Math problems of the week: Traditional vs. IMP trigonometry

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.405-406 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 17-18 [click to enlarge]:

Modern bullies

Re my recent post on differentiated instruction, a reader named Ed posted a compelling response:

Teaching their peers? Are they nuts!

The bullying I received when my teacher tried that because they couldn't find things for me to do.

Plus kids "love" being taught by the geek in the room.

I’ve been thinking a lot about bullying lately because it has become such a ubiquitous topic. Are kids really more mean spirited than they were a generation ago? Or do they simply have more opportunities to be mean?

Thanks to smartphones and “social” media like Facebook and Twitter one can bully anyone anytime and anywhere, even those who do all they can to avoid their tormentors, disseminating insults and gossip to a limitless virtual audience of gawking peers. Particularly lethal--literally so--are doctored images and fake Facebook pages.

A far less appreciated source of growing opportunity, as I’ve argued earlier, is the growing time that students spend working in groups at school. Ironically, some bullying experts view cooperative group work as a way to remedy bullying, but given that bullying (especially when it takes the form of teasing and shunning) can be subtle and difficult to detect, and given that teachers cannot supervise multiple groups simultaneously, group activities often have the opposite effect. Particularly problematic, as Ed suggests, are the increasingly fashionable mixed-ability groups.

Today’s anti-bullying measures can backfire in other ways. Socially savvy kids can take advantage of zero tolerance policies and subtly goad a more socially clueless peer into lashing out. The victim rather than the perpetrator is then the one who gets punished. In whole class discussions in which children are supposed to share their experiences with bullying, the victims may be too uncomfortable to do so, especially if those experiences involved subtle, difficult-to-articulate forms of bullying like shunning, and especially if the vicctims expect subtle reprisals from peers once the adults are out of earshot.

It’s the socially aloof kids, of course, who are most often the true victims. Has the rise in autistic spectrum disorders caused bullying to increase? Perhaps, but a rise in the numbers of organically socially savvy kids may be just as responsible. One of the more compelling reasons for the rise in Asperger’s Syndrome, in particular, is the “assortative mating theory.” This holds that, where today’s mobile, densely interconnected humans are concerned, birds of a feather increasingly mate together. That is, instead of marrying the boy down the street, you’re more likely to marry the boy in your chosen field who shares, say, your analytical interests and distaste for small talk; or your social charms and love of parties. One result, commonly cited by such autism researchers as Simon Baron-Cohen, may be a concentration of autism-related genes. The much less commonly cited flipside is a concentration of sociability-related genes. The likely result? A widening spread of children across the spectrum from socially-aloof to socially-savvy. More victims, yes, but also (among the more mean-spirited of the highly socially-savvy), more sophisticated and powerful perpetrators.

Do we really need high school biology?

I'm still recovering from the week I spent going through all of 9th grade biology with my son to help him study for his final. I was going to write a post complaining about high school biology, but my ideas seemed slighly familiar, so I looked back and realized that I'd written a similar post three years ago on 7th grade biology.

Everything I wrote there applies here, except that this time around there was an even higher ratio of memorization to conceptual understanding

So now I ask: is there any virtue to teaching biology at all in high school? What would be lost if we stuck with chemistry and physics, and prepared a stronger conceptual foundation for college biology?

Or is biology at any level mostly about memorizing things on faith rather than understanding them conceptually? (I was so turned off to high school biology that I never took biology again and therefore don't know the answer to this question.)

Don't just assist, instruct

One aspect of the classroom tech craze is the growth of Augmentative and Alternative Communication  devices (AACs) in special ed classrooms. This raises concerns about whether today's today's schools are teaching language impaired children new linguistic skills as opposed to merely assisting their deployment of existing skills.

At the extreme, there are text-to-speech apps (as well as books on tape) that help make texts accessible to those who can't read, and speech-to-text apps that enable those who can't write to convert spoken words into text. These are especially appropriate for students whose disabilities may prevent them from ever reading or creating text fluently. But for those who do have the potential to read and write, among them many children on the autistic spectrum, such device should never become the be-all and end-all of their reading and writing activities.

Potential aside, teaching AS children can be quite challenging. This creates a strong temptation to turn instead to AACs.

