IMO with a well-designed quiz it shouldn’t matter if the kids bring their calculators or not. As for memorizing frequently-used results like 6 times 9 and log(2), I’m sure it has been extensively studied whether it helps comprehension or not and someone will be along to tell us. It certainly doesn’t hurt, any more than it hurts to memorize the alphabet.
My experience has been the exact opposite, and while still anecdote I have always found that encouraging people to learn why they do something amortizes out over the long run and results in far more flexibility. In fact I cannot think of a single high performing co-worker who didn’t take that approach.
Still anecdote, but I had an extreme negative view of most home schooled co-workers because they were so incapable of moving into new areas, or even solving complex problems that they couldn’t fit on a spreadsheet. While this is almost certainly a false assumption I will be quite impressed the first time I knowingly work with a home schooled student, who had pre-sputnik education, who can keep up.
Especially now that tensors and other high dimensional unimaginable datasets are becoming the norm.
To be clear I am not talking about common core or any other trend, but encouraging children to take the extra step and learn why they are doing something, vs shortchanging their future success with rote memorization is an important goal.
(I am not picking on home-school as a concept, just being realistic that parents have to teach to their own skill level)
I agree. (Anything else is testing a weirdly handicapped skill anyone could be trained to but that never comes up in the modern world. Any idiot already knows how to patiently, mechanically follow the rules of any algorithm you write out before them, with pen and paper and time. Why is it so important that they be drilled to practice some algorithms and carry them out quickly without slip-ups?)
I wonder how many people here have the decimal expansion of log(2) memorized to any length, even the first digit.
As to natural log(2) I cannot say, but if you scroll back a bit you can read a bunch of professional engineers saying they have log[sub]10/sub ≈ 0.3 “burned into [their] mind”, presumably from using it so much, which is the cohort I was thinking of when I wrote my comment.
Ah, of course. I forgot all about those posts, but I suppose lots of people have that approximation in their heads implicitly if not explicitly, in the 1024 ≈ 1000, kilobytes, megabytes, etc., way.
And I could tell you very quickly that the first digit of log(2), regardless of whether you meant decimal or natural, is 0, followed by a decimal point.
Ah, I knew I should’ve guarded against this…
In all seriousness, I can give three more digits of ln(2), but that’s just because I’ve taught/tutored on half-life problems multiple times. Anyone who works with half-lives a lot could probably do the same. And of the majority who don’t work with half-lives (or doubling times or other similar concepts), I suspect that almost none could.
Thanks for using the ISO name of ln(x) which reminded me that log(x) is ambiguous, I went to check it with python using log(x), which is natural, but was thinking of the astronomy log(x) which is the common version, or lg(x).
Related to this topic, logarithm tables seem to use the common logarithm lg(x) while python, my current go to language uses the natural form for log(x). But this demonstrates where rote memorization falls down, as memorizing the tables for log(2) without realizing what is implied by the context would lead to a wrong answer.
For those who don’t want to calculate it.
natural logarithm: ln(2) ~= 0.6931471805599453
common logarithm: lg(2) ~= 0.3010299956639812
This reinforces my belief that learning why you do something is critical. A chemist talking to a biologist, or an engineer talking to a physicist would assume different meanings for log(2) with very different values.
There are explicit applied uses I’ve come across for factoring polynomials but it’s… very abstract and “who cares” for anyone not in a specific field.
In machine learning, we often talk of datasets being “linearly separable”; that is, given a bunch of points belonging to category A and a bunch of points belonging to category B you can draw a line between them. One method of finding this line involves (among other things) minimizing the distance between a line and the points of the two categories nearest each other.
Now, what if you can’t draw such a straight line?
Well, it turns out that due to some math I’m not going to get into you can actually fudge the distance function by first mapping the points you’re comparing into some other dimension where a straight line(/plane) can separate them, and then taking the distance of THAT.
It turns out that one of the secrets to doing this mapping is plain old polynomial factoring. If one of your fudged distance functions can be factored, you can very easily prove the dimensional mapping it represents*.
Like I said, good luck motivating that to a kid, hell, I probably didn’t even motivate that to you super well, but it is a specific applied use of factoring.
