Tall, Dark, and Mysterious

1/10/2006

I assume Piaget’s children were not gifted math students.

File under: Those Who Can't. Posted by Moebius Stripper at 8:28 pm.

I’m still tutoring math these days, but with smaller students.

Today’s pupil was grinning as he declared, “Today after school, I lost a tooth.” He lifted his lip with a graphite-stained finger to show me the gap. “An incisor. I know because that’s what my dentist told me it was when I saw him. It was loose for a long time.”

“An incisor!” I exclaimed. “That’s a pretty big deal.”

He nodded. “I never lost an incisor before. I lost three bottom teeth and two top teeth. This is the sixth tooth I lost. My top ones grew in already, see?” He curled his top lip inward to show me a pair of mismatched front teeth. “I lost them last year.” He then lowered his gaze and scrunched his face thoughtfully. “Does the Tooth Fairy give more for incisors? I know she gives more for top teeth than for bottom teeth.”

“I don’t know,” I said. “I think the rates have changed since I was your age.”

He nodded. “Things are more expensive now. My mom says you used to be able to buy a chocolate bar for fifty cents. Even my brother said that he used to get just fifty cents for his teeth, and I get a dollar.”

“Yeah, that’s called ‘inflation’.”

“My brother told my mom he thinks the Tooth Fairy likes me more than she likes him.”

“How old is your brother?” I asked.

“Thirteen.”

I bit my lip, thinking that the older brother’s going to get whatever he wants for his fourteenth birthday. “Naw,” I said finally, “the Tooth Fairy likes all children equally. It’s just that she’s making more money now and can afford to spend more on kids’ teeth.”

The boy nodded. “Well, I hope I get more than a dollar because there’s a comic that I want and my mom says I have to save up my own money for comics.”

“Well,” I reassured him, “you’ve got another fourteen teeth to lose.”

“Yeah.” He thought for a minute, and then, satisfied, lifted his backpack onto his table and withdrew his homework. “I have a question about this,” he said, pointing. “Do I simplify the fractions before I solve for x or after?”

12/15/2005

So, what IS the point of those introductory college statistics classes, anyway?

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 9:06 pm.

For the past few months, I’ve been tutoring a college student in statistics. This tutee, unlike the other one I took on this summer, is good-natured, engaged, reasonably comfortable with basic mathematics, and in general an absolute pleasure to work with. But I figure there’s a reason that TD&M gets more hits in an hour than there are holds on every copy of Pollyanna in BC’s public libraries put together, and why mess with a winning formula? So enough about that student.

Let’s talk instead about the purpose of these statistics requirements for business and social science majors.

I taught such a course last year. My objective going into the course - and objective that made its way onto the syllabus - was for my students to emerge with a decent ability to assess and interpret quantitative data. Every lesson plan was subordinated to this purpose. I taught the standard intro-stats notation and terminology, but only as a means to an end. Each and every one of the ten quizzes and three tests that my students wrote contained at least one question that required that they answer in plain English. It was not enough that they be able to give me the bounds of confidence interval; to receive full credit, they needed to tell me what it meant. It was not enough for a student to tell me that a sample proportion fell inside the critical region and that we should therefore reject the null hypothesis; they needed to tell me what that meant in terms of a manufacturer’s or politician’s claim.

The class, for the most part, went rather well. Marks weren’t great; but I was confident that a good mark in my statistics class indicated a genuine understanding of statistics, and not just an ability to pluck out numbers that, when plugged into a meaningless equation, will yield the numerical answer that will be marked correct.

