Any article that has me yelling out loud while I read it must be a good one. Highly recommended for several reasons:
http://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-schools
2010-03-25
2010-03-04
Equals Or Not Equals
In my remedial algebra class, I assign these stupendously short homeworks, but require precisely written work -- professional format, one full operation per line, justification in words at each step. (I don't have much evidence that it helps the students, but at one point last year I had an emotional meltdown trying to read students' normal math writing, so there you go -- it's quite literally defending my own sanity.)
So on a lot of the homework, inevitably, some fraction of the class leaves out all of the equals signs. And since I go through and just check off each line as being correct or not, this results in every single line being wrong, resulting in zero points for the entire assignment. Even if everything else was correct, even if all the final results are the same as the answer sheet.
So in one sense, kind of brutal. In another sense, this requirement was made abundantly clear in class, including a fully written example from me on the submission page that they're turning it in on. So if you can't follow that simple requirement, one might argue it's an eminently reasonable end result.
But here's another way of looking at it: Each line of math is, effectively, a sentence. (A highly condensed sentence in specialized notation, but the same nonetheless. It can be re-hydrated back into normal English at any time.) And the equals sign is the verb "to be". It's the most important verb in any language! What if someone were in a writing class and submitted a paper without any verbs? What if they were entirely unable to say "you are", "I am", "he is" anything at all? Would an English teacher totally flip out? You bet they would.
And that's exactly my reaction when I see a paper like that.
Complete sentence: 5 + 3 = 8 ("Five plus three is eight.")
Sentence fragment: 5 + 3 8 ("Five plus three eight.")
Repeat that sentence fragment for 10 or 20 problems per page and see what happens to your eyeballs. Slings and arrows and all that.
So on a lot of the homework, inevitably, some fraction of the class leaves out all of the equals signs. And since I go through and just check off each line as being correct or not, this results in every single line being wrong, resulting in zero points for the entire assignment. Even if everything else was correct, even if all the final results are the same as the answer sheet.
So in one sense, kind of brutal. In another sense, this requirement was made abundantly clear in class, including a fully written example from me on the submission page that they're turning it in on. So if you can't follow that simple requirement, one might argue it's an eminently reasonable end result.
But here's another way of looking at it: Each line of math is, effectively, a sentence. (A highly condensed sentence in specialized notation, but the same nonetheless. It can be re-hydrated back into normal English at any time.) And the equals sign is the verb "to be". It's the most important verb in any language! What if someone were in a writing class and submitted a paper without any verbs? What if they were entirely unable to say "you are", "I am", "he is" anything at all? Would an English teacher totally flip out? You bet they would.
And that's exactly my reaction when I see a paper like that.
Complete sentence: 5 + 3 = 8 ("Five plus three is eight.")
Sentence fragment: 5 + 3 8 ("Five plus three eight.")
Repeat that sentence fragment for 10 or 20 problems per page and see what happens to your eyeballs. Slings and arrows and all that.
2010-02-25
Lottery Puzzle
Let's say there's a lottery run once every month that costs $100 to enter. Chance of winning is 1 in 20,000. The prize is 5 million dollars. Should you play this lottery?
2010-01-28
"Community" TV Show
The NBC comedy "Community" gets way more things right than they have to. For example: The sexpot statistics professor.
But wait, there's more. I've long held a theory about performance that, "No actor can play more intelligent than they are in real life". Largely this is a matter of diction: I hear actors stumbling over, or putting incorrect emphasis on, pieces of vocabulary that they don't really know or use themselves in daily life. "Community" places really great actors throughout the ensemble; they're delivering lines like "as long as we keep the work/fun ratio the same I want to keep seeing you", and "the deception is making the sex 36% hotter" so fluidly that they slide by almost without me realizing that they were meant to be jokey.
But wait, there's more. I've long held a theory about performance that, "No actor can play more intelligent than they are in real life". Largely this is a matter of diction: I hear actors stumbling over, or putting incorrect emphasis on, pieces of vocabulary that they don't really know or use themselves in daily life. "Community" places really great actors throughout the ensemble; they're delivering lines like "as long as we keep the work/fun ratio the same I want to keep seeing you", and "the deception is making the sex 36% hotter" so fluidly that they slide by almost without me realizing that they were meant to be jokey.
2010-01-26
Elementary Teachers
Did you know?
This in a larger article finding:
http://news.yahoo.com/s/ap/us_sci_fear_of_figures
Edit: The article linked above is gone from Yahoo News. But you can still see that via the Internet Archive's Wayback Machine. And here's a citation and link to the original academic article:
Beilock, who studies how anxieties and stress can affect people's performance, noted that other research has indicated that elementary education majors at the college level have the highest levels of math anxiety of any college major.
This in a larger article finding:
But by the end of the year, the more anxious teachers were about their own math skills, the more likely their female students — but not the boys — were to agree that "boys are good at math and girls are good at reading." In addition, the girls who answered that way scored lower on math tests than either the classes' boys or the girls who had not developed a belief in the stereotype, the researchers found.
http://news.yahoo.com/s/ap/us_sci_fear_of_figures
Edit: The article linked above is gone from Yahoo News. But you can still see that via the Internet Archive's Wayback Machine. And here's a citation and link to the original academic article:
Beilock, Sian L., et al. "Female teachers’ math anxiety affects girls’ math achievement." Proceedings of the National Academy of Sciences 107.5 (2010): 1860-1863. (Link)
2010-01-03
Phonics and Bases
Speaking of the "Math Wars", here's an observation I made about teaching the most basic elementary-school subjects.
When I was very young, we were taught reading by "phonics", i.e., sounding out the letters of unknown words, and then thinking about how those sounds related to words we already knew. Sometime after that, phonics was dropped for a "whole word" approach, but it seems like the pendulum might be swinging back these days.
