tag:blogger.com,1999:blog-77184627935169688832016-02-09T11:27:49.181-05:00MadMath"Beauty is the Enemy of Expression"Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger214125tag:blogger.com,1999:blog-7718462793516968883.post-33447042334338717062016-02-05T05:00:00.001-05:002016-02-05T05:00:07.025-05:00Link: Study Time DeclineAn interesting article analyzing the history of reported study time decline for U.S. college students. <br /><ul><li>Point 1: Study time dramatically decreased in the 1961-1981 era (from about 24 hrs/week to 16 hrs/week), but has been close to stable since that time. </li><br /><li>Point 2: In that same early period, it seems that faculty expectations on teaching vs. research flip-flopped in that same early time period (about 70% prioritized teaching over research around 1975, with the proportion quickly dropping to about 50/50 by the mid-80's). </li></ul><div style="text-align: center;"><br /></div><div style="text-align: center;"><span style="font-size: large;"><a href="https://www.aacu.org/publications-research/periodicals/its-about-time-what-make-reported-declines-how-much-college">Alexander McCormick: It's about Time: What to Make of Reported Declines in How Much College Students Study</a></span></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-10740425938555769002016-02-01T05:00:00.000-05:002016-02-02T11:20:27.490-05:00When Dice FailSome of the more popular posts on my gaming blog have been about how to check for balanced dice, using Pearson's chi-square test (<a href="http://deltasdnd.blogspot.com/2009/02/testing-balanced-die.html">testing a balanced die</a>, <a href="http://deltasdnd.blogspot.com/2009/02/follow-up-testing-balanced-dice.html">testing balanced dice</a>, <a href="http://v/">testing balanced dice power</a>). One of the observations in the last blog was that "chi-square is a test of rather lower power" (quoting Richard Lowry of Vassar College); to the extent that I've never had any dice that I've checked actually <i>fail</i> the test. <br /><br />Until now. Here's the situation: A while back my partner Isabelle, preparing entertainment for a long trip, picked up a box of cheap dice at the dollar store around the corner from us. These dice are in the <a href="https://en.wikipedia.org/wiki/Dice#Arrangement">Asian-style arrangement</a>, with the "1" and "4" pip sides colored red (I believe because the lucky color red is meant to offset those unlucky numbers):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-PDZTYZbdntY/Vqhw65Hqk3I/AAAAAAAAD2I/PC1x6prHZUQ/s1600/Image01252016192951.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-PDZTYZbdntY/Vqhw65Hqk3I/AAAAAAAAD2I/PC1x6prHZUQ/s320/Image01252016192951.jpg" width="240" /></a></div><br /><br />A few weeks ago, it occurred to me that these dice are just the right size for an experiment I run early in the semester with my statistics students: namely, rolling a moderately large number of dice in handful batches and comparing convergence to the theoretically-predicted proportion of successes. In particular, the plan is customarily to roll 80 dice and see how many times we get a 5 or 6 (mentally, I'm thinking in my <i>Book of War</i> game, how many times can we score hits against opponents in medium armor -- but I don't say that in class). <br /><br />So when we did that in class last week, it seemed like the number of 5's and 6's was significantly lower than predicted, to the extent that it actually threw the whole lesson under a shadow of suspicion and confusion. I decided that when I got a chance I'd better test these dice before using them in class again. Following the findings of the prior blog on the "low power" issue, I knew that I had to get on the order of about 500 individual die-rolls in order to get a halfway decent test; in this case with a boxful of 15 dice, it seemed was convenient to make 30 batched rolls for 15 × 30 = 450 total die rolls... although somewhere along the way I lost count of the batches and wound up actually making 480 die rolls. Here are the results of my hand-tally sheet:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-ATTC7-dY5Fw/VqhzX2ph1aI/AAAAAAAAD2U/HZ29BTX4S8k/s1600/Image01252016193017.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-ATTC7-dY5Fw/VqhzX2ph1aI/AAAAAAAAD2U/HZ29BTX4S8k/s320/Image01252016193017.jpg" width="240" /></a></div><br />As you can see at the bottom of that sheet, this box of dice actually does fail the chi-square test, as the \(SSE = 1112\) is in fact greater than the critical value of \(X \cdot E = 11.070 \cdot 80 = 885.6\).Or in other words, with a chi-square value of \(X^2 = SSE/E = 1112/80 = 13.9\) and degrees of freedom \(df = 5\), we get a P-value of \(P = 0.016\) for this hypothesis test of the dice being unbalanced; that is, if the dice really were balanced, there would be less than a 2% chance of getting an SSE value this high by natural variation alone. <br /><br />In retrospect, it's easy to see what the manufacturing problem is here: note in the frequency table that it's specifically the "1"'s and the "4"'s, the specially red-colored faces, that are appearing in a preponderance of the rolls. In particular, the "1" face on each die is drilled like an enormous crater compared to the other pips; it's about 3 mm wide and about 2 mm deep (whereas other pips are only about 1 mm in both dimensions). So the "6" on the other side from the "1" would be top heavy, and tends to roll down to the bottom, leaving the "1" on top more than anything else. Also, the corners of the die are very rounded, making it easier for them to turn over freely or even get spinning by accident. <br /><br />Perhaps if the experiment in class had been to count 4's, 5's, and 6's (that is: hits against light armor in my wargame), I never would have noticed the dice being unbalanced (because together those faces have about the same weight as the 1's, 2's, and 3's together)? On the one hand my inclination is to throw these dice out so they never get used again in our house by accident; but on the other hand maybe I should keep them around as the only example that the chi-square test has managed to succeed at <i>rejecting</i> to date. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-89216019020830590082016-01-30T05:00:00.000-05:002016-01-30T05:00:24.225-05:00Link: Tricky Rational ExponentsConsider the following apparent paradox:<br /><br />\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)<br /><br />Of the seven equalities in this statement, <i>exactly which of them are false</i>? Give a specific number between (1) and (7). Join in the discussion where I posted this at StackExachange, if you like:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents">StackExchange: <br />What are the Laws of Rational Exponents?</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-76438375463736351402016-01-28T10:18:00.002-05:002016-01-28T10:22:26.860-05:00Seat Belt EnforcementYesterday in the Washington Post, libertarian police-abuse crusader Radley Balko wrote an opinion piece<a href="https://www.washingtonpost.com/news/the-watch/wp/2016/01/27/profiling-oppressive-fines-and-creating-opportunities-for-escalation-the-case-against-mandatory-seat-belt-laws/"> arguing against mandatory seat-belt laws</a>. He opens:<br /><blockquote class="tr_bq"><i>The ACLU of Florida just released a report showing that in 2014, black motorists in the state were pulled over for seat belt violations at about twice the rate of white motorists... Differences in seat belt use don’t explain the disparity. Blacks in Florida are only slightly less likely to wear seat belts. The ACLU points to a 2014 study by the Florida Department of Transportation that found that 85.8 percent of blacks were observed to be wearing seat belts vs. 91.5 percent of whites. The only possible explanation for the disparity that doesn’t involve racial bias might be that it’s easier to spot seat-belt violations in urban areas than in more rural parts of the state... even if it did explain part or all of the disparity, it still means that blacks in Florida are disproportionately targeted.