Long gone are the days when slide rules, used aboard the Apollo missions putting boot prints on the moon were usable here on earth. Not because they fall short when solving some seriously sophisticated problems but because slide rules don't do the simplest arithmetic operations: addition and subtraction. In the age of slide rules it was a reasonable assumption that a slide rule user could add two or more three or more digit numbers.
We are now confronted with a reality (and a future) where very few if any of the people born in the United States can add or subtract. We're talking the simplest of arithmetic, not math[1]. This has come to the fore in local mass media with a debate on how we teach addition to children, specifically the various journeys between $$ 19 + 27 = ? $$ and $$ 46 $$ Old farts write down 19 then write 27 below aligning along implied decimal points and draw a bar below 27. Then they add 7 to 9 and come up with 16 writing the 6 below the bar and the 7 and then they write the 1 above the 1 in 16. This is called "carry the one"[2] and this is then added to the 1 from 16 and the 2 from 27 with the resulting 4 written to the left of the 6 below the bar. Voila! 46! This approach is well understood by folks who can count points and keep score in bridge. Explains the dearth of Bridge Clubs in Georgia Public Schools, eh?
There are some serious takeaways from the Great Addition Debate. First, we've lowered the bar to the point that we'll celebrate if the average student graduates FROM high school with the ability to add two numbers. Not three nor any more than three. We'll throw a party if more than five out of ten graduates arrive at "46" for the above problem even if they have to copy the answer or have it given them by a teacher. The second issue is the addition algorithm schools are using: $$ 19 + 27 = $$ $$ (10 + 9) + (20 + 7) = $$ $$ (10 + 20) + (7 + 9) = $$ $$ 30 + 16 = $$ $$ 30 + (10 + 6) = $$ $$ (30 + 10) + 6 = $$
and finally $$ 40 + 6 = 46 $$
Execution of this algorithm in a classroom environment pretty much precludes adding numbers with more than two digits due to limitations on time and "Smart Board" space. Should Georgia Public Schools be successful we will pump out graduates who can successfully add any two two digit numbers. Given enough time and paper.
While this situation may have devolved from a more proficient past it is not entirely unintentional and is currently embraced. Asking today's teachers to do a Mad Minute[3] would produce appallingly disappointing results so don't ask them to teach and demand this of tomorrow's leaders (and teachers). Other rationalizations include the bruised and broken self esteems brought on by the Drill and Kill approaches that are pretty much required by the Old School techniques and the fact that the New Math supports "conceptual learning." Or at least the learning of concepts. This is said to include powers of 10[4], as they decompose "16" to "one in the ten's column and six in the one's column" with the illuminating explanation that the "ones column" is "ten to the zeroth power" and the "ten's column" is "ten to the oneth power" at which point we've completely abandoned addition in favour of a hallucinagenic trip down a rabbit hole.
Suppose we demand the basketball couch embrace a concept-centric pedagogy[5]. He would be forced to draw out (with 'x's and 'o's? that almost looks like algebra. or does he need manipulatives? animated power point?) a simple pick and roll play, making sure the team groks the concept. These students also grok dribbling, layups, free throws and general ball handling, even the bounce pass, it's origins, benefits over other forms of passing and the cultural implications of its behind the back use. Now these kids have little or no actual practice and an outstanding performance would be dribbling five times before bouncing the ball off your own foot sending it out of bounds or directly into the hands of an opposing player. Of course the coach is not expected to field a winning team. No, he is going to field a championship team, as good as any in the state, probably the country. All the parents, the principal and even the superintendent of the school system will declare this team "straight A ball players." But honestly, if they ever took the court against a pick-up team of 3rd graders who actually play for 3-4 hours a day who do YOU think would win?
But focusing on concepts over capabilities moves us from the objective world of "the answer is 46 and all other answers are wrong" to the more comfortable subjective world of "critical thinking" and "let's understand the concepts" where flexibility in grading supports the demand for A's. And we also hide behind the calculator excuse (everybody has one, who needs to do their own arithmetic?) to explain our exclusive presentation of the abstract and subjective rather than the concrete and tangible. To be fair the critical thinking part, were it really there, might actually trespass on mathematics, asking and hopefully answering the question: "are these the two right numbers to add and how do we know?"
Similar but perhaps more disturbing is the "nobody really uses math" excuse for cultivated math ignorance. It's almost like Godwin's law, but you cannot have a tweet fest on this topic without someone dragging out "I took calculus at Tech twenty years ago and haven't used it since--it was just a clean out course." That cannot be allowed to slide by without comment. If calculus were your biggest hurdle, you should not have gotten in to Tech. Even twenty years ago. And even if you had Dr. "Death" Wray, calculus doesn't rise to the level of "electromagic", "heat transport" or "p-chem". Those are clean out courses.
