Monday, March 24, 2014

On Subtraction

Perchance you've by now stumbled upon a new method of subtraction that is allegedly being taught in elementary school, as part of the new "Common Core" recommendations:


For anyone used to the "borrow one" method (the "old fashion" way above), the "new" way might seem confusing at first.  Subtraction through addition?

This is not an entirely scandalous idea, if one is somewhat familiar with how for instance computers perform subtraction (using the method of complements).  How this particular method (our "Common Core" method, though I won't say it's named such because I've yet seen a good link to a recent textbook that teaches this) works is by counting upward from our lower number (the "subtrahend") to the higher number (the minuend), in easy-to-grasp intervals.  We add to get a multiple of 5 because we like working in 5s, we add to get to a multiple of 10 for the same reason, and we keep adding until we reach our destination.  All of those numbers we added will get us to the final answer; their sum is the difference between the minuend and the subtrahend, our answer.

This has the benefit of not actually having to teach subtraction.  Subtraction is, after all, just a different way of adding.  It does seem somewhat longer though: in the particular example above, the answer using the borrow one method completes the task in 2 simple steps.  The Common Core method takes around 7 (it could take 5 if we can add directly to get to 10s instead of 5s).  Of course, there are counter examples that are more efficient the latter way: 100-99 would take a couple "borrows" to complete, while the new method would take 1.

The new method is not particularly intriguing or difficult.  But, some, like Hemant Mehta, contend that it is actually easier.  I disagree.

The basic concept of subtraction is not extremely difficult to grasp: Joe has 11 apples, if he eats 3 he has (11 • 10 • 9 •8) 8 apples left.  If you can count up, you can count down.

The contesting difficulties in these methods are those of what it means to "borrow", and the number of steps it takes to solve a given subtraction problem with each algorithm.  We've given the new Common Core method its turn at being explained and justified; we can do what Hemant unfortunately did not do, and explain why the old borrow one method does make sense.  And we don't need to be very difficult with this either.

Our number system is base-10: we have single characters to represent each number up until we reach ten, and then we start forming larger groups of certain sizes, of size ten.  The number of those groups we have, we can count using our original numbers.

If I have, say, twenty-three objects, I can represent them by themselves as twenty-three objects:

• • • • • • • • • • • • • • • • • • • • • • •

and maybe I make up a symbol for twenty-three, maybe Œ.  Or, what I can do is group them:

(• • • • • • • • • •) (• • • • • • • • • •) • • •

and so I have 2 groups of ten, 1 group of 3, and I represent that by an ordered pair of numbers: 23.  23 is the same as 2 of ten, and 3.

This concept will have to be learned at some point in school, and it is not a very difficult concept to grasp.  It is, in fact, adding.  And if I was to say that I could break apart one of my groups into individual pieces,

(• • • • • • • • • •) • • • • • • • • • • • • •

we'd all know what I was saying.

So how does the borrow one method work?  Well, you start taking away like-groups from like-groups, with 1s first.  If you have 8 ones and want to remove 3 of them, easy!  Then you move on.  But if you have 3 ones and want to remove 8, what do you do?  Well, again each larger "group" is merely a collection of a certain number of smaller groups.  Break one apart, like we did to the 2 groups of ten: we now have 3 and ten objects, more than 8, so we subtract.  And then we move on to the next group size, remembering that we just broke one apart.

The act of breaking a group apart is borrowing.  Does everyone think that they would be able to explain this now?

Again, this is not particularly difficult.  This is how kids have been taught how to subtract for a long time; that does not make it the best way of course, but has anyone really had a problem with it?  Has any math teacher been unable to explain the method?

Whether we can appreciate if simple subtraction is easier than simple addition, we can all appreciate that a method that takes fewer steps to implement has an appeal to it.  Now the example given in Hemant's blog of 3000-2999 is quite damning toward the borrow one method, isn't it!  You have to borrow several times, and keep track of extra numbers, you have to add to your groups, so on.

The thing about picking out individual examples because you like them (as you would pick, if you will, cherries) is that you don't really get a good idea of what the general case is.  Is the new Common Core method always better?  Has anyone taken the time to answer that question?

Fear not, I have!  In fact I've also taken the time to figure out how many different calculations you would need to complete subtraction on two numbers using the borrow one method, and the method of complements to boot.  I've completed this work in Excel (I found it more useful than, say, Matlab or R in keeping visual track of the steps of the algorithms, and the conditionals).  My graphs below show the results.