The statistics class that my tutee just completed was quite different. It covered the exact same topics as my class - sampling, measures of central tendency, distribution, probability, estimations of means and proportions, hypothesis testing - but this professor’s teaching and testing style was very different from mine. He gave a lot of assignments, and was fond of breaking questions down into multiple parts that lead a student to the answer. I’m not entirely opposed to some amount of that - hell, I think that such questions are probably the best way, at least initially, to deal with students who freeze when confronted with a problem that they can’t solve immediately - but this prof’s multi-part questions were…ill-conceived, to say the least. In particular:

  • Every question on a certain topic followed exactly the same template. No two topics shared same template. My tutee quickly figured out that an eight-part question in which the first part asked for a sample proportion and the second asked for the claim of the population proportion, was a hypothesis test that called for use of formulas 8.3 and 8.4. She also figured out that you could plug the first, second, third, and fourth numbers given in the question, respectively, into 8.3; 8.4 used the result of 8.3, along with the fifth number given in the question.

  • Even if I had devoted my best efforts to the task, I could not have written questions more leading than this guy’s. For instance:
      …The distribution of student weights is unknown. 42 students are weighed…

      a) …

      b) …

      c) Which of the following applies:

    • We can use Formula 7.3 because the distribution of student weights is normal.
    • We can use Formula 7.3 because the sample size is at least 30.
    • We cannot use Formula 7.3 because the distribution is not normal and the sample size is less than 30.
  • The man was a certifiable jargon/notation fetishist. Tell me, how the hell else do you explain a question of the form “What is the sign (less than, greater than, or not equal to) that appears in the statement of the alternative hypothesis HA?” Or - and this is my bias talking, because I can never for the life of me remember which label goes with which - the query “would this be a Type I or a Type II error?” with no followup.

None of the questions had an “explain in plain English” portion. My tutee, whose term mark was among the highest in her class, could tell me that in Question #4 of Section 9, we reject the null hypothesis because x-bar fell in the critical region, but she could not tell me that what this meant was that the lightbulb manufacturer’s claim was bullshit. She solved the problems on her tests and assignments by pattern-matching on the rigid templates, and on following the leading questions. When I worked with her two nights before her exam, she stated matter-of-factly that she expected to forget everything from the course the next day.

If this what one of the top students in this statistics class has taken from the course, then I think it’s a pretty safe bet that this statistics class is not preparing students to assess and interpret quantitative data.

But I can’t hold this professor responsible, because he seems to be doing a good job under the circumstances: he’s got a jam-packed curriclum to follow, and is responsible for delivering a bevy of content at the expense of skills. Even though this prof he gives plenty of practice problems that prepare for the very predictable tests, and even though he gives excellent notes, and even though he is available for plenty of extra help…despite all that, my student - who has been sick half the term and who, by her own admission, has been slacking off lately - is one of the top students. I’m certainly not going to second-guess what I presume was a conclusion that his students could not handle a more rigorous course, one that aims to train students to assess and interpret non-canned quantitative data.

I can’t blame him for concluding that there’s no way he can deliver such a course successfully, so he might as well not have his students hate him by the end of the term. And if that means that there’s no guarantee that an A student will understand what it means for a poll to be accurate within three percentage points nineteen times out of twenty, then so be it.

And there’s a big problem with that. If there’s one math class in which the question “what’s the point of this?” should never ever come up, surely it is introductory statistics. But I can’t for the life of me see how anyone could justify teaching a statistics course like the one I tutored.

I wish I could design such a course, because having taught it once, I know exactly what I’d do differently if I were granted full control over the format. In two words: less content. Oddly, calling for less content in a math class tends to invite charges of “dumbing down”, and we can’t have that! - nevermind that the textbooks of yore contained vastly less content than the ones of today - but emphasized mastery and application.

Here’s what I’d trim out of a single-semester intro stats class:

  • Most of the probability section. I love probability - so much that I spent far too much time on it last term - but it’s easy to underestimate just how much difficulty students have with it. I’d get rid of everything that isn’t necessary for binomial probability applications, and leave those in only because of the normal approximation to them. (Height of stupidity: spending three weeks on permutations and combinations, and then glossing over connection between probabilty and statistics. Yes, I did that last year.)