</i></blockquote><br />Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.<br /><br />Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book <i>Rise of the Warrior Cop</i> for more. And I'm pretty sensitive to issues of overly-punitive enforcement and fines that are repressively punishing to the poor; back in the 80's I used to routinely listen to Jerry Williams on WRKO radio in Boston, when he was crusading against the rise of seat-belt laws, and I found those arguments, at times, compelling.<br /><br />But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-33744860978090433582016-01-25T05:00:00.000-05:002016-01-25T12:01:34.379-05:00Grading on a ContinuumAnecdote: I had a social-sciences teacher in high school who didn't understand that real numbers are a continuum. <br /><br />On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).<br /><br />Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.<br /><br />And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting <i>again</i> to generate something like: D 60-69.55, C 69.56-79.55, B 79.56-89.55, A 89.56-100. And then of course noticed that there were still gaps between the intervals and went at it for a few more cycles.<br /><br />I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that <i>real numbers are dense</i>? And had he never thought through his grading schema <i>until this very moment? </i><br /><br />I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class. <br /><br />But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-51526112177601563152016-01-18T05:00:00.000-05:002016-01-18T05:00:16.961-05:00Limitations<blockquote class="tr_bq"><i>Whenever one learns a new mathematical operation, it is imperative also to learn the limitations under which the operation may be performed. Lack of this additional knowledge can lead to the employment of the new operation in a blindly formal manner in situations where the operation is not properly applicable, perhaps resulting in absurd and paradoxical conclusions. Instructors of mathematics see mistakes of this sort made by their students almost every day...</i> </blockquote>- Howard Eves, <i>Great Moments in Mathematics</i>, Lecture 32.Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-88603925143548088802016-01-11T05:00:00.000-05:002016-01-11T05:00:10.141-05:00Public ShamingA collection of Twitter messages from the many people who think that a $550 million Powerball payout will let you give $1 million to every person in the U.S.:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://publicshaming.tumblr.com/post/36857566279/won-last-nights-550-million-powerball-lottery">http://publicshaming.tumblr.com/post/36857566279/won-last-nights-550-million-powerball-lottery</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-39359235377202203472016-01-04T05:00:00.000-05:002016-01-05T01:22:54.377-05:00New Year, New NameWith the new year, I've rolled out a new name and domain for this site: <b>MadMath.com</b>. This is something I actually wanted to do at the outset, as it much better captures the "I'm mad about math!" <i>double entendre</i> that I was really going for. So I'm freakishly psyched that I get a chance to use the proper name at this time. Expect to see more frequent blogging at this address, and hopefully some other expansions to the site in the feature. Thanks for reading!<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-65922201095658669052015-12-28T05:00:00.000-05:002015-12-28T05:00:00.434-05:00Dust Mites and Bad Science JournalismMy friend John W. Scott makes a very perceptive takedown of a bad-science story from a few months ago. In 2015, a list of two-dozen international news sites report that leaving beds unmade helps fight dust mites; all refer back to a speculative BBC article from 2005; no confirmation or testing was ever made on the original speculation. Read more here:<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://bostonquad.blogspot.com/2015/10/journalism-odd-lazy-and-bad.html">http://bostonquad.blogspot.com/2015/10/journalism-odd-lazy-and-bad.html</a></span></div>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-11508337038850959152015-12-21T05:00:00.000-05:002015-12-21T05:00:01.128-05:00Short Response to Flipping ClassroomsA short, possible response to the proponents of "flipping classrooms" (as though it were really a new or novel technique): Presumably we agree that students must do <i>some</i> kind of work outside the classroom. Then as the instructor, we might ask ourselves if we are more fulfilled by: (a) leading classroom discussions about the fundamental <i>concepts</i> of the discipline, or (b) serving as technicians to debug work on particular <i>applications</i> of those principles.<br /><br />Personally, in my classes I do manage to include both aspects, but the time emphasis is more heavily on the former. If forced to pick either one or the other, then I would surely pick (a). <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-25726422393388397232015-12-14T05:00:00.000-05:002015-12-14T05:00:06.258-05:00Why m for Slope?Question: Why do we use <i>m</i> for the slope of a line?<br /><br />Personally, I always assumed that we use <i>m</i> because it's the numerical multiplier on the independent variable in the slope-intercept form equation <i>y = mx +b</i>.<br /><br />Michael Sullivan's <i>College Algebra</i> (8th Ed., Sec. 2.3), says this as part of Exercise #133: <br /><blockquote class="tr_bq">The accepted symbol used to denote the slope of a line is the letter <i>m</i>. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word <i>monter</i>. Write a brief essay on your findings.</blockquote>Of course, "monter" is a French verb which means "to climb" or "to go up".<br /><br />But others disagree. Wolfram MathWorld says the following, along with citations of particular early usages (<a href="http://mathworld.wolfram.com/Slope.html">http://mathworld.wolfram.com/Slope.html</a>):<br /><blockquote class="tr_bq">J. Miller has undertaken a detailed study of the origin of the symbol <i>m</i> to denote slope. The consensus seems to be that it is not known why the letter <i>m</i> was chosen. One high school algebra textbook says the reason for <i>m</i> is unknown, but remarks that it is interesting that the French word for "to climb" is "monter." However, there is no evidence to make any such connection. In fact, Descartes, who was French, did not use <i>m</i> (Miller). Eves (1972) suggests "it just happened." </blockquote>The Math Forum at Drexel discusses this more, including a quote from J. Miller himself (<a href="http://mathforum.org/dr.math/faq/faq.terms.html">http://mathforum.org/dr.math/faq/faq.terms.html</a>):<br /><blockquote class="tr_bq">It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In <i>Mathematical Circles Revisited</i> (1971) mathematics historian Howard W. Eves suggests "it just happened."</blockquote>The Grammarphobia site takes up the issue likewise, citing the above, and mostly knocking down the existing theories as lacking support. They end with this witticism by Howard W. Eves (who taught at my <i>alma mater</i> of U. Maine, although before my time): <br /><blockquote class="tr_bq">When lecturing before an analytic geometry class during the early part of the course... one may say: 'We designate the slope of a line by <i>m</i>, because the word <i>slope</i> starts with the letter <i>m</i>; I know of no better reason.' </blockquote>To bring things full circle, I would point out that the English word "multiplication" is spelled identically in French (and nearly the same in Latin, Italian, Spanish, Portuguese, Danish, Norwegian, and Romanian), so lacking any other historical evidence, I don't see why <i>m</i> for "multiplication" isn't considered as a theory in the sources above. (Compare to <i>k</i> for "koefficient" in Swedish textbooks, per Wolfram.)