Not what you do, but how you do it

By Bobby Neal Winters

Before you read any further, you should know that I do not support the Common Core Mathematics.  My reasons for not supporting it have nothing to do with the specifics of the curriculum.  Instead, my lack of support grows from general principles: I believe that teachers should be professionals and allowed to use their professional judgement regarding how to teach.

Students in different parts of the country have different expectations, different levels of preparations, different goals and aspirations. Professional teachers, along with the school district and parents, should be allowed to use teaching techniques appropriate for those particular circumstances.

This having been said, I find myself disturbed by the rhetoric being used to attack the Common Core.  A particular method of attack is being used which I believe has harmful consequences to the level of the debate.

Consider the following problem in subtraction: 342-97. One method to solve this coming from the Common Core is to perform the following calculation:

97+ 3 = 100

100+200=300

300+42=342

Since 3+200+42= 245, it follows that 342-97=245.  There are quicker ways to do this.  There is the standard method that most of us learned that yields that answer more quickly.  I don’t think anyone would argue that.  That having been said, in my opinion, the value of this technique lay elsewhere.

For example, if you get something that cost $15.23 and give the cashier a $20 bill, the cashier will give you 2 cents to make $15.25, then 3 quarters to make $16; then $4 to make $20. It is the same process.

What does this process do?  It allows us to subtract by doing addition.  Someone who knew only the algorithms for addition could pick up this technique and be able to subtract,  They would be doing it  more slowly than someone who knows the standard algorithm, but they would be learning about what the concept of subtraction means as it relates to addition.  They are also learning how to implement a rather simple algorithm that makes good use of the decimal system of numbers.

Lets look at another method that comes in for criticism.  Consider the addition problem 8 + 5.  Think of it as 8+2+3=10+3=13.  Yes, if you know your addition facts, you can just jump to the 8+5=13.  But look at what his technique does.  It shows us that we can manipulate our numbers.  The 5 isn’t just 5; it is 2+3 and that 2 can be very handy in getting the 8 up to 10.

If we go back to our first problem, we can say 342-97= 342-100+3= 242+3=245, to get a completely different way of doing the problem.

These techniques breed familiarity with numbers and provide a gateway for growth.

As a math teacher of many years, I do work with numbers, but not necessarily in the way one might expect.  When I give an exam, I put 100 points worth of problems on it.  I have a class of 40 or more students most of the time and over the years I’ve developed a system for grading these exams.  I grade one page at a time and at the bottom lefthand corner of each page I write the number missed on the page.  

When I am finished with all of the tests I will go through them and calculate the total points.  If a student has missed 14, 8, and 12 points on the first, second, and third pages, respectively, I will proceed with the following calculation:

100-10=90, 90-4=86, 86-8=78, 78-10=68, 68-2=66.  Sometimes, if I am looking ahead, I would combine the last few steps by recognizing that 8+12=20 and calculate that 86-20=66, and often, in a calculation like 86-8, I will hiccup on the usual subtraction fact and, in my head, do 86-8=86-6-2=78.

I do this all in my head without writing a single step down and I do it for 40 to 50 papers at a run.  It rarely takes me more than a few minutes.  The two techniques illustrated above lead to this sort of mental manipulation of numbers.  

Yes, the standard subtraction algorithm is an incredibly useful technique. It should be taught and mastered.  However, these other techniques which I’ve seen used consistently used as examples of why the Common Core is criminally stupid, are  useful techniques when taught correctly by teachers who are professionals and the techniques are given the correct emphasis.

With math, with teaching, and with rhetoric, it’s not only important what you do, but how you do it.   Of these three, the math is the easy part.

In math, we know a variety of techniques and the secret is which one to use at what time.  It is all between us and whatever problem we are working on.

Teaching is more difficult.  We have our techniques, but when we use them, we by necessity have to include the students.  Each student has different abilities, different preparation, and has a different level of support at home.  Rolling down a one-size-fits-all solution and not allowing teachers to use their best judgment is going to cause trouble.

But then we get to the rhetoric.  

One source I read, criticized the particular techniques because they made students cry. Crying is not a bad thing. I cried in long division to the point my mother just did my homework for me.  I cried even worse in trigonometry when she couldn’t help me at all.  Learning is often a struggle.  This appeal to the emotions is effective rhetoric, but it seems to assume we should never stretch ourselves in learning.

