Yearning for Some Joy of X

Last year, I decided to take Algebra I in order to get over my math anxiety. I wrote about it a couple of times here and at in the New York Times. In what anyone but myself could have predicted to be a critical error on my part, I began attending class halfway through the year. I missed the first half of the textbook, never learned how to sing the quadratic equation to the tune of some Christmas carol, and I could only attend class three days out of five because of my teaching schedule, so I was doomed to failure. This year, I have no such excuses. I am all in, from day one, four out of five classes a week.

If I was anxious about my performance in math the first time around, imagine the added pressure I am under now. My students are all aware that I am re-taking the class, and while they finished Pre-Algebra last June, I finished it in June of 1982. My teacher, Alison Gorman, calls on me in class, I do the homework every night at the dining room table while my sons do their homework, I hand in review sheets, and I just took my first test. I planned to take the test along with the rest of the class, but an emergency parent meeting took precedence, so I took my test at home. My sons were fighting over some thing or other at the time, so my test-taking environment was not what you might call calm and stress-free.

That said, things are going really well. Alison allows us to hand our review sheets in early for corrections, and I was the first student to earn a perfect score on the first one. Definitely a new Jess Lahey moment, and the review sheet has a place of honor on our refrigerator. I have been doing well on my homework, and was feeling confident going into my first chapter test. I will hand that in tomorrow, with all fingers crossed.

In the meantime, I've been trying to work out precisely what it is about math that freaks me out. Part of it is certainly the habit I've gotten into, the automatic assumption that I can't execute simple math without using my finger to draw the math problem in the air so I can remember to carry the one. Part of it has to do with my frustration that when when my husband or son multiplies 26 * 4, they simplify the whole endeavor to 25 * 4 for an easy 100, then remember to add on those extra four, for 104. I don't naturally think that way.

Then, last week, I isolated the real source of my frustration. I was doing the following problem:

30 - 5 * 3.

Simple, right? Well, here's what irks me. You have to do the multiplication first, because that's the rule according to the order of operations (PEMDAS - parenthesis, exponents, multiplication/division, addition/subtraction). So...do you do 5*3, which equals 15, then remember that the negative sign is there and take 15 away from 30? Or do you multiply a -5 by 3, which automatically gives you a -15, which you then add (or, as it's negative, take away) the 15 from 30?

Turns out, it works either way. Which is my problem. Alison claims that it doesn't matter. Rules are rules.

Rules. Tricky things, those rules. In Latin and English, rules are slippery creatures, best banned from my classroom. As soon as I tell a student they can rely on a grammatical rule, an exception sneaks out of the recesses of the English language to bite me on the butt and prove me wrong. So this whole trusting of rules thing does not lie within my comfort zone. I brought this up with Alison, and she smiled at my discomfort, reassuring me that I can relax. Rules are the rules, and that's the nice thing about math. Certainty.

My certainty lasted until second period Algebra class. Alison was explaining why numbers to the exponent of zero always equal one - it's a rule I did not understand last year, even after a very sweet student explained it to me repeatedly (see her explanation here). This year, I finally got it. In the middle of the explanation, Alison stopped, got a strange look on her face, and looked at me. She sheepishly apologized to me as she began to write the exception to the rule she had articulated not an hour before. Zero to the exponent of zero is not, in fact, one, but...drumroll...undefined. Not definable. Ahem. An exception. See? Bitten right there in the butt.

And then, an angel appeared. My personal math angel, who, as angels are wont to do, managed to make my life simultaneously simpler and much, much more complicated.

My math angel? Steven Strogatz - sorry, Professor Strogatz. He's a kick-ass teacher, bard of all things mathematical, and a lovely person, as I found out last week. He wrote a series of articles a while back for the New York Times that were amazing, and in advance of the release of his new book, The Joy of X (yeah, I know...so great, you know he did a fist-bump with his agent when he came up with that title), he's doing a new series for the Times. The first piece, on singularities, appeared last week, and I wrote to compliment him on it.

I wrote him an email, he wrote me and email, and we got to chatting about math. I explained my whole distrust of rules issue, especially in light of the zero to the zero debacle, and he replied to that email with his thoughts on the subject. My original email is in black, his are in red (I edited out some of his non-math comments):

Steven,

Sorry to bother you again, but there's a major issue of math you should be made aware of. 