  • The Student’s-t distribution. There’s more than enough you can do with normally-distributed sample sizes, and if we’re going to wave our hands over the Central Limit Theorem anyway, why confuse matters with the rule that samples of size thirty use Table A5 while samples of size 29 use Table A7? This time would be better spent elsewhere.
  • Though not on the “estimating the standard deviation” section. Estimating means is simpler and more relevant, and students still struggle with it.

The leftover time - and really, there isn’t much when you cover the rest of the course at a reasonable pace - can be used with hands-on activities, which are so natural for a statistics course. It can be used to have students design the sorts of questions that usually appear on tests: the data they encounter when they see the latest polls, or when they weigh precisely a bag of apples, provides suitable fodder for a variety of such problems. It can be used to discuss why one researcher would rather risk Type I errors, and another Type II errors. It can be used by emphasizing, over and over and over again, the implications of the material everywhere.

I don’t think that such a course would be at all dumbed-down from the one that my tutee took this year; to the contrary, it would require students to think far more deeply about the material. But such a course would be faithful to what I assume are the reasons for teaching introductory statistics. And if I were to teach it, I’d feel a lot more better about the answers I give to what’s the point of this stuff than I would if I were instead responsible for delivering the more content-heavy statistics class class that nearly every business and social science department requires its students to take.

11/24/2005

It’s not a conspiracy theory if it’s actually true

File under: Righteous Indignation, Those Who Can't. Posted by Moebius Stripper at 10:34 pm.

At the risk of engaging in the premature counting of chickens, it looks like I’m going to be involved in something off-blog that will expose my rants about the fucking graphing calculator to a wider audience. Like, wider by a few orders of magnitude. Exciting stuff. Exciting enough that I spent some time today researching the link between Texas Instruments and the math textbook industry that I wrote about a few months ago.

It’s worse than I thought. It’s scandalous, and everyone who has a stake in what students are taught should be outraged.

Here’s a tiny subset of what ten minutes of Googling got me:

  • An alliance between TI and textbook publisher Pearson Prentice Hall: Pearson Prentice Hall and Texas Instruments to Publish Educational Products for High School Math Market. Never mind the creepy abundance of business jargon: far creepier are the repeated references to “increas[ing] student achievement”, “improv[ing] student performance”, “scientifically researched and standards-based instruction materials”, and the like, all waved around without either specifics or support. Just because you say it, doesn’t mean it’s true. Show me the data, Pearson Prentice Hall and TI.
  • Here’s a beautiful example of technology making simple concepts complicated: …The directions for performing these operations differ from calculator to calculator. The steps for a TI-82 are given on page 663 of the textbooks. For other calculators, you will have to consult the manual for instructions. Learn how to use these important functions… Oh, allow me to present a bold alternative to that shit: graph your bloody STRAIGHT LINE by hand, you goddamned punk.
  • Fostering Children’s Mathematical Power: An Investigative Approach to K-8 Mathematics Instruction. Here’s Activity File 0.1 - ZERO. POINT. ONE - in a book about teaching math to children: It may surprise children to learn that some calculations are too hard for a calculator. Encourage them to explore the limits of their calculators for each of the operations. For example, what is the largest addend that can be added on a Texas Instruments (TI) Math Explorer? I’ve got a word for this approach as a zero point first step toward fostering children’s mathematical power, and it ain’t “investigative”. Also: free cookie to anyone who can tell me why the TI in particular is necessary here. The $10 doodad I use to balance my checkbook could do just as well for this, maybe better.
  • Probably the creepiest material of all comes from the TI site itself. Take this, for instance. What do MTV®, Sesame Street ®, the WalkMan®, the DiscMan®, the Game Boy®, and the Brave New World of Mathematics Education ® have in common? A whole hell of a lot, apparently.
  • More from TI. Just…read the title, which I think is more fitting than the good folks at TI realize. I mean, just tie a bow around a big fat red foam A+, and it probably means about as much as a real A+ in a TI-based math class.

