraction as 1 slice of 2.
The phrase "is the same fraction as" is obviously inappropriate and explicitly circular and self-referential in any definition of "fraction". I don't think you've thought about this as deeply as you think you have.
The teacher would accept 9 or 0 as an answer, not 18. A parent (with math degree even) who was volunteering pointed out that 18 is correct. The teacher ruled against this, arguing that 18 was unacceptable because math with 2-digit numbers hadn't been covered yet.
This was at a "good" school, in a county full of nerds.
Such logic. Poor kids... and people wonder why kids hate math and give up on it.
I suppose the "logic" is that stuff involving 9 could connect up with stuff involving 0 or 9. Since all other 1-digit answers seem more distantly related, and the answer has to be 1-digit because 2-digit hasn't been covered, 0 and 9 are the most attractive answers and therefore correct. ???
Remember, that was a class with smart kids in a school in a good area. Imagine what it might be like in not-so-good areas or in the classes that aren't for smart kids.
People who truly hate math are teaching. My sister-in-law is an example. She got a "degree" in "early childhood education". It's a joke. The hardest math she had to pass was algebra, as is taught in 7th grade. Boy did she complain about that class. She thought it was really hard to pass algebra. I'm horrified she got more than halfway through high school without it, yet there she was, taking it in college.
Teachers have serious credential inflation. Many have Master's degrees... but are those degrees worth anything?
I think the tricky thing is that they are going to encounter fractions in everyday language before they are at the stage of discussing different fractions as being equivalent. I've got a 3 year old and 5 year old. The 3 year old uses 'half' whenever something is divided in two, regardless of the size of the pieces, even if I correct him. The 5 year old on the other hand is capable of doing addition/subtraction with halves, quarters etc. but hasn't encountered that at school yet and hasn't encountered fractions like 2/4 because people never refer to that in real life. By the time they learn properly about fractions in school, he's already going to have had a lot of experience of using them in a non-formal setting which is inevitably linked to concrete concepts. On the other hand, he has got a mother with a PhD in maths, so I suspect he might get introduced to the concept of equivalence classes rather earlier than most kids :-)
(By the way if anybody out there does have children that age, I recommend the game Fraction Formula which has tubes in which you put cylinders in to representing different fractions).
Your desired explanation also would prime kids for introduction to other similar number sets, like the complex plane.. easing understanding significantly in comparison to referring to them as 'imaginary numbers'
I think it's because all my teachers operated off the assumption that kids can't possibly understand abstract definitions, so everything must be explained via analogies. So nobody could tell me that fractions are equivalence classes. Everything was instead explained in terms of pies, which certainly seems like it would be intuitive, and there's really nothing technically wrong with it, but it encouraged me to think of, say, 1/2 and 2/4 as different entities instead of two representations for the same entity (since, after all, cutting a pizza into two slices instead of four is just not the same thing). I didn't "get" fractions until on my own I came to the conception that these are actually kind of abstract concepts that go by many different names.
For me, I was taught too much to identify numbers with concrete objects in the real world. This sort of works for integers (as long as you ignore negative integers and even zero to some extent), but it basically breaks down for all other types of numbers, including rational numbers. So it's a really bad expectation to set in my opinion. But you can hardly find an elementary school teacher who will dare approach this topic.
I honestly think that if someone had just explained that "1/2" is a symbol we've devised to represent the solution to the equation 2x = 1 -- an equation which otherwise has no solutions at all in the integers -- I would've grokked it a lot faster. It would've been clear to me that we're introducing an entire new set of numbers distinct from the integers. These new numbers do not have a direct correspondence with physical objects in reality. And it would've been easier to see that the same solution must go by many other names as well. For instance, multiplying that equation by 2 on both sides gives us a new equation which must have the same solution: 4x = 2. So "2/4" must represent the same entity as "1/2". And so on.
But because US educational culture has decreed that basic algebra is more advanced than fractions, the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom. It is assumed as a basic fact of human life that kids cannot understand the concept of solving an equation before learning about fractions. We'll teach kids to perform mechanical manipulations of decimal expansions before we ever begin explaining what any of those digits actually mean.