- Okay, more strictly:
You don’t actually want to map the vectors into a higher dimensional space, since it’s expensive (or, in the case of some kernels, impossible because they’re infinite-dimensional) to compute. Polynomial kernels are one type of kernel, and they were proven to work via factoring, but you actually don’t want to start with a fudged distance function and work out a mapping, since all you need is the fudged distance function.
A large, large amount of the things past Algebra 1 probably can’t be trivially motivated with direct, applied real world examples. Like, you rarely “use x to do y” so much as “you use x as a trick to manipulate y which does z and that’s why we can make video games” or something abstract like that. Almost everything has a real world use you may come across, but it’s not super obvious to explain. Like, even the example I gave above is kind of tangential to the idea of why you’d want to do that in the first place. If you wanted to be glib you could easily say “you can use polynomial factoring in marketing and business analysis” since that specific technique is a cornerstone of Support Vector Machines which themselves are very common in things like data and business analytics which even someone who never takes a higher level math class may end up working with more directly with modern analytics software (even if they don’t have to understand the complexities of how it works).
Very few things exist solely to be used in higher level mathematics, but they’re so far removed from the level of complexity the students are working with they’re difficult to motivate.
You could equally say factoring polynomials is used in the analysis and design of feedback control systems, or to study population growth dynamics, or whatever. A basic technique like that has myriad uses. It should not be hard to dazzle children by mentioning as many advanced applications as desired. I suppose you could mention them if you feel it would be motivating.
New Math itself was never concerned with any of this, only with finding ways to help young children develop their intuitive sense for working with numbers.
My favorite problem involving factoring polynomials is the Sicherman dice problem (design two nonstandard dice with positive integer number of dots on each face which, when rolled, produce the same probability distribution as standard dice).
There’s a natural correspondence summing independent distributions and multiplying polynomials, so that this turns into a problem about polynomials.
Furthermore, it turns out there is a unique solution. Recognizing this uniqueness requires understanding that integer polynomials actually have essentially unique factorizations into irreducible integer polynomial factors, just like integers themselves essentially uniquely factor as products of primes. (No one ever bothers teaching why this is so, but then again, no one even ever bothers teaching why integers have unique factorization; it is, when mentioned, just handed down as a fact from on high, or assumed by students to be more obvious than it is). But even if one doesn’t care to prove the uniqueness, and just cares about existence/constructing the Sicerman dice, finding a suitable polynomial factorization and playing around with it works.
Even as I dispute the value of drilling students on polynomial factorization, never let it be said that I do not care about polynomial factorization as a potentially interesting, potentially “useful” subject.
And to a computer scientist, to the extent that they care about what base they’re using for their logs at all, log() probably means base 2.
No, that’s cool. I appreciate you mentioning it.
Thanks, that was interesting. However, how will students know how to do factorization unless they are drilled on factorization?
By typing into a computer “factor(2x^2 + 9x + 6)” or whatever. Why is that any different than typing “sqrt(732)”, when that’s what’s needed? Is there any value in teaching students to take square roots without the help of a computer? Or more precisely, is there more value in it than in all of the other things we could be spending classroom time on?
(and actually, these aren’t even two different problems: sqrt(732) is exactly the same problem as factor(x^2-732))
After your first post, I was going to brag that I can tell you the first three significant digits of ln(2), but I could only get log(2) to one digit. Besides half-lives, it’s also used all the time in electrical engineering, with low-pass filters, RF skin depth, and probably more.
I didn’t even have to think - 0.693 just popped into my mind. You run into that number in so many places, that it’s just natural (for me) to remember it. Here’s a link to a thread in which 0.693 popped up http://boards.straightdope.com/sdmb/showthread.php?t=725687 - and if you’ve ever heard of the Rule of 72 (about interest rates Rule of 72 - Wikipedia), that’s another place where .693 comes up - the rule should really be the rule of 69, but 72 is close enough, and makes the math easier in most relevant cases).
Yes, but this seems like you think math classes should just be “Ok, today we are going to learn how to factor 2nd order polynomials. Type this into your calculator ‘factor (2x^2 + 9 + 6)’. The answers are the roots. That’s how you factor 2nd order polynomials. Now, lets move on to square roots”
That’s sort of how my kids are now, and why I was looking for a way to show them they need to learn how to do it.