I'm writing right now about the fact that math "rules" make me nervous. [They should make you nervous. There's a beautiful book called John Stillwell called "Yearning for the Impossible" on this very point.   http://www.amazon.com/Yearning-Impossible-Surprising-Truths-Mathematics/dp/156881254X   It argues that the great advances in math come when people break the rules. Two simple examples: (1) Rule: you can't divide 5 things evenly among 3 people, because 3 doesn't go into 5. True in the world of whole numbers. But if you "yearn for the impossible" and open your mind, break the rule, and invent fractions, suddenly your system of math becomes more free AND more powerful. (2) Rule: You can't take 6 away from 5. True, in the world of positive numbers. But if you yearn for the impossible and enlarge your concept of number and invent negative numbers, you can now work the math of of debts, which you couldn't even conceive of before. I discuss this sort of thing in the beginning of The Joy of x. It's the great narrative arc that runs through all of early math (and later math too, as Stillwell explains).]    I teach grammar and languages and there, "rules" are always in quotations - they are subject to the inevitable exceptions. So, when I come across the homework problem:

30-5*3

I get nervous. Is that -5*3 which is -15 [yes] that I take away [no. You don't take it away. If you're regarding it as -15, you need to add it, not take it away, from 30. You already used the negative sign to make the 15 into negative 15. You don't get to use it a second time to do a subtraction. I'll avoid the temptation to make a bad linguistic joke here about double negatives. Boo!]  from 30, or is that 5*3, and the negative is a subtraction symbol rather than a negative symbol - I mean heck, I don't know enough to know the difference, I managed to take linguistics as a substitute for college math…

Anyway, the fact that the answer is the same either way, and the fact that my Algebra I teacher can explain that it will always work out the same way, no matter what, does not ease my worry. 

There are always exceptions. My husband explained that this is what is lovely and amazing about math, that there are not exceptions. 

Alison Gorman, my lovely Algebra I teacher, explained that this is what is lovely and amazing about math, that there are not exceptions. 

Except…

Except…

I went in to math class the next day and learned why 3^0 is 1. Last year, I wrote about this, about how I had to just accept it as a RULE because I could not SEE it, and math people offered some lovely comments that went over my head. 

But Alison explained it so I understood this year, and I was happy. 

Except…

She then gasped and realized that she was about to make me very unhappy. Just when I had been mollified, when I realized that I could take solace in the rules of math, I could TRUST the rules of math, she offered up the one exception to that rule. 

Damn that 0^0 and its exceptive qualities. [yes, 0^0 is worthy of a column. Very tricky.]

Back to square one and my distrust of math rules. 

This is what I am writing about this evening. Although, as a side note, I did get an A on my first Algebra I chapter review, and I have decided that my new favorite thing to do is simplify. It's quite satisfying. Although when I was just asking my otherwise brilliant husband what the word I was looking for, "What am I trying to say…I simplify like…terms? variables?" he said, absentmindedly, "You simplify like a 'muthah'" which I assume he meant as some sort of compliment. 

Anyway. That's what I'm writing about, and I expect you to fix that whole "exception" issue in math as it appears to be my main stumbling block and source of distrust and unease. [Hope my comments above help. The biggie, which I did not talk about yet, is infinity. The Greeks after Aristotle said the rule is, you can't think about infinity. Not allowed. Too many paradoxes. Math was stuck with that rule for thousands of years, till the 1600s, when the yearning for the impossible got too overwhelming, and the need too great. And the result was calculus, which tamed the infinite and harnessed it to solve all kinds of problems that ultimately made modernity possible.You'll see what I mean when you read the book...]

Jess [Steven]

So you would think that Algebra I would be a breeze, what with Professor Steven Strogatz willing to personally explain the subtleties of my Algebra homework, but I guess not. And now that my teacher, my students, my students' parents, my blog readers and math stud Steven Strogatz are aware of my adventures in Algebra I, the stakes have been raised. Good lord, I hope I did well on my Chapter One test.

P.S. Professor Strogatz's Radiolab segments are some of their best, and they generally produce the best stuff on the radio. His TED lecture on synchronicity in the natural world rocks, too.

P.P.S. After reading this post, he recommended this page, for its discussion of 0^0.

UPDATE: I just got my test back from Alison, and I got a 97/102! Some stupid addition errors and not fully understanding how to simplify some fractions lost me a couple of points, but I'm pretty darn proud of myself today.