<br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-80137737776090289092015-12-07T05:00:00.000-05:002015-12-07T05:00:03.661-05:00That Time I Didn't Get the JobHere's a story that I occasionally share with my students. Back around 2001, I had left my second computer gaming job in Boston, and was still interviewing for other jobs in the industry. I had an interview at a company based in Andover, and it was primarily an hour or two in a conference room with one other staff member. I have no recollection who it was now, or if they were in engineering or production. At any rate, part of it was some standard (at the time) "code this simple thing on the whiteboard" tests of basic programming ability.<br /><br />One of the questions he gave me was "write a function that takes a numeric string and converts it to the equivalent positive integer". Well, it doesn't get much more straightforward than that. I didn't really even think about it, just immediately jotted down something like the following (my C's a bit rusty now, but it was what we were working in at the time; assumes a null-terminated C string):<br /><br /><span style="font-family: "Courier New",Courier,monospace;"> int parseNumber (char *s) {<br /> int digit, sum = 0;<br /> while (*s) {<br /> digit = *s - '0';<br /> sum = sum * 10 + digit;<br /> s++;<br /> }<br /> return sum;<br /> }</span><br /><br />Well, the interviewer jumped on me, saying that's obviously wrong, because it's parsing the digits from left-to-right, whereas place values increase from right-to-left, and therefore the string must be parsed in reverse order. But that's not a problem, as I explained to him, with the way the multiples of 10 are being applied.<br /><br />I stepped him through an example on the side: say the string is "1234". On the first pass, sum = 0*10+1 = 1; on the second pass, sum = 1*10+2 = 12; on the third pass, sum = 12*10+3 = 123; on the fourth pass, sum = 123*10 + 4 = 1234. Moreover, this is more efficient than the grade-school definition; in this example I've used only 4 multiplications; whereas the elementary expression of 1*10^3 + 2*10^2 + 3*10 + 4 would be using 8 multiplications (and increasingly more multiplies for larger place values; to saying nothing of the expense in C of finding the back end of the string first before knowing which place values were which). <br /><br />But it was to no avail. No matter how I explained it or stepped through it, my interviewer seemed irreparably skeptical. I left and got no job offer after the fact. The thing is, part of the reason I was so confident and took zero time to think about it is because it's <i>literally a textbook example</i> that was given in my assembly language class in college. From George Markowsky's "Real and Imaginary Machines: An Introduction to Assembly Language Programming" (p. 105-106):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-JU8lgDiBihY/VhxB-ooSVPI/AAAAAAAADvs/t_uZA9oueow/s1600/MarkowskyConvertBase.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="277" src="http://1.bp.blogspot.com/-JU8lgDiBihY/VhxB-ooSVPI/AAAAAAAADvs/t_uZA9oueow/s320/MarkowskyConvertBase.png" width="320" /></a></div><br />Postscript: That wasn't even the only time I got static for using this textbook algorithm. At some point I shared it in an online forum in response to a question -- and then I had someone claiming they were a producer who wouldn't permit it in the code base, because it was too opaque for normal programmers to understand and maintain. <br /><br />As it turned out, I never worked in computer games or any kind of software engineering again. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-49434070685172925302015-11-30T05:00:00.000-05:002015-11-30T05:00:01.869-05:00Short Argument for TauConsider the use of tau (τ ~ 6.28) as a <a href="http://tauday.com/">more natural unit</a> for circular measures than pi (π ~ 3.14). I have a good colleague at school who counter-argues in this fashion: "But it's only a conversion by a factor of two, which should be trivial for us as mathematicians to deal with. And if our students can't handle that, then perhaps they shouldn't be in college." <br /><br />Among the possible responses to this, here's a quick one (and specific to part of the curriculum that we teach): Scientific notation is a number written in the format \(a \cdot 10^b\). But imagine if instead we had defined it to be the format \(a \cdot 5^b\). The difference in the base is also only a factor of 2, but consider how much more complicated it is to convert between standard notation and this revised scientific notation.<br /><br />Lesson: Consider your choice of basis carefully. Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com1tag:blogger.com,1999:blog-7718462793516968883.post-87916056913718976702015-11-23T05:00:00.000-05:002015-11-23T05:00:07.566-05:00A Bunch of Dumb Things Journalists Say About PiA lovely rant by Dave Renfro, via Pat Ballew's blog, here:<br /><div style="text-align: center;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><span style="font-size: large;"><a href="http://pballew.blogspot.com/2010/03/guest-blog-rant-from-dave-renfro.html">http://pballew.blogspot.com/2010/03/guest-blog-rant-from-dave-renfro.html</a></span></div><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-29165681699488240102015-11-16T05:00:00.001-05:002015-11-16T05:00:07.521-05:00Joyous ExcitementDid you know that this week is the 100th anniversary of Einstein's completion of General Relativity? Specifically it was November 18, 1915 when Einstein drafted a paper that realized the final fix to his theories that would account for the previously unexplainable advance of the perihelion of Mercury. The next week he submitted this paper, "The field equations of gravitation", to the Prussian Academy of Sciences, which included what we now refer to simply as <a href="https://en.wikipedia.org/wiki/Einstein_field_equations">"Einstein's equations"</a>. <br /><br />Einstein later recalled of this seminal moment:<br /><blockquote class="tr_bq"><i>For a few days I was beside myself with joyous excitement.</i></blockquote><br />And further: <br /><blockquote class="tr_bq"><i> ... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. </i></blockquote><br />(Quotes from <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/General_relativity.html">"General Relativity"</a> by J.J. O'Connor and E.F. Robertson at the School of Mathematics and Statistics, University of St. Andrews, Scotland). <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-67131474895848830302015-11-09T05:00:00.000-05:002015-11-12T18:36:08.598-05:00Measurement GranularityAnswering a question on StackExchange, and I came across some very nice little articles by the Six Sigma system people on Measurement System Analysis: <br /><blockquote class="tr_bq"><i>Establishing the adequacy of your measurement system using a measurement system analysis process is fundamental to measuring your own business process capability and meeting the needs of your customer (specifications). Take, for instance, cycle time measurements: It can be measured in seconds, minutes, hours, days, months, years and so on. There is an appropriate measurement scale for every customer need/specification, and it is the job of the quality professional to select the scale that is most appropriate.</i></blockquote><br />I like this because this issue comes up a lot in issues of the mathematics of game design: What is the most convenient and efficient scale for a particular system of measurement? And what should we be considering when we mindfully choose those units at the outset?