And there’s more. Much, much more. The skills-lite, calculator-heavy approach to mathematics education, which produces top high school students that are completely unprepared to do college-level math, won’t last forever. I just hope I’ll be around to bury it.

10/20/2005

Show me the data.

File under: Righteous Indignation, Sound And Fury, Those Who Can't. Posted by Moebius Stripper at 10:56 pm.

What’s that you say? You’re sick of all those long-winded education rants that never go anywhere? Me too! However, I can’t pass up this opportunity to commend the California Commission on Teacher Credentialing for so succinctly summarizing everything that’s wrong with elementary-school-level mathematics education today. And in an exam that every prospective teacher in the state is required to write, to boot - now that’s efficient delivery!

Many people believe children will never learn mathematics if allowed to use pocket calculators. Having spent countless hours memorizing multiplication tables and doing long-division problems unaided by any mechanical device, many adults cannot conceive of anyone acquiring this knowledge without similar effort and practice. ______________. What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts. Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate. Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.

Which sentence, if inserted into the blank line, would best focus attention on the main idea of the passage?

(A) It is true that mathematics is not the easiest subject in the typical elementary school curriculum.

(B) Many of you have doubtless heard about the bitter classroom experiences of students who learned mathematics this way.

(C) There is much to be said for instilling this kind of discipline in students.

(D) Although it was clearly not fun, students trained in this manner rarely forgot what they had learned.

(E) Such views, however, seem to reflect a resistance to change rather than a rational approach to mathematics instruction.

(F) Contrast this instance of common sense with the hallucination that follows.

Just kidding! (F) isn’t an option. The correct answer is (E).

Yes, yes: fish in a barrel are targets for amateur marksmen, I know. But what we have here is a barrelful of tranquilized guppies that been given free reign over early mathematics education, so let’s have a go at it before any more metaphors get mixed:

What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts.

You know, I skewered this one months ago. I even used the word evolved, for crying out loud. My readers found this -

Yes, if there’s one field that has evolved beyond recognition in the past quarter-century, it’s introductory calculus. Why, back when I was a tot, the area of a rectangle was length plus width, the derivative of sin x was 5, and we only had whole numbers, so whenever we needed to compute the area of a circle we had to use pi=3, AND YOU NEVER HEARD US COMPLAIN.

- to be hysterically funny, because it was so over-the-top. Except that…it wasn’t. It was very much beneath the top. It was smack dab in the middle of what those enlightened educators, unlike “many people”, have succeeded in understanding.

Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate.

They don’t? Really? I’m not convinced. Let’s see what a cognitive scientist has to say about this!

It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary.

… By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

But what does that guy know, anyway? The times (tables), they are a-changin’! Mathematics is evolving! Get with the program!

I’m curious about something, though: is this anti-repetition view unique to mathematics education? Have the musicians among my readers, for instance, noticed a similar trend in music pedagogy? Music is EVOLVING! It is not a FIXED BODY OF FACTS! We have new-fangled technomology that enables students to bypass all that boring stuff, like learning scales! Or…do music students still practice scales, even though scales really aren’t that much fun to practice? I know that when I teach pottery, I spend a fair bit of time focusing on the basic skill of centering the lump of clay, even though it’s more rewarding to throw teapots.

Onward:

Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.

Joanne Jacobs speaks for me:

I’ve spent countless minutes (that seemed like hours) waiting for students I’ve tutored to multiply 4 times 5 or 3 times 6. Oddly enough, they weren’t adept at understanding mathematical concepts or solving problems.

The idea that students who don’t waste their time learning the boring basics will have freed up valuable time and brain space to become creative problem-solvers is one I’ve seen cited often. It’s an idea that every single mathematics educator with whom I have ever communicated - and I have communicated with hundreds, across several continents, both in person and through this blog - considers to be utter hogwash. But it’s possible that we’re just outliers. It’s possible that on the whole, students who don’t learn their times tables beyond mastery are brighter, bolder, and more creative problem solvers than their predecessors. And if that’s the case, then there should be plenty of statistics to bear that out. And surely the proponents of calculator-based curricula have plenty of rigorous statistical studies at their fingertips.