But suppose we entertain the notion of this "don't teach it if they ain't be usin' it" model of "education" for just a bit. How many times do you find yourself in a sales presentation only to realize this is the perfect time for that Shakespeare sonnet you had to memorize in your Junior year? Not so much, eh? Then why teach Shakespeare? Or Chaucer. Or Mark Twain. Or Steinbeck. Really want to touch a PC nerve? Toss out Maya Angelou. What about history? Just because one person way back when said "those who do not learn from history are doomed to repeat it" does not mean we need to study history. First we keep hearing about the good old days and second what makes everyone think only the bad things will be repeated? Isn't it equally likely that only the good things will repeat? Ditch history. Then there is "Art" and whatever that is it isn't something you do unless forced. Ditch it too. Then there is English or Language Arts or whatever PC fluff name used these days. Here is where the calculator excuse applies. No one needs to know grammar or how to spell--your word processor does that for you. Don't know what a word means? Look it up online. Ditch Language Arts too. If we only teach what kids will need to know in the real world we can start handing out high school diplomas after kindergarten.
Want to really piss folks off and get a clear understanding of what is really going on at the same time? Ask how many high school football players make it into the NFL. That low? Then ditch all football programs in public school. Now you've hit a hard stop. Folks, mostly parents, will come out of the woodwork touting the benefits, mostly indirect (like building character) offered by sports competition. Perhaps so. It certainly fills a man-made void created when classroom academic competition was declared verboten. So if we're going to justify sports and all the fuzzies based on their secondary carry-over benefits then just what makes math so special that it can be excluded as "unused in the real world?"
The excuses presented sound a lot like mass societal self-inflicted stupidity. Until we recognize how we really do use math on a daily basis[6] and stop accepting "I'm just not good at math" as an excuse to not only fail a math course but to avoid taking them altogether we as a society will continue to Suck At Math.
We are now confronted with a reality (and a future) where very few if any of the people born in the United States can add or subtract. We're talking the simplest of arithmetic, not math[1]. This has come to the fore in local mass media with a debate on how we teach addition to children, specifically the various journeys between $$ 19 + 27 = ? $$ and $$ 46 $$ Old farts write down 19 then write 27 below aligning along implied decimal points and draw a bar below 27. Then they add 7 to 9 and come up with 16 writing the 6 below the bar and the 7 and then they write the 1 above the 1 in 16. This is called "carry the one"[2] and this is then added to the 1 from 16 and the 2 from 27 with the resulting 4 written to the left of the 6 below the bar. Voila! 46! This approach is well understood by folks who can count points and keep score in bridge. Explains the dearth of Bridge Clubs in Georgia Public Schools, eh?
There are some serious takeaways from the Great Addition Debate. First, we've lowered the bar to the point that we'll celebrate if the average student graduates FROM high school with the ability to add two numbers. Not three nor any more than three. We'll throw a party if more than five out of ten graduates arrive at "46" for the above problem even if they have to copy the answer or have it given them by a teacher. The second issue is the addition algorithm schools are using: $$ 19 + 27 = $$ $$ (10 + 9) + (20 + 7) = $$ $$ (10 + 20) + (7 + 9) = $$ $$ 30 + 16 = $$ $$ 30 + (10 + 6) = $$ $$ (30 + 10) + 6 = $$
and finally $$ 40 + 6 = 46 $$
Execution of this algorithm in a classroom environment pretty much precludes adding numbers with more than two digits due to limitations on time and "Smart Board" space. Should Georgia Public Schools be successful we will pump out graduates who can successfully add any two two digit numbers. Given enough time and paper.
While this situation may have devolved from a more proficient past it is not entirely unintentional and is currently embraced. Asking today's teachers to do a Mad Minute[3] would produce appallingly disappointing results so don't ask them to teach and demand this of tomorrow's leaders (and teachers). Other rationalizations include the bruised and broken self esteems brought on by the Drill and Kill approaches that are pretty much required by the Old School techniques and the fact that the New Math supports "conceptual learning." Or at least the learning of concepts. This is said to include powers of 10[4], as they decompose "16" to "one in the ten's column and six in the one's column" with the illuminating explanation that the "ones column" is "ten to the zeroth power" and the "ten's column" is "ten to the oneth power" at which point we've completely abandoned addition in favour of a hallucinagenic trip down a rabbit hole.