<br /><br />One key example in my D&D gaming, is that at the outset, units of encumbrance (weight carried) were ludicrously set in <i>tenths-of-a-pound</i>, so tracking gear carried by any characters involves adding up units in the hundreds or thousands, frequently requiring a calculator to do so. As a result, D&D encumbrance is infamous for being almost entirely unusable, and frequently discarded during play. My argument is that this is almost entirely due to an incorrect choice in measurement scale for the task -- equivalent to measuring a daily schedule in seconds, when what you really need is hours. I've recommended for a long time using the flavorfully archaic scale of "stone" weight (i.e., 14-pound units; <a href="http://deltasdnd.blogspot.com/2010/09/stone-encumbrance-detail-example.html">see here</a>), although the advantage could also be achieved by taking 5- or 10-pound units as the base. Likewise, I have a tendency to defend other Imperial units of weight as being useful in this sense (see: <a href="https://en.wikipedia.org/wiki/Human_scale">Human scale measurements</a>), although I might be biased just a bit for being so steeped in D&D (further example: a league is about how far one walks in an hour, etc.).<br /><br />The Six Sigma articles further show a situation where the difference in two production processes is discernible at one scale of measurement, but invisible at another incorrectly-chosen scale of measurement. See more below:<br /><ul><li><a href="http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-system-analysis-resolution-granularity/">Measurement System Analysis Resolution, Granularity </a></li><li><a href="http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/proper-data-granularity-allows-stronger-analysis/">Proper Data Granularity Allows for Stronger Analysis </a></li><li><a href="http://www.isixsigma.com/tools-templates/measurement-systems-analysis-msa-gage-rr/measurement-systems-analysis-process-industries/">Measurement Systems Analysis in Process Industries </a></li></ul><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com2tag:blogger.com,1999:blog-7718462793516968883.post-46651750291859277342015-11-02T05:00:00.000-05:002015-11-12T18:31:56.318-05:00On Common CoreAs people boil the oil and man the ramparts for this decade's education-reform efforts, I've gotten more questions recently about what I think regarding Common Core. Fortunately, I had a chance to look at it recently as part of CUNY's ongoing attempts to refine our algebra remediation and exam structure.<br /><br />A few opening comments: One, this is purely in regards to the math side of things, and mostly just focused on the area of 6th-8th grade and high school Algebra I that my colleagues and I are largely involved in remediating (see the standards here: <a href="http://www.corestandards.org/Math/">http://www.corestandards.org/Math/</a>... and I would highlight the assertion that "Indeed, some of the highest priority content for college and career readiness comes from Grades 6-8.", <a href="http://www.corestandards.org/Math/Content/note-on-courses-transitions/courses-transitions/">Note on courses & transitions</a>). Second, we must distinguish what Common Core specifies and what it does not: it does dictate <i>things to know at the end of each grade level</i>, but <u>not</u><i> how they are to be taught</i>. In general:<br /><blockquote class="tr_bq"><i>The standards establish what students need to learn, but they do not dictate how teachers should teach. Teachers will devise their own lesson plans and curriculum, and tailor their instruction to the individual needs of the students in their classrooms.</i> (<a href="http://www.corestandards.org/about-the-standards/frequently-asked-questions/">Frequently Asked Questions</a>: What guidance do the Common Core Standards provide to teachers?)</blockquote>Specifically in regards to math: <br /><blockquote class="tr_bq"><i>The standards themselves do not dictate curriculum, pedagogy, or delivery of content. (<a href="http://www.corestandards.org/Math/Content/note-on-courses-transitions/courses-transitions/">Note on courses & transitions</a>)</i></blockquote>So this foreshadows a two-part answer:<br /><br /><h3>(1) I think the standards look great.</h3>Everything that I've seen in the standards themselves looks smart, rigorous, challenging, core to the subject, and pretty much indispensable to a traditional college curriculum in calculus, statistics, computer programming, and other STEM pursuits. I encourage you to read them at the link above. It includes pretty much everything in a standard algebra sequence for the last few centuries or so. <br /><br />I like the balanced requirement to achieve both conceptual understanding <i>and</i> procedural fluency (<a href="http://www.corestandards.org/Math/Practice/"> http://www.corestandards.org/Math/Practice/</a>). As always, my response in a lot of debates is, "<i>you need both</i>". And this reflects the process of presenting higher-level mathematics theorems: a careful proof, and then applications. The former guarantees correctness and understanding; the latter uses the theorem as a powerful shortcut to get work done more efficiently. <br /><br />Quick example that I came across last night: "<i>By the end of Grade 3, know from memory all products of two one-digit numbers.</i>" (<a href="http://www.corestandards.org/Math/Content/3/OA/">http://www.corestandards.org/Math/Content/3/OA/</a>). That's not nonsense exercise, that's a necessary tool to later understand long division, factoring, fractions, rational versus irrational numbers, estimations, the Fundamental Theorems of Arithmetic and Algebra, etc. I was happy to spot that as a case example. (And I deeply wish that we could depend on all of our college students having that skill.) <br /><br />I like what I see for sample tests. Here are some examples from the nation-wide PARCC consortium (by Pearson, of course; <a href="http://parcc.pearson.com/practice-tests/math/">http://parcc.pearson.com/practice-tests/math/</a>): I'm looking at the 7th- and 8th-grade and Algebra I tests. They all come in two parts: Part I, short questions, multiple-choice, with no calculators allowed. Part II, more sophisticated questions, short-answer (<i>not</i> multiple choice), with calculators allowed. I think that's great: <i>you need both</i>. <br /><br />New York State writes their own Common Core tests instead of using PARCC, at least at the high school level (<a href="http://www.nysedregents.org/">http://www.nysedregents.org/</a>): here I'm looking mostly at Algebra I (<a href="http://www.nysedregents.org/algebraone/">http://www.nysedregents.org/algebraone/</a>). Again, a nice pattern of one part multiple-choice, the other part short-answer. I wish we could do that in our system. Now, the NYS Algebra I test is all-graphing-calculator mandatory, which sets my teeth on edge a bit compared to the PARCC tests. Maybe I could live with that as long as students have confirmed mental mastery at the 7th- and 8th-grade level (not that I can confirm that they do). Even the grading rubric shown here for NYS looks fine to me (approximately half-credit for calculation, and half-credit for conceptual understanding and approach on any problem; that's pretty close to what I've evolved to do in my own classes). <br /><br />In summary: Pretty great stuff as far as published standards and test questions (at least for 7th-8th grade math and Algebra I).