In autism, the most commonly used AACs aren't for reading and writing, but for language--devices like the DynaVox, which gives users a menu of common vocabulary and phrases to select in order to communicate basic needs. The devices can revolutionize a child's basic functioning and psychological well-being, improving substantially his or her classroom behavior and teachability.

But no one should see AACs as a panacea for language instruction. However much they assist children in deploying their current language skills, it's far from clear that they actually teach them new ones.

DynaVoxs and the like may also give teachers (and parents) the illusion that the child is operating at a higher linguistic level than he or she actually is. This is because, across many domains of language use, non-linguists tend to confuse simple recognition of "key words" and set phrases with true sentence comprehension. For example, when the dog rushes to the door when asked "Do you want to go on a walk?", people tend to assume that the dog understands the question as such, when it's more likely to be simply the key word "walk,"or the set phrase "Do you want," that's cluing it in. Relatedly, many  software programs--from foreign language programs like Rosetta Stone and Pimsleur to remediation programs like Laureate Learning--often end up teaching key word recognition rather than syntactic processing.

So, when a child pushes "JUICE" on his or her DynaVox, many may see this as the child's intended shorthand for a full-fledged grammatical sentence--"I want juice"--or question--"Can I have juice?"--when he or she hasn't actually acquired general subject-verb-object order or the syntax of question inversion. Clicking on preset key words and phrases rather than constructing one's own phrases and sentences from scratch may mean that one has learned simple associations between stimuli and sounds, but not the linguistic skills prerequisite for intentional linguistic communication and thinking in full-fledged propositions.

The ease of AACs, and how much the help teachers manage a classroom full of AS students, may sometimes breed complacency about the students' continuing needs for direct instruction in language. Managing AS kids is one thing, but, when it comes to meeting their legally-mandated educational needs, it is only the first step.

Math problems of the week: traditional vs. IMP trigonometry

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.403-404 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 15-16 [click to enlarge]:

How far can you get on key words alone?

Recently someone queried me about the possibility of writing a blog post on the subject of education for veterans. She felt that, given the focus of my blog, her post would fit in perfectly. I couldn’t figure out what she was talking about, but when she sent me a follow-up reminder, I clicked on the link to the post she had said was particularly inspiring, and discovered that it was one in which I had made reference to a “veteran education reporter.” Aha. The aspiring writer had done a keyword blog search on and was emailing every blog owner who’d appeared to use the term "veteran[s'] education."

While this is an extreme example of focusing on key words or phrases at the expense of whole sentences, I’ve wondered how many people still read sentences carefully--especially ones that exceed the length of your typical text message. Partly the issue is attention. One thing I learned about ADHD while designing my course on language and reading disabilities is that it has pervasive effects on reading comprehension--largely because reading requires sustained attention. Intermittent attention means taking in words and phrases, but not whole sentences in which these words and phrases may interact in complex ways. While the true incidence of ADHD is hard to know, the incidence of distracted reading is higher than ever. As writers compensate by writing simpler sentences (either on their own initiative or at the request of textbook editors), the cycle continues.

As long as sentences don’t involve modifiers, qualifiers, and parentheticals that interact in complex ways (e.g., in which a modifier negates or places conditions on the application of a phrase somewhere else in the sentence), you can get pretty far with a piecemeal focus on phrases. But some thoughts are too complex to be captured in sentences that avoid this kind of attention-demanding complexity. It’s alarming to think that sentences that express such thoughts are no longer accessible to many readers. Even more alarming is the possibility that people are too distracted to even think them on their own.

Father's Day

In honor of fathers and their longevity, I thought I'd link to a couple of recent findings.

First, growing evidence suggests a positive correlation between father's age and child's autism.

Second, new evidence suggests a positive correlation between father's age and child's longevity.

Of course, we shouldn't see this as a black and white tradeoff between autism and longevity. In autism, there are many shades of grey, and, as Temple Grandin reminds us, a world without autism spectrum people would be a world of technically backwards babblers potentially free of scientists.

Of course, keeping our supply of scientists flowing requires the persistence not only of certain types of brains, but of certain types of classrooms. Regarding this, I have a proposal out on Slate that could use your vote!