Let’s see them. Put up or shut up.

Show me the data.

Show me a single peer-reviewed study that indicates that students who were raised with calculators show greater facility and more creativity in problem-solving than those who were raised without. I want to see test scores. I want to see comparisons of performances in university-level mathematics classes between students who were made to memorize and practice their times tables, and those who weren’t. I want to see evidence that learning the basics and learning advanced mathematics are negatively correlated.

Until then, I’m going to rationally resist change, and teach mathematics in the way that my peers and I actually learned it.

9/28/2005

This is why my little college-math-ed blog has so many readers:

File under: Those Who Can't, Home And Native Land, I Read The News Today, Oh Boy. Posted by Moebius Stripper at 4:05 pm.

Because I am Everycollegeinstructor.

In U.S. report released this month, 40 per cent of professors who were surveyed said that most of the students they teach lack the basic skills for university-level work. Further, the survey conducted by the Higher Education Research Institute at the University of California at Los Angeles found that 56 per cent cited working with unprepared students as a source of stress.

Count me among both groups.

The rest of the article is essentially a quantitative, snark-free summary of everything I’ve written about teaching over the past year. Incoming university students, reports the Globe, are woefully unprepared for the demands of university; it’s hard to know how to deal with them; their high school grades mean nothing. Nothing new here, but I’m glad this issue is getting national attention.

Unfortunately, I think the remedies described in the article - more remedial classes! extra help for students who lack basic skills! diagnostic tests for students whose math marks are below 70% or whose English marks are below 80%! - are remarkably short-sighted, and contribute to the unfortunate trend of students paying universities to learn what they used to be able to learn in high school, for free. The main problem, as I see it, is an increasingly incoherent high school curriculum that is quickly diverging from the goals of a university education. And this problem won’t be solved until high school and university educators start talking to one another.

I’ve got a lot to say about this piece, and I’m finding that my thoughts are all over the place, so bear with me. Or don’t, I guess.

First, a quote from Ann Barrett, managing director of the University of Waterloo’s English language proficiency program, that dovetails with the experience of every single college calculus instructor who’s ever taught students who took calculus in high school:

“I have seen students present high school English grades in the 90s, who have not passed our simple English test.”

And the proposed reasons for this?

Some officials blame grade inflation at the high school level. Others say that in this primarily visual world, there’s little focus on the written word. And one professor points to the high school curriculum being so jam-packed with content that teachers have no time to instruct on the basic skills.

My thoughts on these, respectively, are: kind of, but that’s beside the point; give me a break; and ok, now we’re getting somewhere.

Let’s start with grade inflation, becuase it’s the most frequently-cited cause for students obtaining A’s in high school and flunking out of college. I stand by what I wrote on the subject back in January, but I think that the A-students-flunking issue is a lot more complicated than that. If grade inflation were the main culprit, then we could say that a student who gets an A has what may once have been considered C-level understanding of the material; that is, an A in 2005 is equivalent to a C in (say) 1995.

I disagree. My A-minus student does not have a C-level understanding of the grade twelve course that I took a decade ago, the one that prepared me reasonably well for my university math classes. He doesn’t even have a D-level understanding of such material. To say that an A-minus means anything in terms of a student’s understanding of the math they need to succeed in university is to say that there’s any correlation whatsoever between college level math and grade twelve math as it’s taught in BC. And there isn’t.

My student’s A-minus is a in fact pretty accurate reflection of his knowledge. My student does indeed have an A-minus grasp of the material taught in grade twelve math in BC. My student has acquired A-minus-level proficiency at storing formulas in his fucking graphing calculator and memorizing the solutions to homework problems so that he can recall them when he faces the test. He’s quite good at all that, really. It’s just that this proficiency would help him not one whit if he were to take a university-level math class, taught by professors who naïvely expect their A-minus students to be minimally numerate, not to mention vaguely proficient in reasoning mathematically.