I can understand why these two particular techniques were attacked.  They appear to make difficult what can be done easily another way. Indeed, I was taken in myself at first until I sat down with pencil and paper and worked through the first technique I cited.  It looked stupid to me, someone who’s been teaching math for over 30 years, until I worked it through once.

It is easier to attack a technical package like this by pulling out the complicated looking bits and making sport of them out of context to a non-technical audience.  It is effective; is sews a lot of confusion; it’s harmful to the level of debate.

It’s not just the result that is important. It’s how you do it.  This is not a healthy way to argue against the Common Core.

Too Far from the Average

By Bobby Neal Winters

Buster Williams was sitting in his Elementary Statistics class struggling not to go to sleep.  Sometimes he bit his lip, sometimes he slapped himself on the neck--a trick he’d learned while driving all night to get to Panama Beach--but nothing he did seemed to work.  The problem was at the front of the classroom: the Standard Deviation, or Dr. Bottlebutt as he was known in polite society.

Professor Bertram Bottlebutt, covered from head to foot with chalk dust, wrote one last equation on the board.  He turned for the first or second time during class and asked, “Are there any questions?

Twenty pairs of glazed eyes, Buster’s included, didn’t even blink in response. This was a trick considering he’d been drifting off to sleep before.

“In that case,” he said, “class dismissed.”

Buster felt a rush of adrenaline. His heart soared.  Existence was worth continuing again.

For the first time in an hour, the students in the room showed life. Notebooks were shut, slid into backpacks, and motions were made toward the door.

“Wait, wait, wait,” he called out.  It was something students had heard before.  Indeed several of them did passible imitations of Dr. Bottlebutt, and competed against each other with their “Wait, wait, waits”.

“I need to talk to ... uh... Williams and ... uh ... Young,” he said.

Buster, hearing this, uttered an expletive to himself.

Since being in Dr. Bottlebutt’s class, he’d gotten used to being called “Uh...Williams.” Everyone who took a class from Bottlebutt had the same first name. Students also worked uhs into their imitations of Dr. Bottlebutt.  This was a man who could give hour upon hour of lecture of mathematics composed entirely of abstruse equations on the blackboard without any notes at all but could remember Bob, Emily, or Buster.

The room emptied of students with the expressions of those who, having eaten their morning live frog, knew everything would be better for the rest of the day.  All but Buster and Young who remained.

“Uh...Young” was actually named Betty.  Buster had been sitting in a desk one behind and to the left all semester just because he wanted to be able to see her.  She was...gorgeous. Everything that nature equipped young women with to get the attention of young men was present in Betty not in abundance but in perfect proportion.  And everything was presented in such a way that Buster could find no way to ignore.  As he stood, working on an inventory of Betty starting at her ankles, working up her calves to her thighs...

“Mr. Williams,” came Dr. Bottlebutt’s voice. “Could I please have your attention?’

“Uh...” oh God it must be catching he thought, “Sorry.”

“I was saying my assistant lost your and Miss Young’s grades,” said the professor, seeming quite chagrinned. “I always put the papers in alphabetical order, and apparently he lost the two on the bottom.”

Professor Bottlebutt had his own way of dealing with exams.  He would grade them himself and work up the statistics for the class.  He would then hand back the papers to the class for a few minutes to allow the students to look them over.  Then he would put them into alphabetical order and give them to his assistant to record.

“If you tell me your correct scores, I will have my assistant record them,” he said.

Buster was astonished.  Dr. Bottlebutt was willing to take his word.  This was an opportunity for him to improve his grade.  He was straining through his brain for a number when he heard Betty’s voice from next to him.

“Ninety-eight,” she said.

Mesmerized by the sweet sound of her voice and not being the sharpest tool in the said, Buster followed suit.

“Yeah, me too, ninety-eight,” he said.,

The professor, who was sitting at his desk, glanced upward at both of them with a look on his face that was not indicative of credulity.

“Ninety-eight,” he echoed.

“Uh...,” Buster began but he didn’t get far.

“You know,” Bottlebutt said, “I don’t think that is exactly believable.  Let’s see.  The class average, which I calculated before your papers were lost, was about 70 and the standard deviation was about 6.  This means 98 is just over 4 standard deviations from the mean. That is unlikely for any population, but we have only 20 students in class.  By Chebyshev’s Rule, only 1 over 4 squared of the data can be that far from the average, that only 1 in sixteen. Now 20/16 is between 1 and 2.  So there is some chance that one of you can have that grade, mathematically speaking, but by Chebyshev’s it is mathematically impossible for both of you to have a grade that far from the average.”