<br /><br /><h3>(2) The implementation is possibly suspect. </h3>Having established rigorous standards and examinations, these don't solve some of the endemic problems in our primary education system. Granted that "Teachers will devise their own lesson plans and curriculum, and tailor their instruction to the individual needs of the students in their classrooms." (above):<br /><br />Most teachers in grades K-6, and even 7-8 in some places (note that's specifically the key grades highlighted above for "some of the highest priority content for college and career readiness") are not mathematics specialists. In fact, U.S. education school entrants are perennially the <a href="http://qz.com/493971/inside-chipotles-extremely-intense-39-point-checklist-for-good-management/">very weakest of all incoming college students in proficiency and attitude towards math</a> (also: <a href="http://www.angrymath.com/2014/12/academically-adrift.html">here</a>). If the teachers at these levels fundamentally don't understand math themselves -- don't understand the later algebra and STEM work that it prepares them for -- then I have a really tough time seeing how they can understand the Common Core requirements, or effectively select and implement appropriate mathematical curriculum for their classrooms. Sometimes I refer to students at this level as having "anti-knowledge" -- and I find that it's much easier to instruct a student who has <i>never heard of algebra ever</i> (which sometimes happens for graduates of certain religious programs) than it is to deconstruct and repair incorrect the conceptual frameworks of students with many years of broken instruction. <br /><br />Before I go on: The best solution to this would be to massively increase salary and benefits for all public-school teachers, and implement top-notch rigorous requirements for entry to education programs (as done in other top-performing nations). A second-best solution, which is probably more feasible in the near-term, would be to place mathematics-specialist teachers in all grades K-12.<br /><br />The other key problem I see is: how are the test scores generated? We already know that in many places students take tests, and then the test scores are arbitrarily inflated or scaled by the state institutions, manipulating them to guarantee some particular high percentage is deemed "passing" (regardless of actual proficiency, for political purposes). For example, the conversion chart for NYS Algebra I Common Core raw scores to final scores for this past August is shown below (from NYS regents link above): <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-T7T1hNwm1RM/VghDVbpYFHI/AAAAAAAADtM/npVWf15jkLY/s1600/NYSCCAlgebraI-ConversionChart.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-T7T1hNwm1RM/VghDVbpYFHI/AAAAAAAADtM/npVWf15jkLY/s320/NYSCCAlgebraI-ConversionChart.png" width="320" /></a></div><br />Now, this is a test that had a maximum total 86 possible points scored. If we linearly converted this to a percentage, we would just multiply any score by 100/86 = 1.16; it would add 14 points at the top of the scale, about 7 points at the middle, and 0 points at the bottom. But that's not what we see here -- it's a nonlinear scaling from raw to final. The top adds 14 points, but in the middle it adds 30 or more points in the raw range from 13 to 40. <br /><br />The final range is 0 to 100, allowing you to think it might be a percentage, but it's not. If we consider 60% be minimal normal passing at a test, for this test that would occur at the 52-point raw score mark; but that gets scaled to a 73 final score, which usually means a middle-C grade. Looking at the 5 performance levels (more-or-less equivalent to A- through F- letter grades): A performance level of "3" is achieved with a raw score of just 30, which is only 30/86 = 35% of the available points on the test. A performance level of "2" is achieved with a raw score of only 20, that is, 20/86 = 23% of the available points on the test. And these low levels (near random-guessing) are considered acceptable for awards of a high school diploma (<a href="http://www.p12.nysed.gov/assessment/reports/commoncore/tr-a1-ela.pdf">www.p12.nysed.gov/assessment/reports/commoncore/tr-a1-ela.pdf</a>, p. 19): <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-4CU-Lvlz2M0/VghGWI7mZwI/AAAAAAAADtY/r0Y1BbinbPA/s1600/RegentsPerformanceLevels.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="211" src="http://3.bp.blogspot.com/-4CU-Lvlz2M0/VghGWI7mZwI/AAAAAAAADtY/r0Y1BbinbPA/s320/RegentsPerformanceLevels.png" width="320" /></a></div><br />In summary: While the publicized standards and exam formats look fine to me, the devil is in the details. On the input end, actual curriculum and instruction are left as undefined behavior in the hands of primary-school teachers who are not specialists, and rarely empowered, and frequently the very weakest of all professionals in math skills and understanding. And on the output end, grading scales can be manipulated arbitrarily to show any desired passing rate, almost entirely disconnected from the actual level of mastery demonstrated in a cohort of students. So I fear that almost any number of students can go through a system like that and not actual meet the published Common Core standards to be ready for work in college or a career. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-15816536022658974552015-10-26T05:00:00.000-04:002015-10-26T05:00:02.002-04:00Double Factorial TableThe <a href="https://en.wikipedia.org/wiki/Double_factorial">double factorial</a> is the product of a number and every <i>second</i> natural number less than itself. That is:<br /><br /><div style="text-align: center;">\(n!! = \prod_{k = 0}^{ \lceil n/2 \rceil - 1} (n - 2k) = n(n-2)(n-4)...\)</div><br />Presentation of the values for double factorials is usually split up into separate even- and odd- sequences. Instead, I wanted to see the sequence all together, as below: <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Ua53g6RQ9j4/Vf2fRankanI/AAAAAAAADsw/q2cUs29962c/s1600/DoubleFactorialTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Values of the double factorial function" border="0" height="320" src="http://1.bp.blogspot.com/-Ua53g6RQ9j4/Vf2fRankanI/AAAAAAAADsw/q2cUs29962c/s320/DoubleFactorialTable.png" title="Double Factorial Table" width="151" /></a></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-3716737898871287132015-10-19T05:00:00.000-04:002015-10-19T05:00:05.958-04:00Geometry Formulas in TauHere's a modified a geometry formula sheet so all the presentations of circular shapes are in terms of tau (not pi); tack it to your wall and see if anybody spots the difference.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-Z2yzqWUGFxw/Vf2bZHmeRvI/AAAAAAAADsk/L__jdLNQDN4/s1600/GeometryFormulasWithTau.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-Z2yzqWUGFxw/Vf2bZHmeRvI/AAAAAAAADsk/L__jdLNQDN4/s320/GeometryFormulasWithTau.gif" width="256" /></a></div><br />(<a href="https://www.pinterest.com/pin/377176537513172091/">Original sheet here.</a>)Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-19306243693066243732015-10-12T05:00:00.000-04:002015-10-12T11:14:09.008-04:00On ZerationIn my post last week on hyperoperations, I didn't talk much about the operation under addition, the zero-th operation in the hierarchy, which many refer to as "zeration". There is a surprising amount of disagreement about exactly how zeration should be defined. <br /><br /><a href="https://en.wikipedia.org/wiki/Peano_axioms">The standard Peano axioms defining the natural numbers stipulate a single operation called the "successor".</a> This is commonly written S(n), which indicates the next natural number after n. Later on, addition is defined in terms of repeated successor operations, and so forth. <br /><br /><a href="https://en.wikipedia.org/wiki/Hyperoperation#Definition">The traditional definition of zeration, per Goodstein, is: \(H_0(a, b) = b + 1\).