Suppose we demand the basketball couch embrace a concept-centric pedagogy[5]. He would be forced to draw out (with 'x's and 'o's? that almost looks like algebra. or does he need manipulatives? animated power point?) a simple pick and roll play, making sure the team groks the concept. These students also grok dribbling, layups, free throws and general ball handling, even the bounce pass, it's origins, benefits over other forms of passing and the cultural implications of its behind the back use. Now these kids have little or no actual practice and an outstanding performance would be dribbling five times before bouncing the ball off your own foot sending it out of bounds or directly into the hands of an opposing player. Of course the coach is not expected to field a winning team. No, he is going to field a championship team, as good as any in the state, probably the country. All the parents, the principal and even the superintendent of the school system will declare this team "straight A ball players." But honestly, if they ever took the court against a pick-up team of 3rd graders who actually play for 3-4 hours a day who do YOU think would win?
But focusing on concepts over capabilities moves us from the objective world of "the answer is 46 and all other answers are wrong" to the more comfortable subjective world of "critical thinking" and "let's understand the concepts" where flexibility in grading supports the demand for A's. And we also hide behind the calculator excuse (everybody has one, who needs to do their own arithmetic?) to explain our exclusive presentation of the abstract and subjective rather than the concrete and tangible. To be fair the critical thinking part, were it really there, might actually trespass on mathematics, asking and hopefully answering the question: "are these the two right numbers to add and how do we know?"
Similar but perhaps more disturbing is the "nobody really uses math" excuse for cultivated math ignorance. It's almost like Godwin's law, but you cannot have a tweet fest on this topic without someone dragging out "I took calculus at Tech twenty years ago and haven't used it since--it was just a clean out course." That cannot be allowed to slide by without comment. If calculus were your biggest hurdle, you should not have gotten in to Tech. Even twenty years ago. And even if you had Dr. "Death" Wray, calculus doesn't rise to the level of "electromagic", "heat transport" or "p-chem". Those are clean out courses.
But suppose we entertain the notion of this "don't teach it if they ain't be usin' it" model of "education" for just a bit. How many times do you find yourself in a sales presentation only to realize this is the perfect time for that Shakespeare sonnet you had to memorize in your Junior year? Not so much, eh? Then why teach Shakespeare? Or Chaucer. Or Mark Twain. Or Steinbeck. Really want to touch a PC nerve? Toss out Maya Angelou. What about history? Just because one person way back when said "those who do not learn from history are doomed to repeat it" does not mean we need to study history. First we keep hearing about the good old days and second what makes everyone think only the bad things will be repeated? Isn't it equally likely that only the good things will repeat? Ditch history. Then there is "Art" and whatever that is it isn't something you do unless forced. Ditch it too. Then there is English or Language Arts or whatever PC fluff name used these days. Here is where the calculator excuse applies. No one needs to know grammar or how to spell--your word processor does that for you. Don't know what a word means? Look it up online. Ditch Language Arts too. If we only teach what kids will need to know in the real world we can start handing out high school diplomas after kindergarten.
Want to really piss folks off and get a clear understanding of what is really going on at the same time? Ask how many high school football players make it into the NFL. That low? Then ditch all football programs in public school. Now you've hit a hard stop. Folks, mostly parents, will come out of the woodwork touting the benefits, mostly indirect (like building character) offered by sports competition. Perhaps so. It certainly fills a man-made void created when classroom academic competition was declared verboten. So if we're going to justify sports and all the fuzzies based on their secondary carry-over benefits then just what makes math so special that it can be excluded as "unused in the real world?"
The excuses presented sound a lot like mass societal self-inflicted stupidity. Until we recognize how we really do use math on a daily basis[6] and stop accepting "I'm just not good at math" as an excuse to not only fail a math course but to avoid taking them altogether we as a society will continue to Suck At Math.
[1] We'll not here go down the rat hole of "arithmetic IS math" but instead will deal with the corrosive impact of "Term and Topic Inflation" in education in a separate diatribe.
[2] We leave it as an exercise to the reader to PROVE that when adding only two numbers with an arbitrary number of digits the most that will be carried from one column to another is 1. Extra credit if you can prove that the maximum carry-over when adding "n" numbers is "n-1".
[3] A test with approximately 20 arithmetic problems (addition, subtraction, multiplication and division) to be completed in one or two minutes. Often used as a mental warmup/stretching exercise at the beginning of a math class.
[4] Set aside for the moment that exponentiation is special form of multiplication and if multiplication is reduced to iterative addition that is still higher order arithmetic than the problem being solved. Just. Don't. Go. There.
[5] Can you apply the term "pedagogy" to coaching? What the hell, if educrats can call "arithmetic" "math" let's abuse one of their words.
[6] If you want an excellent introduction to how mathematics is involved in every thing we do every day Jordan Ellenberg's "How Not To Be Wrong: The Power Of Mathematical Thinking," is highly recommended.
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