My point was - With all of this talk (by reformers in general) of all of this untapped mathematical understanding in children, why not, at the very least, tap into it in algebra class and actually teach algebra?
Also, I wasn't addressing the validity of *actually* teaching students about logarithms and exponents early. I was addressing this paper's premise that our ability to readily use "inches", "feet", "yards" and "miles" (orders of magnitude) in context has anything to do with thinking about the math behind the mechanism, or the mechanism itself for that matter.
I am addressing this...
"The tendency of 'uneducated' people to compress the number scale for big numbers is actually an admirable way of measuring the world, says Philip Ball."
This is written as if people (in general, uneducated or not) actively and *thoughtfully* compress number scales. They do nothing of the sort. In fact, even when you explain the connection between these number scales and the math behind them, the majority still do not understand, even a smidgen, yet they can use the principle in conversation just fine. Similarly, and I hate bursting anyone's bubble, but the desert ants' dead reckoning skills are not the result of the thoughtful application of trigonometry to navigation.
Most of my thoughts on this originated over the last 30 years in my study of A.I. and cognition. It is only recently (the last 5 years or so) that I have applied this to education. I had long ago distinguished between reasoning and the other 95% (100% for most) of what we do day to day. I am not scoffing at that 95%. It is that 95% that is the ultimate goal for A.I. The language. Understanding context. Operating machines. But reasoning, abstract thought and mathematics is not in that 95%. Mathematics, at least beyond arithmetic, is in that other 5%. It requires one to think.
With regards to your suggestion for teaching more advanced topics earlier, sure. That would work fine for the math-smart kids and it should be done anyways. Compared to what we had I think many smart kids today are being defrauded. But in the context of the vast majority that don't get it, that wouldn't work at all. I have watched them try to teach algebra a dozen different ways. With pictures. With computers. With writing. With modeling. With social studies. With peer instruction. With discovery. Etc. Etc. Etc. And they fail. My conclusion...
Algebra is too hard for these students because these students are not thinking. My solution. Make sure that they can think first, before you teach them algebra. Stop taking for granted what we know is not true! And that doesn't mean teach them formal logic. They will suck at that just as well. Math-smart kids can take these courses and thrive because they think. The others wilt and die because they don't. Courses like algebra work great for kids already thinking. That is historically what that track was about. Putting everyone in algebra might have been a noble thought a long time ago, but at this point with what we now know, it is nothing but irresponsibility and negligence.
When kids don't get algebra it means THEY DON'T GET ALGEBRA. Back up and work on their thinking skills. And also remember, very few people use algebra in a profession. There are other pre-algebra skills that are universally used.
I have kept a pretty good record of teaching mathematics to my son. He is only one case but a fairly typical one. He is not overly enthused by mathematics, but his enthusiasm grows year by year. I won't say that he is as challenging as a student living in poverty or a dysfunctional home, but he also isn't like teaching me either. My son is applicable to this conversation because getting him into mathematics was as much about getting him to think as it was about teaching him mathematics. I don't think you can teach someone to think but you can certainly coach them and I certainly have to coach my son at times. I can tell just by how he answers a question, even if technically correct, that he is wrong. I can (most of the time) tell what he is thinking, and almost all the time, what he isn't thinking. And I am not too bashful to point these things out to him. I am not teaching him to think. I am coaching him to use what ability he has to think to its fullest. I am not trying to! say he is a slouch, but he certainly won't be taking on the old man at this sport.:)
On Aug 24, 2013, at 4:28 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> R Hansen says >> Teach students logarithms in algebra. > > Why not wait till calculus, or perhaps what used to be (still is?) called "advanced calculus". You can't give a proper accounting till then. > > But of course children can understand the basic idea long before then. > > Cheers, > Joe N