To briefly explain how each algorithm is carried out:

Borrow One: you subtract the ones-digit of the subtrahend from the ones-digit of the minuend.  If it is smaller, easy-peasy, one step; if it is larger, you (1) subtract one from the next digit, (2) add 10 to the current digit, and (3) subtract.  You then move on.  Trivial subtraction is counted as well (so, it takes a step to subtract 0 from 6, or 5 from 5).

Common Core: you add to the right-most non-zero digit to increase the next digit by one; you do this until the digits to the left of your current digit in the subtrahend match those in the minuend.  Then you add going to the right, to increase to the minuend's digits.  Then you add all of those numbers you added (if you added n numbers, you perform n–1 more additions).  An example, 84680–59391:

59391 (then add 9)
59400 (then add 600)
60000 (then add 20000)
80000 (then add 4000)
84000 (then add 600)
84600 (then add 80)
84680

then add 9 + 600 + 20000 + 4000 + 600 + 80: 11 steps altogether.

Method of Complements: you first find the 9s–complement of your subtrahend (if your subtrahend XXXX has 4 digits, the 9s-complement is 9999–XXXX), which is 4 steps here (again, trivial subtraction is counted; this also technically uses the borrow one method, but you never have to borrow since you're starting with 9 each time).  Then you add that complement to your minuend (here, 4 steps more, or 5 since you'll have to carry), then knock off the leading 1 (same as subtracting 10000 here; 1 step), then add one to the result.  To demonstrate generally why this works, define your complement C, subtrahend S, minuend M, and difference D:

C = 9999 – S
S = 9999 – C

D = M – S
D = M – 9999 + C
D = M + C + (1 – 10000)

So which is more efficient?  If we're working with 5-digit numbers (leading zeros count), and draw a sufficiently large sample of pairings (10000 here) with the larger of the pair subtracting the smaller, then we can see that the borrow one method is much more efficient in general:
The Common Core method has a wider spread, with some very few pairings needing only 1 step; the minimum for the borrow one method is 5.  At the same time, the borrow one method has an average step count of ~9, whereas the average step count for the Common Core method is about 14-15, with a heavy skew toward higher numbers (15/17).  It is much less efficient.

But these kids are only barely learning subtraction, of course!  Let's keep it to numbers with only 2 or fewer digits.  This is a list we can fully exhaust, there are only 5050 unique pairings.  The results change somewhat: the method of complements is unequivocally the least efficient:
Still, the new method is inefficient, though it does have advantages to the simplest problems (the ones where it's actually kind of silly to think in terms of "algorithms" anyway).

This method isn't wrong by any means; I personally find it to be rather ad-hoc (especially for instance the movement to multiples of 5, and then multiples of 10... why?) and lengthy, and not really that much easier to grasp than the idea of simple subtraction or powers of ten (which students will have to learn anyway if they want to have a good understanding of decimals, non-linear equations, so on).  And I hope that I've helped to explain why.

I find Hemant's commentary funny:
"But none of that matters to the people who would rather complain about the "new" math without taking a second to understand what they're even looking at."
I guess I agree with that, yes.

Wednesday, March 12, 2014

David Silverman and Conservatism

Dave Silverman from American Atheists stuck his foot in his mouth this past weekend when he commented that we should try to attract more conservative atheists to the movement and that there are secular arguments for things like abortion. As you might expect, there was a strong negative reaction from a lot of others in the movement. Here is my two cents:


Thursday, December 5, 2013

Exploring Gravity Trains, in MatLab

Many of us American children have probably wondered at some point in our lives if it would be possible to dig a hole to China: would we be able to actually travel through it to the other side of the planet?  While China is not on the exact opposite side of the Earth from us, this need not be only the musing of young children.  A way to travel through the Earth, instead of over it in the fashions we do now, would take advantage of the shortest direct route.

A gravity train is such a (theoretical) form of transportation, that takes advantage of the Earth's natural gravitational pull downward to accelerate into the planet, and that also takes advantage of that same pull to slow down on approach to the destination.  Ideally, the train would move through a frictionless environment, so that it doesn't have to provide its own thrust.  While such a means of transportation is very likely not feasible because of the amount of energy needed to maintain a several thousand kilometer-long vacuum tube (and electromagnetic rail for levitating, so on), we can still model how it would behave if it existed.