Reducing this issue to grade inflation suggests that the problem lies in the evaluation of students, not in the choice or presentation of material. Absolute mastery of BC’s garbage grade 12 math curriculum doesn’t prepare students for university, because BC’s garbage grade 12 math curriculum is virtually disconnected from university. My colleagues and I have griped amongst ourselves about this, but as far as I can tell, there is no communication between high school curriculum developers and university educators. Tweaking grades won’t fix that.

On to the next idea - we live in a visual world, with little emphasis on the written word, so no wonder Johnny can’t read - am I missing something here? Did our world become significantly more visual in the last decade - a time during which universities have reported tremendous increase in unprepared students? The high school texts I’ve seen are jam-packed with the written word.

What I do see is this: I see students calling me over to their desks to ask about a word problem, and half the time me reading the word problem aloud to them is enough to answer their question. I see students skimming over paragraphs of text (not that I blame them) and then asking me what they really needed to read in order to solve the problem. I seldom see any indication that students are reading their textbooks beyond skimming over the examples so that they can match them to the homework questions. I’ve lost track of the number of students I’ve tutored, or fielded during office hours, who did not avail themselves of the indices of their textbooks. The reason they couldn’t show that two events were mutually exclusive was because they didn’t know what “mutually exclusive” meant, nor did they think to look it up.

When I was in high school, my English teachers routinely gave marks for producing drafts of essays. Producing the draft was worth half marks; the rest of our marks came from the quality of the actual essay. An incoherent essay could easily earn a B if the writer produced a draft. When I was in grade twelve, we had to submit one or more essays every week. There was plenty of emphasis on the written word; our ability to use it well, however, was virtually irrelevant.

Things have gotten worse in my home province, according to a former camper of mine. This camper was a brilliant math and science student; by his own account, he was “average” in English - and he wanted to improve. But he was having trouble doing so, because he was never assigned essays as homework. A few years earlier, he told me, teachers were reporting a rise of internet plagiarism. The school board’s solution: stop assigning essays for homework. In 2002, the only essay-writing experience that high school English students had, consisted of sitting in class for eighty minutes or so and producing an unedited, unresearched paper. It’s not hard to imagine a student who excels at writing those sorts of papers, flunking out of a class that requires long, researched papers.

There’s plenty of emphasis on the written word. There’s virtually none on developing the skills required to use it effectively.

Moving along - I am a lot more sympathetic to the third proposed explanation for the increase in unprepared university students: the emphasis on content over skills. Erin O’Connor, from whom I pilfered the original link, puts it well:

What [the article strongly implies] is that the problem stems in no small part from an ideology of progressive education that is famously hostile to skills acquisition (which requires such child-stifling practices as memorization, drill, repetition, and so on).

This certainly rings true in math, where I labour endlessly to disabuse my students of the notion that if only they memorized more formulas, more examples, they’d be doing a lot better in my class. The idea that there is a smallish set of basic skills that, solidly understood and correctly applied, will carry them through more difficult work, is alien to them. Pointing out that they can use material in Chapter n-k to solve a question in Chapter n risks an uprising. (True story: the precalculus 2 prof last year had a student in his office ask how to find a hyperbola’s asymptote. The prof reminded the student how to find equations of straight lines, and was met with a blank stare. “We did that last term,” she explained earnestly. “You didn’t show us how to do it this term.”) Last April, I talked to my then-department head to suggest completely reworking the curriculum for the terrible precalculus class. He was more than receptive, and took notes as I ranted. One idea that came up: teaching half the content, but taking time to make sure that students had a solid grasp on everything that was taught. It interests me, thought it doesn’t surprise me, that Erin and the English professor quoted in the article have come to similar conclusions about the courses with which they have experience: those courses too display an emphasis on content to the exclusion of skills that can be more broadly applied.