</a> Now when I first saw this, I was surprised and taken aback. All the other operations start with \(a\) as a "base", and then effectively apply some simpler operation \(b\) times, so it seems odd to start with the \(b\) and just add one to it. (If anything my expectation would have been to take \(a+1\), but that doesn't satisfy the regular recursive definition of \(H_n\) when you try to construct addition.) <br /><br />As it turns out, when you get to this basic level, you're doomed to lose many of the regular properties of the operations hierarchy. <a href="http://math.eretrandre.org/tetrationforum/showthread.php?tid=122">So there's nothing to do but start arguing about which properties to prioritize as "most fundamental" when constructing the definition.</a><br /><br />Here are some points in <b>favor</b> of the standard definition \(b+1\): (1) It does satisfy the recursive formula that repeated applications are equivalent to addition (\(H_1\)). (2) It does looking passingly like counting by 1, i.e., the Peano "successor" operation. (3) It shares the key identity that \(H_n(a, 0) = 1\), for all \(n \ge 3\). (4) Since it is an elementary operation (addition, really), it can be extended from natural numbers to all real and complex numbers in a fashion which is analytic (infinitely differentiable). <br /><br />But here are some points <b>against</b> the standard definition (1) It is not "really" a binary operator like the rest of the hierarchy, in that it totally ignores the first parameter \(a\). (2) Because of its ignoring \(a\), it's not commutative like the other low-level operations n = 1 or 2 (yet like them it is still associative and distributive, or as I sometimes say, collective of the next higher operation). (3) For the same reason, it has no identity element (no way to recover the value \(a\), unique among the entire hyperoperations hierarchy). (4) It's the only hyperoperation which doesn't need a special base case for when \(b = 0\). (5) I might turn around favorable point #3 above and call it weird and unfavorable, in that it is misaligned in this way with operations n = 1 and 2, and it's the only case of one of the key identities being <i>added</i> at a lower level instead of being lost. See how weird that looks below?<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/--kwtOW8ccho/VfyFj-RI_jI/AAAAAAAADsQ/a8TUBy0C_Is/s1600/ZerationIdentities.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/--kwtOW8ccho/VfyFj-RI_jI/AAAAAAAADsQ/a8TUBy0C_Is/s1600/ZerationIdentities.png" /></a></div><br />So as a result, a variety of alternative definitions have been put forward. I think my favorite is \(H_0(a, b) = max(a, b) + 1\). Again, this looks a lot like counting; I might possibly explain it to a young student as "count one more than the largest number you've seen before". Points in <b>favor</b>: (1) Repeated applications are again the same as addition. (2) It is truly a binary operation. (3) It is commutative, and thus completes the trifecta of commutativity, association, and distribution/collection being true for all operations \(n < 3\). (4) It does have an identity element, in \(b = 0\). (5) It maintains the pattern of <i>losing</i> more of the high-level identities, and in fact perfects the situation in that <i>none</i> of the five identities hold for this zeration (all "no's" in the modified table above for \(n = 0\)). Points <b>against</b>: (1) It isn't exactly the same as the unary Peano successor function. (2) It's non-differentiable, and therefore cannot be extended to an analytic function over the fields of real or complex numbers.<br /><br />There are vocal proponents of related possible re-definition: \(H_0(a, b) = max(a, b) + 1\) if a ≠ b, \(a + 2\) if a = b. Advantage here is that it matches some identities in other operations, like \(H_n(a, a) = H_{n+1}(a, 2)\) and \(H_n(2, 2) = 4\), but I'm less impressed by specific magic numbers like that (as compared to having commutativity and the pattern of actually <i>losing more identities</i>). Disadvantage is obviously that the possibility of adding 2 in the \(a+2\) case gets us even further away from the simple Peano successor function. <br /><br />And then some people want to establish commutativity so badly that they assert this: \(H_0(a, b) = ln(e^a + e^b)\). That does get you commutativity, but at that point we're so far away from simple counting in natural numbers that I don't even want to think about it.<br /><br /><br />Final thought: While most people interpret the standard definition of zeration, \(H_0(a, b) = b + 1\) as "counting 1 more place from b", it makes more sense to my brain to turn that around and say that we are <i>"counting b places from 1"</i>. That is, ignoring the \(a\) parameter, start at the number 1 and apply the successor function repeatedly b times: \(S(S(S(...S(1))))\), with the \(S\) function appearing \(b\) times. This feels more like "basic" Peano counting, it maintains the sense of \(b\) being the number of times some simpler operation is applied, and it avoids defining zeration in terms of the higher operation of addition. And then you also need to stipulate a special base case for \(b = 0\), like all the other hyperoperations, namely \(H_0(a, 0) = 1\). <br /><br />So maybe the standard definition is the best we can do, and the closest expression of what Peano successor'ing in natural numbers (counting) really indicates. Perhaps we can't really have a "true" binary operator at level \(H_0\), at a point when we haven't even discovered what the number "2" is yet. <br /><br />P.S. Can we consider defining an operation one level even lower, perhaps \(H_{-1}(a, b) = 1\) which ignores <i>both</i> parameters, just returns the natural number 1, and loses every single one of the regular properties of hyperoperations (including recursivity in the next one up)? <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com6tag:blogger.com,1999:blog-7718462793516968883.post-40354459098553366992015-10-05T05:00:00.000-04:002015-10-05T05:00:04.163-04:00On HyperoperationsConsider the basic operations: Repeated counting is addition; repeated addition is multiplication; repeated multiplication is exponentiation. Hyperoperations are the idea of generally extending this sequence. This was first proposed as such, in a passing comment, by R. L. Goodstein in an article to the <i>Journal of Symbolic Logic</i>, "Transfinite Ordinals in Recursive Number Theory" (1947):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-6bLiFM91yvM/VfuCoLClA7I/AAAAAAAADq4/nGoQWpl4ij0/s1600/Goodstein-Hypoeroperations.png" imageanchor="1"><img alt="" border="0" height="190" src="http://4.bp.blogspot.com/-6bLiFM91yvM/VfuCoLClA7I/AAAAAAAADq4/nGoQWpl4ij0/s320/Goodstein-Hypoeroperations.png" title="Goodstein's definition of hyperoperations (1947)" width="320" /></a></div><br />At this point, there are a lot of different ways of denoting these operations. There's \(H_n\) notation. There's the Knuth up-arrow notation. There's box notation and bracket notation. The Ackerman function means almost the same thing. Conway's chained arrow notation can be used to show them. Some people concisely symbolize the zero-th level operation (under addition) as \(a \circ b\), and the fourth operation (above exponentiation) as \(a \# b\). <a href="https://en.wikipedia.org/wiki/Hyperoperation#cite_ref-nega_16-1">Wikipedia reiterates Goodstein's original definition like so</a>, for \(H_n (a,b): (\mathbb N_0)^3 \to \mathbb N_0\):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-HQZmXXVpbGE/VfuEv_aHK6I/AAAAAAAADrE/3bNRkA4GX9s/s1600/HyperoperationsDefinition.