I'll walk through the equations governing the train's motion and how I attempted to model said motion using the math-based coding language MatLab, and also explore some possible density distributions for a given planet and how that affects the travel time for the train.


Friday, September 27, 2013

Nate Phelps: "The Uncomfortable Grayness of Life"

The Michigan Secular Student Alliance is proud to host quite a spectacular speaker this upcoming October 10th: Nate Phelps, secular and LGBT activist, will be coming to Ann Arbor for a presentation at Rackham Auditorium.

Nate is the son of Pastor Fred Phelps, infamously known for founding and heading the Westboro Baptist Church (in the astoundingly rare event that you do not know about this church, here ya go).  Nate became the estranged son of Fred at the age of 18 and left the church in 1980, as his culminating separation from religious belief in the years following concluded with the attacks on September 11th.  He has since worked as a secular activist for both the Center for Inquiry Canada and Recovering from Religion, and quite contrary to his namesake's church, has become a passionate LGBT activist.  Salon interviewed Nate almost a year ago to the day; it is worth the read.

And, he will soon be giving a talk here in Ann Arbor at Rackham, titled "The Uncomfortable Grayness of Life."  As we understand it, Nate plans to talk about his life within the Westboro Baptist Church, and his escape from it and religion in general.  The talk ought to serve as a window into some of the details of his troubled life, and serve as a prologue for his biographical book that he plans to finish writing in the near future.

The Michigan Secular Student Alliance would like to extend a warm and open invitation to anyone that would be able to attend this event.  The talk will start at 7:00pm on October 10th, but the doors will be open to the general public starting at 6:30pm.  Here is our Facebook event page for it, which includes more details such as parking locations.

As with all of our events to date, there will be no cost for attending.  We appreciate the help of the Center for Inquiry, Michigan Chapter, for being our co-sponsor of the event and for acting as our intermediary with Nate; and we also appreciate anyone who at the event wishes to donate to our group, so that we may continue to host speakers and plan large events of interest.  Any contributions can go a long way to improving our group and expanding our influence on – and off – campus.

If you want to attend and are able to, we encourage you to make the trip.  If you want to attend but cannot, we encourage you to help us spread the word to others that might be interested in listening to an inside perspective on the Westboro Baptist Church, and to the life story of someone who has struggled against that church since dissolving such close connections with it.

I look forward to participating in this with you!

- Alexander Coulter
President, Michigan Secular Student Alliance

Saturday, September 21, 2013

State of MI on Same-Sex Marriage: Insultingly Stupid

[please see edit at bottom]
There is an ongoing lawsuit in the state of Michigan right now to overturn our state's ban on same-sex marriage, Deboer v. Snyder.  A lot of hubbub has been raised in the media about a statement in this document filed by the State defendants, claiming:
"One of the paramount purposes of marriage in Michigan—and at least 37 other states that define marriage as a union between a man and a woman—is, and has always been, to regulate sexual relationships between men and women so that the unique procreative capacity of such relationships benefits rather than harms society. The understanding of marriage as a union of man and woman, uniquely involving the rearing of children born of their union, is age-old, universal, and enduring." [p. 15]
Quite a bit has been said already (HuffPo, MLive, ThinkProgress, Detroit News – no coverage from national mainstream media it seems, cursory searches aren't turning up articles), but what is rather strange is how not many people have caught on to a very basic detail about this statement, which comes before how contemptuous this is or how detached from the people's understanding of marriage is:

It's just fundamentally not true.

The argument is insultingly stupid.  There are zero consummation requirements for married couples in Michigan, there are zero requirements for married couples to have or raise kids, there are zero regulations on the types of sex that married couples may have (barring sexual misconduct of 1st through 4th degrees, which apply to all people – but hey actually there are provisions for married couples to not be covered by those laws, like here here and here!  I guess if regulation and lack of regulation can be considered the same thing...).  There is not even a requirement for married couples to undergo testing for diseases and infections, as exists in some other states, only that educational material on these be provided.  Of the laws in Michigan pertaining to marriage, none relate to sex.

So, the purpose of marriage in Michigan is to regulate sexual relationships?  There are no regulations of sexual relationships for married couples in Michigan.  And even if there were laws in Michigan regulating sexual relationships of married couples (which again, there aren't!), allowing same-sex couples to get married cannot possibly reduce the efficacy of said regulations.