Algebra I: Still Hazy After All These Years

This isn't good. One homework problem took me 32 minutes and two entire sheets of paper. As there were about thirty problems in the entire homework assignment, it should only take me...um...sixteen hours to do my homework. Did I do that right? 32 times 30, then divide that by 60? Good lord, I hope that's right, because if I can't figure that out, I am dead in the water in Algebra I. The scan above is the cleaned-up version of the problem, the version I re-copied so I could actually how the heck I got to a final answer.

I knew I would struggle in Algebra I, when I returned to class in order to get over my math anxiety, and not just because I have a lot of trouble visualizing how numbers work. The bigger problem is that due to my overloaded teaching schedule (I teach two sections of composition 8, one of composition 7, English 8, English 7, Latin 6, Latin 7, and Latin 8), I can only attend two of the four Algebra I classes held in a week. I do my best to keep up, and as I have mentioned, my students fill me in on the important lessons I miss.

The biggest hurdle I face at the moment is the lack of continuity. I jumped into Alison Gorman's Algebra I class halfway through the year, and missed all of the lessons that inform our current chapter (11). When Alison says, "Don't forget to use the discriminate factorability test! You will have to keep it in mind when you do tonight's homework!" I have to quickly and surreptitiously flip to the index in the back of the book to understand what she's talking about. D...D...Directly proportional...Disappearing variable...Discriminant...ah - Discriminant, test for factorability, 436.

"A quadratic [oh, crap, look that up, too] trinomial can be factored if an only if the discriminant [...and that, too] is a perfect square."

Lovely. As clear as swamp effluvia.

My biggest worry at the moment is that I am going to need the quadratic formula to do some of the work in this chapter. I remember learning it once, long ago...something squared over something else with b and 4ac and...

Q....Q...Quadratic degree...Quadratic equations...solving by factoring - here we are. Formula, 214. Way the heck back in chapter six.

Okay. I kind of remember that. But come on, it was thirty years ago. Apparently, Alison teaches the kids a song about the formula. I remember hearing them singing during class back in December. I have to remember to ask my students to teach it to me the quadratic formula song on Monday.

What has been fun is the opportunity to see my students in a new context. Alison witnesses different aspects of these kids in her math class than I see in English and Latin. I am getting an education in my own students, and that alone has been worth the effort I've put into Algebra. This new knowledge has really paid off this trimester. One of my students was having trouble in math, and when I met with Alison to discuss what was going on, she was able to show me in the students' math notebook exactly what mistakes she was making. I never would have understood the nature of her mistakes before, and it allowed me to sit down with my advisee and offer up more than general advice about diligence and "doing your best." We were able to untangle her specific issues with the unit and devise a game plan. We laughed together over her misunderstandings and I empathized over the time spent working through the more confusing problems on her last test.

But she's still in better shape than I am. She had never spent 32 minutes on one problem.

This weekend's homework: catch up on relative rates problems. Alison calls them "sitcom problems" because whenever a sitcom character wants to freak another sitcom character out with a complex math problem, they give a relative rate problem as an example.

"A kisses B goodbye and leaves home on his bicycle, going 30 km/h. Half an hour later, B realizes that A left her briefcase at home and so he starts from the same house, going after A in his car, at 50 km/h. How long does it take B to catch up with A?"

Alison insists that these problems only sound complex, but are really just a matter of common sense. Alison's reassurances make one huge assumption, however. The small detail of common sense, which I fear I lack. As there are around thirty problems assigned for this weekend's homework, I'd better get to work. Monday is only 72 hours away.

Part VI of my math odyssey can be found here.

Exponent Negativity

An 8th grade student approached me yesterday after third period. I had just taught composition class - the composition class that conflicts with the Algebra I class I have been attending in order to get over my math anxiety.

"Mrs. Lahey, I we did some hard stuff in math today, and we were really worried that you'd get behind, so I agreed to take notes for you and teach you the lesson. I can come by during lunch today and explain what we learned, if you want."

This reversal just slays me. My students are worried for me. Worried about my ability to keep up in math class. How sweet is that? I was really touched. One of the things I love most about my school is the sense of community, but until yesterday, this community had always been a "them" and "us" community. We adore our students, mind you, but as much as we'd like for our community to be one, big, fuzzy "us," it's not. Students are students and teachers are teachers, and never the twain shall meet.