High school curricula are disjointed. We get a topic here, an application there - and we get nothing to tie them together. There’s no overarching theme for any course, no concept to unify the incredible mass of content. Students are understandably hard-pressed to recall any skills they learned in high school. And I can’t blame them for wondering, on occasion, “what’s the point of all this stuff?” I’m not even sure the people who designed their courses know.

At the end of the day, we’re left with two facts that are increasingly troubling, and increasingly at odds with one another:

1. High school students are discouraged from pursuing post-secondary options other than university; but

2. A high school education does not prepare one for university.

The first of these is seldom challenged among high school teachers and guidance counselors; the second is addressed at the university level alone. Unless high school and university educators start working together to figure out what they’re trying to accomplish, and how best to accomplish it, we’re still going to have unprepared students scrambling when they enter university, and we’re still going to have short-staffed universities rushing to endow them with the skills they should have acquired in high school. That’s not education; that’s damage control.

9/17/2005

What passes for success

File under: Sound And Fury, Those Who Can't, Queen of Sciences, Know Thyself. Posted by Moebius Stripper at 6:06 pm.

I realize that I don’t post many stories about my positive experiences with students. Part of it is because those experiences don’t require the sort of catharsis that the horror stories demand. Part of it is that my darkly comical style of writing doesn’t lend itself to expositions of, like, students actually learning stuff from me. And part of it is because sometimes, the success stories depress me even more than the failures.

I’ve been tutoring a high school kid for the past two months. The kid’s in grade 12; when I met him, he was doing math at a grade two or three level. This is not an exaggeration: he couldn’t add or multiply single-digit numbers without a calculator. And this wasn’t just rustiness, as this inability extended to not being able to compute things like 6+0, 5*1, or 3*0. In other words, he didn’t know what numbers were. Not surprisingly, he couldn’t solve linear equations, add fractions, or make heads or tails of the most simple word problem.

I met with him every other day, two hours at a time. And, to his credit, what he lacked in mathematical skill, he more than made up for in persistence. He worked diligently, if not terribly successfully, on his homework. We spent a lot of time on the basics - fractions, simple algebra, the meaning of equations. We also spent a lot of time - far, far more than I’d have liked - on how to use the fucking graphing calculator to perform tasks that every student should know how to do with a pencil and paper.

He’s two thirds of the way through the course now, and he’s pulling an A-minus.

And he’s learned a lot in this class, and I’m glad that his effort will almost certainly earn him the C he needs by the end of October. But little of what he knows falls into the category of “mathematics”; the bulk falls under the umbrella of “tricks that will get a student through a grade 12 math class in BC.” He’s vaguely familiar with basic algebra now, but he still falters when simplifying an expression (”Am I allowed to cancel out the x’s in (x+5)/x?”). He cannot immediately identify which methods to use in solving an equation - he’ll use the quadratic formula if someone tells him he’s dealing with a quadratic equation, but he needs that push. Meanwhile, he’s a whiz at using his graphing calculator: he’s mastered all of the fancy features, and can program all of the rules into it. (He’s managed to fit, in the calculator’s memory, examples of every type of question he might see, along with solutions.) After each test, he purges both his calculator’s memory and his own of everything we’d studied in the previous chapter.

Two weeks ago we wrapped up the trig unit, which vexed him even more than the previous four chapters had. Following some introspection, he was able to identify the source of his frustration: why, he wondered, did those bastard curriculum designers make him do this shit? Didn’t they know that the TI-83+ could neither do proofs, nor provide exact values of sin (pi/3) and the like? During one particularly trying session in which I was explaining, for the third time, how to find those values, he declared that he wasn’t going to learn “that triangle shit”, and I found myself, for the first time ever, raising my voice with a student. He relented. He got an A on the trig test. The next week, reminiscing, he asked, “trig - we did that already, right? Was that the stuff with the logs?”