Buster stood with his mouth open.  He noticed that Betty didn’t move a muscle. Not even her eyes moved.  Her chest was still moving in and out with a beautiful rhythm...

Butster looked down at the professor and noted that he was still staring up at them.

“Does either of you have anything more to say?”

Buster glanced Betty’s way and her expression was as innocent as a dove.  It made him ashamed of lying, but before he could own up, the professor continued.

“It might be well for you to know that since I’d figured up the statistics for the class  I was able to figure out some information from the rest of the papers,” he said. “The sum of your scores is 140.”

Buster thought of the score he’d really made, 70, and then, in spite of not being the sharpest tool in the shed, figured what Betty’s score had to be. She’d made 70 too.  Two thoughts raced to share his mind. One: They’d made the same score and had something in common.  Two: She was lying her gorgeously perfect backside off.

Buster looked at Betty.  The expression was so slight that he didn’t know whether it was there or whether he was imagining it.  It was in the eyes; it was around the curve of her beautiful lips. Please don’t give me away.

“Uh...Dr. Bottlebutt, did I say 98? I must have misspoke,” Buster said as he frantically tried to do mental arithmetic. “My real score is...”

“Forty-two?” the professor interrupted him.

Buster did a quick 98+42=140 and answered, “Uh...yeah.”

“There is a problem with that,” Dr. Bottlebutt said. “Forty-two is also more that four standard deviations from the average.  There can only be one that far away, high or low.  Do you want to stick with that?”

Buster looked at Betty, who was the most beautiful example of the human female that he’d stood in the presence of.  She was still holding that expression. It was so subtle, but so compelling.

“Yes,” he said, “sure.”

Bottlebutt looked up at both of them.

“You know,” he said, “I could take ten or twenty minutes and work this out. It’s not hard. I’ve got all the information I need.  But I think there is a learning opportunity to be had here, so I am going to take both of your words as true.  Good day.”

With that, they were dismissed.

Buster walked into the hall with Betty.

“Betty,” he began.  His next words were going to be about maybe going to get a latte or something, but no one was listen.

Betty walked over to the tallest, most broad shoulder man Buster had ever seen.  She put her arm around him and he put his hand in her back pocket. And the strolled away.

Buster, more than a bit crestfallen, got kind of a bitter taste in his throat. He turned and saw Bottlebutt who had clearly taken in the whole scene.

Buster stood there wordless and didn’t expect the professor to say anything, but he did.

“There are some lessons harder in the learning that math,” he said.

Converting Fahrenheit to Celsius

By Bobby Neal Winters

I am still in the process of learning Brazilian Portuguese. When you are learning a new language, numbers are always a problem because they are entities that require the same sort of manipulation that words do and everything gets clogged up.  This happens especially when the numbers you are dealing with are temperatures and you are going to a country where they measure temperature in Celsius instead of Fahrenheit.  

There are simple formulas for converting from one to another:

F=(9/5)*C+32

C=(5/9)(F-32)

where F is degrees Fahrenheit and C is degrees Celsius.  I say simple. To a mathematician, these are simple because they are linear. Linear is just one level of complication above constant.  The problem is that if you are trying to work with a new language, there is probably too much going on in your head to do this.  And to be frank, there is probably too much even if you are working in your own language because there are fractions.

The simple way to deal with the Celsius scale is to learn it on its own terms.  We will be get with some facts.  In Celsius, 0 is freezing and 38 is body temperature.  Also, 50 degrees Fahrenheit is 10 degrees Celsius; 68 Fahrenheit is 20 degrees Celsius;86 Fahrenheit is 30 Celsius.  Finally, both scales are the same at -40.  

So think about it like this.  At 40 degrees Celsius, you are above body temperature, so it’s pretty hot.  It is a typical summer day in Oklahoma or Kansas or Rio.  Folks from the north or from the coasts, wonder how people live in such weather.  They come to the conclusion they can’t, so whatever lives in it can’t be human.

At 30 degrees, an Okie will say it’s pleasantly warm, and a Californian will complain and, perhaps, pray for death.  At 20 degrees, the girls in Rio will put on light sweaters over their summer dresses; Kansas girls will think they are in paradise; Californians will complain because it’s cold.

