Funny that I was just teaching that lesson about fraction comparison last Saturday morning, but I was using fraction pairs that are a bit more challenging (from the excellent textbook Algebra by the late I.M. Gelfand and Alexander Shen).
I just put problem 40 from the book, which I taught last week to children of third-grade to fifth-grade age, into Wolfram Alpha's natural language interface.
The Wolfram Alpha input and output is convenient for making the teaching point, and could spark a discussion about problem 41, which is
41. Which is greater, 12345/54321 or 12346/54322?
Of course a sensitive mathematics teacher is supposed to recognize at once that what is really being asked for by the second problem is a way to generalize when a/b is greater than (a+1)/(b+1) and when it is not. I will wrap up that part of the lesson next week.
I have posted recently here on HN that "There will continue to be an important role for in-person teachers,"
even after online teaching tools become much more fancy. I recommend some good tools (I don't think Khan Academy is the best available online mathematics teaching tool, but its price point is appealing) in that comment. I also include links in that comment to thoughtful recent articles about improving mathematics education. A skillful teacher will teach learners how to use tools, when tools are suitable for getting the answer, and how to use the unaided human brain and speech when that should be enough to get (and EXPLAIN) the answer. A big part of mathematics learning is learning how to use appropriate tools and methods in different circumstances. I don't decry online mathematics learning tools; I model in the classroom using the good-old human brain, sometimes with some help from pencil-and-paper or whiteboard-and-marker calculations, to puzzle through challenging mathematical problems
and get reality checks on whether the procedure used to reach a solution is correct or not.
AFTER EDIT: After posting this comment, I asked my Facebook friends (who include a number of professional and amateur mathematics educators, including homeschooling parents who have brought up International Mathematical Olympiad gold medalists) about the blog post submitted here, and one of those friends suggested that the blog post author look closely at the Art of Problem Solving
model of online mathematics education. "The medium is not the message, because the medium is only stepping in to do (interactively) what you would do in person if you could, and instead distributing the teaching resource more widely, but basically in the same mode." I agree with that suggestion, and with that comment on whether or not the medium is the message if online mathematics education is well done.
This problem was enough hard for me that I want to post my solution.
(a+1)/(b+1) = (a * (1 + 1/a))/(b * (1 + 1/a))
Which in turn, is the same as (a/b) * ((1+1/a)/(1+1/b))
This can only be greater than a/b if (1+1/a) / (1+1/b) > 1.
A fraction is greater than 1 if the numerator is greater than the denominator, which means 1/a > 1/b. Dividing by a greater number leads to a smaller result, so this happens if a < b.
12345 < 54321, so 12346/54322 is greater.
Did I overlook a much easier way to solve this, or is third grade much better at math than I am now?
I suck at pure math, so I tend to think of this in some sort of analogy.
Suppose the fraction a/b is some statistic you are trying to measure, say, a batting average or percentage of correct notes played in Guitar Hero. (a+1)/(b+1) would be the new fraction after you got the next one right. By getting the next one right, did you improve your score?
Of course, if you have a perfect record already, getting an additional 1-for-1 won't change anything. And if a>b, then you'd have to somehow score more than 1 point per attempt in order to maintain the same ratio, so (a+1)/(b+1) would be lower.
I used to be a mathematician, and I think your comment is everything that math education in school should aspire to be :-) Another example in the same vein is Terry Tao's airport puzzle (http://terrytao.wordpress.com/2008/12/09/an-airport-inspired...), scroll down to Harald Hanche-Olsen's comment for the best explanation.
That is a very nice explanation. It shows why intuitively we expect (a+1)/(b+1) to be greater. But note that there are some unstated assumptions. Namely that both a, b > 0. And so we see that intuition at times carries unstated assumptions that can be a trap in a generalized situation.
Tokenadult and codehotter didn't mention the case when a or b < 0.
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
I just put problem 40 from the book, which I taught last week to children of third-grade to fifth-grade age, into Wolfram Alpha's natural language interface.
http://www.wolframalpha.com/input/?i=Is+10001%2F10002+greate...
The Wolfram Alpha input and output is convenient for making the teaching point, and could spark a discussion about problem 41, which is
41. Which is greater, 12345/54321 or 12346/54322?
Of course a sensitive mathematics teacher is supposed to recognize at once that what is really being asked for by the second problem is a way to generalize when a/b is greater than (a+1)/(b+1) and when it is not. I will wrap up that part of the lesson next week.
I have posted recently here on HN that "There will continue to be an important role for in-person teachers,"
http://news.ycombinator.com/item?id=3555728
even after online teaching tools become much more fancy. I recommend some good tools (I don't think Khan Academy is the best available online mathematics teaching tool, but its price point is appealing) in that comment. I also include links in that comment to thoughtful recent articles about improving mathematics education. A skillful teacher will teach learners how to use tools, when tools are suitable for getting the answer, and how to use the unaided human brain and speech when that should be enough to get (and EXPLAIN) the answer. A big part of mathematics learning is learning how to use appropriate tools and methods in different circumstances. I don't decry online mathematics learning tools; I model in the classroom using the good-old human brain, sometimes with some help from pencil-and-paper or whiteboard-and-marker calculations, to puzzle through challenging mathematical problems
http://news.ycombinator.com/item?id=2760663
and get reality checks on whether the procedure used to reach a solution is correct or not.
AFTER EDIT: After posting this comment, I asked my Facebook friends (who include a number of professional and amateur mathematics educators, including homeschooling parents who have brought up International Mathematical Olympiad gold medalists) about the blog post submitted here, and one of those friends suggested that the blog post author look closely at the Art of Problem Solving
http://www.artofproblemsolving.com/
model of online mathematics education. "The medium is not the message, because the medium is only stepping in to do (interactively) what you would do in person if you could, and instead distributing the teaching resource more widely, but basically in the same mode." I agree with that suggestion, and with that comment on whether or not the medium is the message if online mathematics education is well done.