Until yesterday.

Before I start getting attacks from readers who think I am trying to create inappropriate relationships with my students - relationships I am supposed to view as rigidly hierarchical and hopelessly lopsided as a power structure - know that I am not trying to be my students' friend. I just think it's good for them to see adults not know things, and not be afraid to not know, and not run from not knowing.

I talk and talk about the importance of viewing education as a lifelong process rather than a means to some calligraphy-on-parchment end, and my attempt to work through my math anxiety is proof of statement. I really mean it. I love to learn - and not just the stuff that comes more easily. Even the stuff that makes me want to give up and run screaming in the other direction.

Stuff like this:

As I am a newly minted Algebra I student, let me break this explanation down for you.

[Silence, eventually the sound of fingers tapping idly on the keyboard, as I attempt to think of words appropriate to the material contained in the scanned image, above.]

Okay. Here's what I know. I must accept the fact that any number to the power of zero = 1. Ellie, my English-student-cum-Algebra-mentor explains that concept in her red notes above. I don't understand it, but I accept that it's a rule, and follow it.

The other thing I learned is this: if an integer is negative, you can stick it at the bottom of a fraction under a one, and it magically becomes positive. I don't know why, but if I simply accept this and apply it, my homework answers are right. I got through a difficult problem set today through the blind application of these two rules.

I'm not proud of this reality; I'm just owning it. I just don't get it. Alison Gorman, my colleague, friend, and math teacher is patient and kind and generous with her time, and I feel as if I get it for a second or two after she explains it to me, but then, poof, it's gone.

This has always been my problem with math. I try to understand the whys and wherefores of the rules, I really do, but the why just goes over my head. I don't know if it's because I don't care, or because I know that in the end, if I just decide to accept the rule and use it, I can get by through sheer grit and application of the rules I don't really understand. That's how I got through high school math.

Alison is one of the most effective, organized, creative, and dedicated teacher I have ever met, and yet, I think she may have met her match in me. She's extremely patient, and more than a little entertained by my efforts, but I'm afraid she's going to realize that I have severe limitations where numbers are concerned.

I will continue to try to understand, because I hate that I don't. I hate not knowing. I hate butting up against my limitations.

It's time to bust through the negative and transform those integers into their positive form once and for all.

Monday: Powers of 10 and Scientific Notation.

Part V of my math odyssey can be found here.

Algebra I: I Scream

Algebra I

Day I

I am an English teacher with a math anxiety, and I have gone back to Algebra. I explain it more fully here.

Linear functions, scattered data, and probability, and apparently, I have just about the worst timing ever. The chapter test is tomorrow, so the students were reviewing. As I can't remember what a linear function is, I was at a slight disadvantage.

Today's problems concerned coin flips, lottery problems, factorial problems (another concept I can't remember) and dice rolls, complete with histograms.

Okay - here's my first question. Histology is the study of organic tissue. Histo- comes from the Greek and means "a warp or web," literally "anything that stands upright" from histasthai, "to stand." Medical writers in the 19th century decided that this root was the best candidate for medical terminology concerning itself with "tissue." So how the heck do we get to a place where a histogram means a mathematical graph?

I did not want to raise my hand in the first five minutes of my first Algebra class. I wisely sensed that an etymology lesson might just derail poor Alison from her homework corrections, so I wrote it down in the corner of my notebook and did the research later on.

A mathematical histogram is a visual representation of the distribution of data, specifically for probability. According to the research I did after class, Karl Pearson, who coined the term in 1895, took histo- as "anything set upright, as the mast of a ship or the bars of a loom" and gramma-, meaning "drawing, record, or writing." Another source says that Pearson just went with histo- from "historical" and gramma- "drawing," and meant to create a drawing that shows a historical representation of statistical data.

Whatever. My point is that my first question in math class was an etymology question. You can see what I am up against here.

I liked the next problem. Vermont license plates use a combination of three letters, followed by three numbers. How many possibilities are there for Vermont licence plates? Well 26* 26* 26 10*10 *10 (I am just going to assume here that I did not represent that correctly; give me a break, it was my first day) = 17,576,000 possibilities for Vermont license plates. As there are about 28 people living in Vermont at the moment,* I think the state is in the clear where motor vehicle registrations are concerned. 

The third problem really threw me for a loop and not because of the math - no, that threw me, too, but only because I could not get to the math part.