At 10 degrees, it’s cool. The girls in Paraguay are wearing every stitch of clothing that they own.  The Californians are filing lawsuits. A Kansan has put on a windbreaker. The folks in Wisconsin are wearing shorts.

Zero degrees is cold.  The Brazilians are experiencing hypothermia, but are hoping in their hearts that they might see snow before they die. The Californians are curled up in little balls, scratching their wills into the frost. The Kansans have on long sleeves under their windbreakers. The guys from Wisconsin are still wearing shorts.

Front Porch Math

Front Porch Math 

By Bobby Neal Winters
The following is a real problem that a carpenter with a great respect for trigonometry came to me with.  Say you are going to add a porch of width L to the side of a house and want to over it by adding to the roof as depicted in the picture below:

The question is how long do you want your rafter to  be, i.e. what is the length of the segment D.  As it is there is not enough information in the problem.  You have to know that the slope of the current roof is 7/12, (i.e. over a run of 12 feet the roof will rise 7) and they desire the porch roof to have a slope of 3/12 (i.e. over a run of 12 feet it will rise 3.)  Yes, we could say that is a slope of 1/4, but these are how the numbers came to me, and it will all come out in the wash anyway.

I must confess that this problem took longer for me to solve than it might have because I was guided initially by the carpenters love of trigonometry.  It can be solved that way, but this is a nice case study of how imposing a coordinate system can simplify the problem.  Consider the following:

We've put on coordinates so that the outside point of the porch roof is a (0,0), the point where the old roof touches the house is (L,0), and the point where the porch roof meets the old roof is (x*, y*).

The equation for the line representing the porch roof is y=mx and for the old roof is y=nx-nL.  It is an exercise to work out that x*= nL/(n-m) and y*=mnL/(n-m).  We may use the Pythagorean Theorem (or the distance formula) to calculate the value of D2=(x*)2+(y*)2 which after substitutions simplifies to:
D2=(1+m2)n2L2/(n-m)2   or D=(1+m2)1/2 nL/(n-m).

For the given values of n and m, this works out to D=2.02598L.

Having done it this way, I also worked it by trigonometry, but I despair of reproducing those calculations without specialized mathematical software.

A Squirrel’s Tale

By Bobby Neal Winters

I received the following report from the Bureau of Unlikely Seeming Happenings, Bulsh for short.

Cletus T. Gnasher was out hunting squirrels one day in order to put a little meat on the table when he came upon a strange sight. He saw a squirrel go into one end of a ten foot long hollow limb.  He took out his trusty 22 rifle and aimed it toward the squirrel.  One minute later the squirrel came out the other end.  

Cletus was about to squeeze off a round when the squirrel ducked back into the hollow limb.  Not about to give up this easily, Cletus kept his eyes on the tree.  At the end of 30 seconds, the squirrels head was out of the other end.  Again, before Cletus could squeeze off a shot, the squirrel had ducked back inside the hollow limb.

The squirrel went again to the other end of the hollow limb and this time in only took 15 second before the little head poked out the other side.

It is at the point some began to doubt the veracity of the report because Cletus said that something funny began to happen. The squirrel continued to go back and forth  between the ends of the hollow limb until at one point it appeared he was looking out of both ends at once.  How long did this take?

This was presented to the Bulsh board and reports were requested.  The first report came from a mathematician.  It was very succinct.  This is an infinite series problem.  To determine the time expended, sum one minute plus one-half minute plus one-fourth minute and so on, i.e. 1+ 1/2 + 1/4+ ... .  This is a geometric series whose sum is 2.

The second response was from another mathematician who didn’t offer a solution, but said that this one couldn’t be correct because it assumed the squirrel would be transversing the tree limb an infinite number of times.  As the length of the limb was fixed, being 10 feet, transversing it at infinite number of times would require the squirrel to go an infinite distance.  That is to say that while the infinite series for time-passed converged, the series for distance traveled diverged.

At this point, a physicist chimed in, saying that the problem was reached long before infinity.  The first time the squirrel went through the limb, he was going at a speed of 0.1136 miles per hour.  However, by halving the length of time through the limb each time, he was doubling his speed. Taking the liberty of converting this to 0.05784 meters per second to make sure no one forgot he was a physicist, he said that after 33 more times through the limb, the squirrel would be going 4.36 times 10 to the 8 meters per second which is greater than the 3.0 times 10 to the 8 meters per second that light travels.  At this point, one of the mathematicians corrected him to 2.998 times 10 to the 8 meters per second.  The physicist wrote back something that best not be repeated here and a flame war ensued.  