In fact, if these fantastical regulations actually worked to effect their stated goal, you'd think that they'd want homosexual couples to marry, so that their children would grow up in more stable households!  Heck, you'd think they'd even take the advice of the people that they reference on p. 16 of their report, who all argue that a married parental structure is better for children than a simply cohabiting couple of parents, or a single parent.

You'd think, anyway.  But these aren't thinking people.

EDIT – correction

It's been pointed out to me by a more-astute-than-myself reader that in Michigan, adultery is still technically illegal, as is cohabiting by divorced parties (I think this means two people who divorce and then still live together, rather than two individuals from separate divorced marriages).  The law has been on the books since 1931, but it has not been enforced since 1971; it's my understanding that, like failure to consummate a marriage, adultery can and likely has been used in divorce filings, but apparently it's possible to be prosecuted for adultery, which it is a felonious offense.

Considering too how this type of sexual behavior probably will have drastic effects on a marriage, this is probably a paramount example of a regulation on sexual relationships of married people.  And yes, it has not been enforced for over 4 decades, but it's technically on the books and I suppose the state could decide to prosecute any time it chooses (backlash notwithstanding).

Tuesday, September 17, 2013

The "Fallacy" of Infinite Regress

There is a preacher on our campus Diag:
(h/t to Monica Harmsen, our former President and current Historian.  Sign reads "It's easy to be an atheist when you don't think about where everything (including God) came from.")

that I just had a discussion with regarding an argument that he tried to use to support the existence of a deity.  I had to go turn homework in for a class so I had to cut off our conversation, and we only got as far as arguing over whether or not infinite regression of causes is a fallacy.  I wanted to talk a bit on this before I went about my day again.

The pamphlet that I was handed has, amongst its 40 (40!) footnotes, an explanation of the fallacy of infinite regress that I will quote (stretched across 2 footnotes):
"Positing (infinite) prior dependencies to account for subsequent ones is not the solution: it is the problem. Fn. 11 [so, going off of footnote 11, which reads...] E.g. no matter how many dominos you add to a line or group, they will never "fall" by themselves because every faller is completely dependent upon its prior.  That's why dominos only fall when some outside force makes it happen.  All the more so with domino existence.  See also Tyson & Goldsmith Origins pg. 44 [link], Vilenkin pg. 204 [no link to page, but some discussion and Amazon has notes for p. 204], Scientific American pg. 50 (inset) [link], F. Collins The Language of God 2006 pg. 54-67 [no suitable link].  Plus, using various interpretations of quantum mechanics and/or special/general relativity and/or singularity theories (no mass = no thing) to ignore that and/or negate the fallacy of infinite dependent regress and the necessities of source/production is a composition fallacy, category mistake, and a red herring. (Craig pgs 150-156)."
I don't know why the Scientific American article is referenced.  Best guess is that the inset says "Expansion probably accelerated early in cosmic history as well, erasing almost all traces of the preexisting universe, including whatever transpired at the big bang itself," and thus that's just like arguing that the universe had an infinite prior universes.  It's not clear that was the intent behind the statement though.  The reference to Tyson and Goldsmith is just for a quote of them off-handedly positing that there might be multiverses or that the universe popped into existence from nothing we could see.

The important thing here is that it's being claimed that asserting there is an infinite number of explanatory events is inherently fallacious – in particular this preacher asserted that it's a "vicious infinite regress," which I can only satisfactorily define as a regression that posits new explanations to account for a cause, explanations that themselves require explanations.  There are two main points to be made here:

Thursday, August 1, 2013

Reza Aslan, and What Constitutes Appeal to Authority

Reza Aslan was recently interviewed by FOX News' Laura Green for his new book Zealot: The Life and Times of Jesus of Nazareth.  The interview was an intellectual train wreck for Green, who decided that the best line of questioning about the book was how Aslan could possibly be interested in Jesus, being a Muslim instead of a Christian.  The interview is below.


While I have my own reservations as to whether or not this series of "questions" was actually intended to be an interview at all, I still think that this, in particular the complaints it garnered toward Aslan's response, serves as a useful event to help us examine appeals to authority, and develop a better understanding of where it can lead to a fallacy and its progenitors in discussion.