3. Chris Cross si 14 months old, and is just learning how to walk. He zigzags across the 12-by-15 foot room shown in the figure, eventually falling down. The ice cream cone he is carrying spills either on the 9-by-12 foot rug or on the floor around the rug.

Okay. Before I can even begin to think about the math, what sort of parent gives a 14-month old an ice cream cone and lets him walk around on a carpet? This entire mathematical quandary could have been avoided with better parenting.

But, as I am here in remedial math class due to my superhuman ability to avoid math, I forced myself to get past this appalling lack of judgment. I gamely participated in the problem and did not even point out that any intelligent mom would have that kid on the hardwood floor OUTSIDE the perimeter of that carpet. Oh - and hey! I know what the perimeter of the carpet is!

1. Bill Ding [clever] constructs quality homes. He figures that a house built on a particular lot should be sold for $373,000 if it has a floor space of 3,000 square feet. A 4,000 square foot house constructed on the same lot should be sold for $483,000. The selling price is the sum of the construction cost, at a fixed number of dollars per square foot, and the cost of the lot.

The problems after the example tested me. I was able to figure out the number of dollars per square foot it costs to construct the house, and the cost of the lot. I was actually quite proud of myself. However, two minutes later, I had to ask another student explain the terms "domain" and "range," terms I must have known when I was 13 or so when I had Mr. Davin for Algebra. I tried to sketch the graph, but my graph was just sad and looked nothing like the students'.

In the end, I'm okay with how things went. My students saw me go out on an intellectual limb and dare to admit I don't know something, and they got to feel good about teaching me something. It's win-win.

Items to look up tonight: factorial, linear function, domain, range, cleaning products that remove dairy from carpet.

*also look up how many people actually live in Vermont.

Part III of my math odyssey can be found here.

Quantifying the Unknown

I have a recurring dream: I am in college, or law school - sometimes high school - and I have not attended classes all semester. I don't know where my locker is, let alone the combination, and I'm not sure where the classrooms are. I certainly don't know what material we have covered. It's exam time, and I know I won't even be able to find the classroom so I can take the exam. I know that if I don't pass the exam, I will not be able to graduate, so I have to find the main office and I can ask them where my classes are. I wander around the school, often in that slow, stunted, underwater way, and get nowhere as the minutes of the exam period tick by.

One fun twist on last night's version of the dream is that while I was digging around in my materials, looking for any clue as to my class schedule, I found an application for an internship with Billy Collins. Despite that nice surprise, I still woke up nervous and flummoxed.

The class I fail in my dreams is usually math, which is absolutely predictable if you know me at all. I have never done well in math, and as a result, I have serious math anxiety. I can grade papers and calculate percentages, but anything more complex than that brings on the shakes. So when I woke up this morning at 4:32 A.M., drenched in sweat, I decided to return to the place of my defeat (again) and go back to school.

Fortunately, the math teacher at my school is amazing. She's the math teacher I wish I'd been lucky enough to have when I was in middle school. She's organized, demanding, witty, and kind, and every time I poke my head into her classroom, her students are enjoying themselves.

I know! That was a new concept for me, too.

I checked the middle school schedule and realized I have a prep period during her Algebra I classes on Wednesday and Friday, so starting next week, I'm back in school. I will have my own textbook, and I will even try to do some of the homework once I catch up to the kids (if that even happens).

It's time to put my money - well, my pride, I suppose - where my mouth is. I encourage my students to be brave, diligent, and never back down from an intellectual challenge. I try to model a love of education, a genuine thirst for knowledge that drives what I read, watch, and listen to. They know I am a frequent buyer at The Teaching Company, and that I listen to lectures on my iPod when I stack wood, fold laundry, and chop vegetables for dinner. Currently, I'm listening to a course on the evolution of the English language, and I love to share the tidbits I've gleaned with my students. They know I listen to courses on English, history, and science, but they also know about my math aversion. They joke about it. I joke about it. But it's not really that funny; it's actually quite sad.

I am hoping that my efforts to remedy my math anxiety will put weight behind my words. I hate math, yes. I hate the obsessive attention to plusses and minuses. I hate the inflexibility of numbers. But the excuse that "my brain just does not work that way" doesn't cut it when I simultaneously tell my students that they can do anything - anything - they want to do.

Well, dammit, so can I.

Part II of my math odyssey can be found here.