After the verbal abuse subsided, the physicist calculated that, because of relativity, a one pound squirrel would weigh 1.45 pounds after 32 trips through the limb.  One of the mathematicians asked him whether he meant a one kilogram squirrel would have the mass of 1.45 kilograms, but fortunately the irony was lost on him, and then he was distracted by the following question from an engineer.

The engineer noted that Cletus only said that it looked as if the squirrel were looking out of both ends at the same time.  An image persists on the retina for one-sixteenth of a second.  According to his calculations, it would only take 10 trips through the limb to be making it in that amount of time.  This is a speed of 58 meters per second.

The first mathematician then calculated that this would take 1.998 minutes which was only 2 one thousands off his original estimate. The second mathematician said, “Yeah, your original WRONG estimate.”

This was beginning to degenerate into something nasty then the physicist said, “You know this would require an acceleration of almost 12 times the force of gravity to accomplish.  The squirrel would be  mush.”

At that point, one of the animal-rights folks on the committee said this was something you couldn’t even talk about doing not even to a theoretical squirrel. As this was a supposedly real squirrel, an investigation would have to be launched.

One of the mathematicians said, “By academic freedom..."

And the reply was that he could be free to seek another academic job if he said one more word.

On One Hand But Then on the Other

By Bobby Neal Winters

I was approached yesterday by a friend of mine at church in Opolis in the following way.

“You’re a math professor, so I got a question for you,” he said.  “It’s from my grandson.”

I remained calm outside, but on the inside I had the reaction a gunfighter in the Old West had whenever someone said, “They say you’re pretty fast.”  You never know when the person asking the question might be faster.

I decided to take my chances and listen.  He held out his right hand and began to count extending his fingers one by one:

“One, two, three, four, five,” he said, extending his pinky last. “And five is ten,” he said extending all of the fingers of his left hand at once.

I nodded because I knew we weren’t at the hard part.

He then held out the fingers of his left hand and started extending them thumb first:

“Ten, nine, eight, seven, six,” he said, giving emphasis to the six. “And five is eleven, so you got eleven fingers.  How can I explain to my grandson why this is wrong.”

And my friend knows it’s wrong.  But knowing it’s wrong and being able to explain why are two different things.  And I could go on to a digression about politics here, but my point is math, or at least the explaining of math.

My friend knows this is wrong because we have ten fingers and ten does not equal eleven, no matter how fast you talk.

Let’s first analyze how this is presented because that is very important to how the confusion comes in.  We start off with something that is true: one, two, three, four, five, and five is ten.   What has happened there?  We’ve listed off the names of the first five numbers that we learned when we learned how to count.  We’ve set up a correspondence between those names and our fingers.  

The next part is where something subtle is done which sets us up for the confusion. When he says, “And five makes ten,” he’s made a very subtle shift.  He’s using the name five to refer to the quantity of fingers on his left hand.”  The five he said when he held out his pinkie was referring to a place in order; the five he said when he held out all his fingers at once was referring to the quantity of fingers.  Mathematics call the first one ordinality and the second cardinality; think of these as order and quantity.

In listing the numbers in standard order, “one, two, three, etc,” the name of the last number listed is also the name of the quantity of the items in the list.

This does not work when you count backwards, and that is one of the things that the example my friend brought me illustrates. In counting backwards from ten, there is no point at which the number counted with also be the quantity of things counted. (Ironically, if he’d started counting at eleven it could be made to work, but you can’t do that because you have ten fingers. It is the center of this trick that you start counting backwards at ten because you know there are ten fingers.)   

When you hold out the pinkie on your left hand as you count backwards from ten until you get to six, you are giving its order as if all the fingers were counted beginning with the other hand.  It can refer to a quantity, but that quantity would be the pinkie itself and the fingers on the left  hand.

Now, you must understand that I didn’t tell my friend all this.  I said, “It’s a confusion between cardinality and ordinality.  Don’t let your grandson play poker with Bill.”

Bill is a man who’s given me poker lessons. After $15 worth, I learned: don’t play cards will Bill. But that is a different story.