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>Fractions are like groups of pencils or tables of girls and boys. No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table o

• simiones 6y ago
  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 you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests. All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered. The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it. There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability. > I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".
  • macspoofing 6y ago
    >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 arithmetic. If that's all we wanted, we could just give them an algorithm and an abacus and that's all they would ever need. Fractions can be used to think about ratios and things like that but they come with WAY more baggage because they are more than that. They are abstract concepts that underpin further abstract mathematics. >It's a bit contrived, but say I ask the pupils at the table to send someone away. They You can always add an ad-hoc extension to whatever the analogy you're using. I'm not saying that's wrong. I'm making no judgement call on how to teach mathematics to fourth-graders either (I don't know what the best way is). But I do want to point out that your extension to the analogy (negative fractions being equivalent to sending a student away) is because you've reached the limits of your initial abstraction. It's not a clean mapping either. For one thing, you just assumed the numerator is negative (student being away), but it is just as valid that the denominator may be negative (or both!). And it isn't clear that a negative numerator and a positive denominator is equivalent to a positive numerator and negative denominator. You either have to prove that from first principles using formal methods or you simply have to assert that to the fourth-grader and justify it with "it's just the way things are"
    • tsimionescu 6y ago
      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.