Last week we started on the combinatorics chapter, and I braced myself for the damage. Combinatorics, unlike most of the rest of the course, requires some creativity: formulas are few, and variations on themes are many. Every question must be read carefully, and even the simplest ones require students to think about how to set them up. However, my student took to this section surprisingly well, and found it to be a lot less stressful than the others. He even asked me some questions that were related to the material, and not just to how to pass the next test.

The last section of the combinatorics unit covered the Binomial Theorem, which provides a shortcut for expanding expressions of the form (a+b)^n. I always thought that this material was the easiest part of the combinatorics chapter: it’s completely algorithmic, and demands no creativity.

It does, however, demand a knowledge of basic algebra.

“Here’s what we’re going to do in this section,” I explained. “We’re going to find a shortcut for expanding things like (a+b)^n - n is a whole number.”

“Which whole number?”

Sigh. We’ve been through this, many times. He always gets tripped up by questions that require him to generalize anything, even slightly. Has to do with skimming over the directions, and failing to understand that variables…can vary. Last month’s lesson didn’t sink in, apparently.

I explained it again. “N can be any whole number,” I said. “We’re going to introduce a formula that will let us expand (a+b)^1, (a+b)^2, (a+b)^3…and (a+b)^n for any whole number n.”

And away we went. We had worked out the first several lines of Pascal’s Triangle on the previous page, and I made sure that that was accessible as I had him expand the powers of binomials the long way.

He started having trouble with (a+b)^0. “Zero?” he asked. Even though he’d raised things to powers of zero in the chapter on geometric series. And the one on exponents. And a few others.

“What’s anything raised to the power of zero?”

He reached for his calculator and tried a few values. Frustrating, but at least he knew that he could figure out the answer by trying some values. He hadn’t known that in July. “One,” he declared.

The expansion of (a+b)^1 proceeded without incident. He was just as quick to give an answer for the next one: “(a+b)^2 = a^2+b^2,” he said.

We’d gone over this one a dozen times in the previous month. He’d made that mistake a dozen times. Each time we went over the basic rules of algebra, and tried plugging in numbers for a and b that showed that (a+b)^2 doesn’t always equal the same thing as a^2+b^2. But that was last month. He was supposed to know this still?

And it wasn’t over. You mean ab and ba are the same? So we can collect them? And we can’t collect ab and a^2? Why not?

Anyway, enough. My point is: in the BC high school math metric, this is A-minus work. A good graphing calculator and a mediocre short-term memory are sufficient to achieve excellence in high school math classes; most students will take the path of least resistence and develop little else. Grades of C or higher are required for the college program this kid’s trying to get into. Grades of D in this course are sufficient to gain entry into the precalculus class I taught at Island U last year. If a failure to grasp basic algebra doesn’t stand in the way of getting an A-minus, then what on earth does one have to know in order to get a C? Or a D? Why not just eliminate the middle man and admit anyone into a college math class, rather than wasting everyone’s time on this extended TI-83+ how-to seminar? This sort of thing is part of why I usually have little sympathy for students who claim that, no, really, they do know the material, it’s just that they do badly on tests. In my experience, it’s more frequently the opposite: they do far better on tests than their actual understanding of the material reflects. And then they sink like stones when they take a college class with me, in which I design the tests and don’t allow them to use their graphing calculators.

Well, not all of them sink. I had a student last year, who earned a C on the first test and then quickly adapted to my expectations. He had the aptitude to do so; it’s just that he’d never been required to use it to full capacity. He pulled an A-minus on the final exam, and wrote me a note at the end. I transcribed it, and I go back and read it every now and again when I’m particularly frustrated:

I hope I did as well on this exam as I think I did. I know it took me awhile to get going this term, but I worked really hard and I finally feel like I get this stuff. I always used to do well in math class in high school, but you actually made me think. Even though I got good marks in math class before, I feel like this is the first time I really learned math.

Thank you.

I guess that’s success. I expect a thank-you from my current tutee, who certainly would not have achieved the required C without me, and who will almost certainly get or surpass it next month. But I doubt I’ll feel all that good about it.

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