MOST BERKELEY STUDENTS DO NOT KNOW THAT 2^5=32 AND 2^10≈1000To which I can only say: Brad DeLong has not been paying attention!
How can they expect to survive in the modern world without knowing these things?
And why haven't they learned them?
I've just started reading the comments. Calculators in grade school aren't faring well thus far.
From the first Comment:
The best students are great, but the quantitative reasoning skills of even the average student at a good university are worse than those of a typical waiter/waitress 40 years ago.
invhand
I bet the CS students know that--most of them probably know a bunch of the powers of 2. But anything higher than 2^4 I have to think out (or did, until I started doing a lot of math with my kids, and also I'm married to a CS guy who does know them cold).
ReplyDeleteI went to Cal in the old days before we all had calculators and never took any math higher than trig. Humanities types don't have to take that much math.
The first few commenters are all directly connecting knowledge of arithmetic with mathematical reasoning.
ReplyDeleteHas no one heard of the math wars?
ReplyDeleteBut I do not quite understand why the specific case of knowing the powers of two should be so important. Why do we all need to know that for survival in the 21st century? And why is it implied that everyone did used to know it?
ReplyDeleteIts a strange thing and in my opinion they should take reference from online tutoring sites because its easy to acces them from anywhere.
ReplyDeleteI didn't know that 2^10 was 1000. I always thought it was 1024.
ReplyDeleteThey don't know it off the top of their heads? Or they cannot calculate it quickly? IMHO, there is a *BIG* difference between the two. I'm not sure why a student would need to have either of those two facts memorized (the squares and cubes are the ones I learned back in the day).
ReplyDeleteGlen,
ReplyDeleteI was thrown by that, too. Then I noticed that the original has an approx. sign (2^10≈1000), not an equal sign.
oh! sorry - that's a typo - !
ReplyDeleteI'll change it
Well, it seems that this guy actually does not get what he is saying because these are not vital in math. In reality, algebra forms the basis for most of calculus and specific exponents are not nearly as important as being able to work with the general notion of one. If he has mastered algebra and trignometry, he would be just fine in calculus. Furthermore, statistics has started to become essential to many fields of study and it may be more viable as a field of study. I am not a professional in any sense because I am currently in college as a full time student but if I had to pick something that would be disturbing, it would be if a student did not know powers of ten. They are extremely important in any field of science in the form of scientific notation.
ReplyDeleteAs for the partial products algorithm, it could actually have a good educational value at the lower secondary level, which would be around 9th grade. This is because it could be used to demonstrate how to prove that an algorithm is equivalent to another one. The partial products algorithm actually comes directly from the standard algorithm if we write out both of the factors in extended form. This can be proven with the distributive property.
I was just going to let this one go, but oh, well....
ReplyDeleteIf your conceptual knowledge of "higher arithmetic" is solid, you can turn it into a real-time thinking tool by having a few numbers memorized (some squares, square roots, cubes, sin & cos, logs, powers of two...), having some estimation shortcuts memorized, and using the whole set frequently for quick mental images, estimation, and back-of-the-envelope calculations.
Many people still think this way, though far fewer than in the days before calculators. Interviews for technical jobs often include estimation questions to see if the candidate is the sort who is comfortable "thinking in math."
On their own, the powers of two, like most of the other numbers, aren't worth memorizing. You don't often have to answer the question, How much is 2^x?, and when you do, you can just figure it out.
But in combination with the right concepts and other memorized facts, the powers of two pop up all the time in the middle of questions about other things. A question isn't about powers of 2 per se, yet math is all of a piece, and the powers of two end up being part of the solution process.
For example, the other day I wondered how much 60^10 would be, roughly. I noticed (memorization is the seed; noticing is the fruit) that 60 was almost 2^6. So my answer would be about (2^6)^10, which would be (2^10)^6, or about 1000^6, or about 10^18.
If the question had been 80^x, I might have called 80 "2^3 * 10" and solved it that way. If 50^x, I might have called 50 "1/2 * 10^2". If 30^x, I would have called 30 "2^5", and so on.
And powers of two come up even more often in questions involving discrete math ("what are the chances of...?", "how many different cases would we have if...?", "how many SKUs could we represent in our online catalog using only...?", etc.)
Of course, memorization has a point of diminishing return for everybody, but you can do a lot with a memorization burden equivalent to learning another 12x12 times table. Expecting Berkeley students to have twice the memorized constants of a third grader and to know how to use them doesn't seem that out of line to me.
Unfortunately, the K-12 school system talks a lot about "understanding concepts," but overlooks using those concepts in daily thinking, which makes memorizing the constants which would greatly simplify such thinking seem useless.
I frequently count in binary on my fingers (it is particularly convenient when what you are counting are numbers). To convert back to decimal, I need to know the powers of 2 up to 2^10, but essentially all computer scientists have memorized those values years ago.
ReplyDelete