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="105" src="http://1.bp.blogspot.com/-HQZmXXVpbGE/VfuEv_aHK6I/AAAAAAAADrE/3bNRkA4GX9s/s320/HyperoperationsDefinition.png" title="Wikipedia's definition of hyperoperations" width="320" /></a></div><br />Let's use Goodstein's suggested names for the levels above exponentiation. Repeated exponentiation is tetration; repeated tetration is pentation; repeated pentation is hexation; and so forth. Since I don't see them anywhere else online, below you'll find some partial hyperproduct tables for these next-level operations (<a href="http://www.superdan.net/download/blog/angrymath/HyperoperationTables.ods">and ODS spreadsheet here</a>). Of course, the values get large very fast; you'll see some entries in scientific notation, and then "#NUM!" indicates a place where my spreadsheet could no longer handle the value (that is, something greater than \(1 \times 10^{308}\)). <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-4-wH3LalKdQ/VfuKeX4ZjFI/AAAAAAAADr0/YhPh8llVFjM/s1600/TetrationTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Values of repeated exponentiation" border="0" height="221" src="http://1.bp.blogspot.com/-4-wH3LalKdQ/VfuKeX4ZjFI/AAAAAAAADr0/YhPh8llVFjM/s320/TetrationTable.png" title="Hyperoperation tetration table" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Xs24vrw9x3g/VfuKFhmzX5I/AAAAAAAADrs/IfCmlebhWjU/s1600/PentationTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Values of repeated tetration" border="0" height="227" src="http://2.bp.blogspot.com/-Xs24vrw9x3g/VfuKFhmzX5I/AAAAAAAADrs/IfCmlebhWjU/s320/PentationTable.png" title="Hyperoperation pentation table" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-fftHCHmAjhc/VfuJgYNFqiI/AAAAAAAADrY/Le31DE6naYE/s1600/HexationTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Values of repeated pentation" border="0" height="233" src="http://3.bp.blogspot.com/-fftHCHmAjhc/VfuJgYNFqiI/AAAAAAAADrY/Le31DE6naYE/s320/HexationTable.png" title="Hyperoperation hexation table" width="320" /></a></div><br />From this point forward, the hyperoperation tables look passingly similar in this limited view. You have some fixed values in the first two rows and columns; the 2-by-2 result is eternally 4; and everything other than that is so astronomically huge that you can't even usefully write it in scientific notation. Here are some identities suggested above that we can prove pretty easily for all hyperoperations \(n > 3\):<br /><ol><li>\(H_n(a, 0) = 1\) (by definition)</li><li>\(H_n(a, 1) = a\)</li><li>\(H_n(0, b) = \) 0 if b odd, 1 if b even</li><li>\(H_n(1, b) = 1\)</li><li>\(H_n(2, 2) = 4\)</li></ol>One passingly interesting question is how many of these master identities hold true in the lower operations (n = 1 to 3; addition, multiplication, and exponentiation); in short, each step further down the hierarchy loses more of these identities, as summarized here:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ZblrA9ngVro/VfuOI8Ud-EI/AAAAAAAADsA/XIc8ToRiYks/s1600/HyperoperationIdentities.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Key identities are lost in lower operations" border="0" src="http://2.bp.blogspot.com/-ZblrA9ngVro/VfuOI8Ud-EI/AAAAAAAADsA/XIc8ToRiYks/s1600/HyperoperationIdentities.png" title="Identities in lower operations" /></a></div><br />Now, to connect up to my post last week, recall the basic properties of real numbers taken as axiomatic at the start of most algebra and analysis classes. Addition and multiplication (n = 1 and 2) are <b>commutative</b> and <b>associative</b>; but exponents are not, and neither are any of the higher operations. <br /><br />Finally consider the general case of <b>distribution</b>, what in my algebra classes I summarize as the "General Distribution Rule" (<a href="http://www.angrymath.com/2012/07/power-rules.html">Principle #2 here</a>). Or perhaps based on last week's observation I might suggest it could be better phrased as "collecting terms of the next higher operation", like \(ac + bc = (a+b)c\) and \(a^c \cdot b^c = (a \cdot b)^c\), or in the general hyperoperational form: <br /><br /><div style="text-align: center;">\(H_n(H_{n+1}(a, c), H_{n+1}(b, c)) = H_{n+1}(H_n(a, b), c)\)</div><br />Well, just like commutativity and associativity, distribution in this general form also holds for n = 1 and 2, but <b>fails for higher operations</b>. Here's the first counterexample, using \(a \uparrow b\) for exponents (\(H_3\)), and \(a \# b\) for tetration (\(H_4\)):<br /><br /><div style="text-align: center;">\((2\#2)\uparrow (0\#2) = 4 \uparrow 1 = 4\), but</div><div style="text-align: center;">\((2 \uparrow 0)\#2 = 1 \# 2 = 1\). </div><br />Likewise, what I call the "Fundamental Rules of Exponents" (Principle #1 above, <a href="http://www.angrymath.com/2012/08/fundamental-rule-of-exponents.html">or also here</a>) works only for levels \(n \le 3\), and fails to be meaningful at higher levels of hyperoperation. <br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-88490026909415087592015-09-28T05:00:00.000-04:002015-09-28T10:31:27.146-04:00Why Is Distribution Prioritized Over Combining?So I've come up with this question that's been bothering me for weeks,and I've been searching and asking everyone and everywhere that I can. I suspect that it may have no answer. The question is this:<br /><blockquote class="tr_bq"><i>Consider the properties of real numbers that we take for granted at the start of an algebra or analysis class (commutativity, association, and distribution of multiplying over addition). Granted that the last one, distribution (the transformation \(a(b+c) = ab + ac\)), is effectively equivalent to what we might call "combining like terms" (the transformation \(ax + bx = (a+b)x\)). It seems like the latter is more fundamental and easier to intuit as an axiom, since it resembles simple addition of units (e.g., 3 feet + 5 feet = 8 feet). So historically and/or pedagogically, what was the reason choosing the name and order we have ("distribution", \(a(b+c)=ab+ac\)), instead of the other option ("combining", \(ax+bx = (a+b)x\)) for the starting axiom?</i></blockquote>I suspect now that there simply isn't any reason that we can document. Some expansion on the problem:<br /><br /><a href="http://ocw.mit.edu/courses/mechanical-engineering/2-25-advanced-fluid-mechanics-fall-2013/dimensional-analysis/Rayleigh_similitude_1915_.pdf">In dimensional analysis, some call the idea of only adding or comparing like units the "Great Principle of Similitude".</a> Which provides some of my motivation for wishing that we would start with this ("combining") and then derive distribution (using commutativity a few times). Note that this phrase is in many places erroneously attributed to Newton; in truth the earliest documented usage of the phrase is by Rayleigh in a letter to <i>Nature</i> (No. 2368, Vol. 95; March 18, 1915). I could probably write a whole post just on the hunt for this quote. Big thanks to Juan Santiago who teaches a class by that name at Stanford (<a href="http://explorecourses.stanford.edu/search?view=catalog&academicYear=&page=0&q=ME&filter-departmentcode-ME=on&filter-coursestatus-Active=on&filter-term-Autumn=on">link</a>) for helping me track down the article. <br /><br /><a href="http://mathforum.org/library/drmath/view/52599.html">The Math Forum at Drexel discusses some history of the names of the basic properties.</a> The best that Doctor Peterson can track down is that terms such as "distribution" were first used in the late 1700's to 1800's (starting in French in a memoir by Francois Joseph Servois). No commentary on a <i>reason</i> for why this was picked over alternative formulations. But perhaps the fact that the original discussion was in terms of functions (not binary operators) provides a clue. (For the full French text, see <a href="http://jeff560.tripod.com/c.html">here</a> and search "commutative and distributive"). <br /><br /><a href="http://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining">Here's me asking the question at the StackExchange Mathematics site.