Why do "phonics" make sense as an instructional strategy? Because our system of writing is a technology based on exactly that principle. The whole point to our alphabet is that it is phonogrammatic, i.e., written symbols represent spoken sounds. This system of writing is meant for there to be an obvious connection between what we write, and what we say. (Of course, this is totally different from logographies such as Chinese whole-word characters and Egyptian heiroglyphics, but that's an entirely different story.) In teaching a child to read, why would you not use the language tool the way it was designed to be used?
Similarly, the traditional way of teaching arithmetic is to memorize a small number of basic facts (addition and multiplication tables), and then learn fundamental written procedures for adding, subtracting, multiplying, etc., large (many-digit) numbers. More recently, we've had to deal with the "Math Wars" is which training in time-tested prodecures has been frowned upon as too authoritarian (or something). Rather, there appears to be extensive time spent in base-10-system conceptual understanding, and then inventive or creative exhortations to make up your own multifarious addition, subtraction, etc., algorithms.
Why do "procedures" make sense as an instructional strategy? Again, because our system of written numbers is a technology based on exactly that principle. The whole point to our place-value system (base-10) is that it is intended to make the specific procedures for adding, subtracting, and multiplying simple, straightforward, and consistent for everyone! Consider the history of written numbers: With ancient systems like, again, Egyptian numerals or Roman numerals, there was no way to get addition or multiplying accomplished by simple writing or mental effort. Without a fixed base system, numbers don't "line up" the way they do for us. To get any arithmetic done (including total sales or tax calculations), you had to go to a licensed counter (think: public notary) with their counting board or abacus tool for use as a mechanical calculator, and trust that the computations they did there were correct. You were entirely at the mercy of this elite, cryptic profession, just to do simple addition.
At some later point, our Hindu-Arabic numeral system was invented and made its way to the West. This system of writing numbers is meant for there to be an obvious connection between the numbers we write, and how to add and multiply them, via a specific written procedure. Kings and princes were astounded at the prospect of people being able to do arithmetic on paper, or in their head, without using an abacus as a calculator. It's like magic! It's not some kind of accident that we use a positional-number system, it was engineered that way only so that we could use a specific adding and multiplying algorithm. In teaching a child to do arithmetic, why would you not use the written number tool the way it was designed to be used?
In summary, it's fascinating that both reading & writing instruction have taken almost exactly parallel paths in the last few decades in America. In each case, they have abandoned the rationale for which the writing tool was designed in the first place. It's like showing someone a power saw for the first time and asking them, "Can you think of something you might use this for?" To leave out the intention of the tool is to miss the whole point of it. We should play to the strengths of our writing technology, and not frustrate ourselves fighting against it.
When I was very young, we were taught reading by "phonics", i.e., sounding out the letters of unknown words, and then thinking about how those sounds related to words we already knew. Sometime after that, phonics was dropped for a "whole word" approach, but it seems like the pendulum might be swinging back these days.
Why do "phonics" make sense as an instructional strategy? Because our system of writing is a technology based on exactly that principle. The whole point to our alphabet is that it is phonogrammatic, i.e., written symbols represent spoken sounds. This system of writing is meant for there to be an obvious connection between what we write, and what we say. (Of course, this is totally different from logographies such as Chinese whole-word characters and Egyptian heiroglyphics, but that's an entirely different story.) In teaching a child to read, why would you not use the language tool the way it was designed to be used?
Similarly, the traditional way of teaching arithmetic is to memorize a small number of basic facts (addition and multiplication tables), and then learn fundamental written procedures for adding, subtracting, multiplying, etc., large (many-digit) numbers. More recently, we've had to deal with the "Math Wars" is which training in time-tested prodecures has been frowned upon as too authoritarian (or something). Rather, there appears to be extensive time spent in base-10-system conceptual understanding, and then inventive or creative exhortations to make up your own multifarious addition, subtraction, etc., algorithms.
Why do "procedures" make sense as an instructional strategy? Again, because our system of written numbers is a technology based on exactly that principle. The whole point to our place-value system (base-10) is that it is intended to make the specific procedures for adding, subtracting, and multiplying simple, straightforward, and consistent for everyone! Consider the history of written numbers: With ancient systems like, again, Egyptian numerals or Roman numerals, there was no way to get addition or multiplying accomplished by simple writing or mental effort. Without a fixed base system, numbers don't "line up" the way they do for us. To get any arithmetic done (including total sales or tax calculations), you had to go to a licensed counter (think: public notary) with their counting board or abacus tool for use as a mechanical calculator, and trust that the computations they did there were correct. You were entirely at the mercy of this elite, cryptic profession, just to do simple addition.
At some later point, our Hindu-Arabic numeral system was invented and made its way to the West. This system of writing numbers is meant for there to be an obvious connection between the numbers we write, and how to add and multiply them, via a specific written procedure. Kings and princes were astounded at the prospect of people being able to do arithmetic on paper, or in their head, without using an abacus as a calculator. It's like magic! It's not some kind of accident that we use a positional-number system, it was engineered that way only so that we could use a specific adding and multiplying algorithm. In teaching a child to do arithmetic, why would you not use the written number tool the way it was designed to be used?
In summary, it's fascinating that both reading & writing instruction have taken almost exactly parallel paths in the last few decades in America. In each case, they have abandoned the rationale for which the writing tool was designed in the first place. It's like showing someone a power saw for the first time and asking them, "Can you think of something you might use this for?" To leave out the intention of the tool is to miss the whole point of it. We should play to the strengths of our writing technology, and not frustrate ourselves fighting against it.
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