</a> Unfortunately, most commentators considered it to be uninteresting. When it got no responses, I cross-posted to the Mathematics Educators site -- which is apparently a huge <i>faux pas</i>, and immediately got it down-moderated into oblivion. The only relevant answer to date was from Benjamin Dickman, who pointed to a very nice quote from Euclid: when he states a similar property in geometric terms (area of a rectangle versus the sum of it being sliced up), it happens to be in the same order as we present the distribution property. But still no word on any reason <i>why</i> it should be in that order and not the reverse. <br /><br />Observations from a few textbooks that I have lying around:<br /><ul><li>Rietz and Crathorne, <i>Introductory College Algebra</i> (1933). Article 4 shows combining like terms, and asserts that's justified by the associative law (which is nonsensical). The distributive property isn't presented until later, in Article 8.</li><li>Martin-Gay, <i>Prealgebra & Introductory Algebra</i>. In the purely numerical prealgebra section, this first shows up as distribution among numbers (Sec 1.5). But the first time it appears in the algebra section with variables it is in fact written and used for combining like terms (Sec 3.1: \(ac + bc = (a+b)c\), although still called the "distributive property"). Combining like terms is actually done even earlier than that on an intuitive basis (see Sec. 2.6, Example 4). Only later is the property presented and used to remove parentheses from variable expressions.</li><li>Bittinger's <i>Intermediate Algebra</i> shows standard distribution, followed immediately by use for combining like terms. Sullivan's <i>Algebra & Trigonometry</i> does the same. </li></ul>So my point with those sources is that even though distribution is usually presented in a removing-parentheses format, in practice many textbook authors find themselves unable to escape the need to use combining like terms at some earlier point in their presentation (Rietz and Crathorne, Martin-Gay). This observation bolsters my growing instinct that it would be more intuitive to present the property in that format in the first place (as Martin-Gay does, the first time it appears with variables), and then derive what we call distribution from that. <br /><br />Another thought is that while you can point to distribution as justifying the standard long-multiplication process (across decimal place value), the interior additions are implied and not explicit, and so they don't really serve to develop intuition in the same way that simple unit addition does. <br /><br />Therefore, I find myself fantasizing about the following. Write a slightly nonstandard algebra textbook that starts by assuming commutativity, association, and the combining-like-terms-property (and shortly after deriving the distribution property). Perhaps for a better name it could be called "collection of like multiplications inside addition", or something like that.<br /><br />Do you think this would be a better set of axioms for a basic algebra class? Can you think of a solid historical or pedagogical reason why the name and presentation were not the other way around, like this? Likely some more on this later.<br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com4tag:blogger.com,1999:blog-7718462793516968883.post-29545847217629596542015-09-21T05:00:00.000-04:002015-09-21T23:52:31.261-04:00Rational Numbers and Randomized Digits<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-8nGbt4i0wZ4/Ve3Md4bRJ4I/AAAAAAAADqk/e4FMYyd3pNM/s1600/d10.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="http://3.bp.blogspot.com/-8nGbt4i0wZ4/Ve3Md4bRJ4I/AAAAAAAADqk/e4FMYyd3pNM/s200/d10.png" width="200" /></a></div>Here's a quick thought experiment to develop intuition about the cardinality of rational versus irrational decimal numbers. We know that any rational number (a/b with integer a, b and b ≠ 0) has a decimal expansion that either terminates or repeats (and terminating is itself equivalent to ending with a repeating block of all 0's).<br /><br />Consider randomizing decimal digits in an infinite string (say, by using a standard d10 from a roleplaying game, shown above). How likely does it seem that at any point you'll start rolling repeated 0's, and nothing but 0's, until the end of time? It's obviously diminishingly unlikely, so effectively impossible that you'll roll a terminating decimal. Alternatively, how probable does it seem that you'll roll some particular block of digits, and then repeat them in exactly the same order, and keep doing so without fail an infinite number of times? Again, it seems effectively impossible.<br /><br />So this intuitively shows that if you pick any real number "at random" (in this case, generating random decimal digits one at a time), it's effectively certain that you'll produce an irrational number. The proportion of rational numbers can be seen to be practically negligible compared to the preponderance of irrationals. <br /><br /><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com24tag:blogger.com,1999:blog-7718462793516968883.post-31304500464992126692015-09-14T05:00:00.000-04:002015-09-14T05:00:01.013-04:00Algebra for CryptographyCryptography researcher Victor Shoup recently gave a talk at the Simons Institute at Berkeley. Richard Lipton quotes him in one of his interesting observations about cryptography:<br /><blockquote class="tr_bq"><span style="font-size: large;"><i><span style="color: ”#0066cc?;"><span style="color: ”#000000?;">He also made another point: For the basic type of systems under discussion, he averred that the mathematics needed to describe and understand them was essentially high school algebra. Or as he said, “at least high school algebra outside the US.” </span></span></i></span></blockquote><a href="https://rjlipton.wordpress.com/2015/08/14/cryptography-and-quicksand/">Quoted here.</a>Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0tag:blogger.com,1999:blog-7718462793516968883.post-75557101157202718732015-09-07T05:00:00.000-04:002015-09-07T05:00:09.099-04:00The MOOC Revolution that Wasn'tThree years ago I wrote a review of "Udacity Statistics 101" that went semi-viral, finding the MOOC course to be slapdash, unplanned, and in many cases pure nonsense (<a href="http://www.angrymath.com/2012/09/udacity-statistics-101.html">link</a>). I wound up personally corresponding with Sebastian Thrun (Stanford professor, founder of Udacity, head of Google's auto-car project) over it, and came away super skeptical of his work. Today here's a fantastic article about the fallen hopes for MOOCs and Thrun's Udacity in particular -- highly recommended, jealous that I didn't write this.<br /><blockquote class="tr_bq"><i>Just a few short years after promising higher education for anyone with an Internet connection, MOOCs have scaled back their ambitions, content to become job training for the tech sector and for students who already have college degrees... <br /><br />"In 50 years,” Thrun told Wired, “there will be only 10 institutions in the world delivering higher education and Udacity has a shot at being one of them.”<br /><br />Three years later, Thrun and the other MOOC startup founders are now telling a different story. The latest tagline used by Thrun to describe his company: “Uber for Education.”</i></blockquote>I want to quote the whole thing here; probably best that you just go and read it. Big kudos to Audrey Waters for writing this (and tip to Cathy O'Neil for sharing a link).<br /><br /><div style="text-align: center;"><span style="font-size: large;"><a href="http://kernelmag.dailydot.com/issue-sections/headline-story/14046/mooc-revolution-uber-for-education/">The MOOC Revolution that Wasn't</a></span></div><br />Deltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.com0