Math problems of the week: traditional vs. IMP trigonometry

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.401-402 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 12-14 [click to enlarge]:

A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form

One of the mathematicians most often cited when Reform math advocates critique traditional math is Paul Lockhart’s A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. His book opens with an allegory about a musician, who awakens from a nightmare in which the “curious black dots and lines” that “must constitute the ‘language of music’” become the center piece of what has become a universally mandated music curriculum:

It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.

In Lockhart’s nightmare, the mission of the primary and secondary schools is:

To train students to use this language-- to jiggle symbols around according to a fixed set of rules: "Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."

As for the higher grades:

The pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. "It's a lot for them to learn, but later in college when they finally get to hear all this stuff, they'll really appreciate all the work they did in high school."

But luckily:

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. "Of course!" he reassures himself, "No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!"

Lockhart stacks the deck just a tad by imagining that no one except the music professionals actually get to hear what music sounds like. Transposed to mathematics, this would be the equivalent of not telling students that the symbol 10 stands for the number of fingers on their two hands, or that the symbol + stands for the process of combining two different quantities into a single quantity. In this respect, Lockhart’s nightmare is completely irrelevant.

But in nearly all other respects Lockhart's nightmare was my reality. During my 12-odd years of piano lessons, I had to:

*practice meaningless drills until I learned how to read those curious black dots and lines to the point of mindless automaticity

*practice meaningless scales to the oppressive tick of the metronome

*practice even more meaningless and repetitive finger exercises to the point of mindless automaticity and a punishing buildup of finger muscles.

Surely true musical artists like Vladimir Horowitz and Misha Dichter weren't forced through such mindless, trivial, time-consuming, soul-crushing tedium. Surely, from the first day they lay finger tips to keyboard, they grasped immediately the beautiful patterns in the lines and dots before them, and, as their souls communed with Scarlatti and Chopin, the music coursed seemlessly out of the tips of their fingers.

Just imagine how many other people might have discovered the joys of piano playing if they, too, had enjoyed the same freedoms as the professionals must!

Separating the M from STEM

My older son is heading off to college soon, hoping to pursue a major in chemical engineering. It seems a potentially promising field for him, not just because of its generally favorable job prospects, but because chemistry and math were his best subjects in high school. My biggest concern is whether he’ll make it through one of the prerequisites: organic chemistry. I’ve heard from multiple sources--including here on this blog--about how the premeds that predominate ruin this course for everyone else. Their collective eyes focused on the MCATS rather than on intellectual enlightenment, they turn it into a cutthroat memorization fest in which earning a decent grade means sacrificing conceptual understanding for cramming. This is a sacrifice my son may not be prepared to make.

How many potential chemical engineers, I wonder, abandon the field because of premeds?

People are trying in all sorts of ways--many of them misguided--to encourage kids to go into STEM fields. Overlooked in all the talk of science and technology festivals and webstreaming of scientists doing field work in exotic places and testimonials from celebrity STEM professionals about how great STEM is are two things. One, of course, is the need to improve basic math and foundational science instruction in our K12 schools. The other is a straight-forward pragmatic strategy for colleges--one which some colleges are actually quietly considering but which I’ve never heard anyone publicize. Separate the E and S kids from the M kids, and, thus, the cram-fest science courses from the conceptual ones.

Autism diaries XXXIV: But is he a sociopath, II

He doesn't torture little animals and cut them into bits, as bonafide sociapaths are said to do. Nor has he done even once what some 'normal' little boys I know have done regularly: burn bugs with a magnifying glass.

But he doesn't hesitate to throw a mat over an errant mouse and squash it. Nor, when we return from a trip and discover that our cat has left a dead mouse "welcome home gift" on our basement floor (as we did the other day), do we hesitate to ask him to pick it up and flush it down the toilet (J being the only of us human residents who doesn't flinch at the prospect).

And then there are some suggestive things he's said recently. A few weeks ago: "Earlier, I thought it was OK to kill people, until you told me that you can go to jail for that." This echoes a disclosure I blogged about earlier: "Sometimes when I want to do something bad part of my brain says 'no' because it knows I will get in trouble."

Being bad makes you feel sad. This is the refrain I repeat whenever he shares with me a mischievous "what if I" scenario. And this little rhyme, evoking the feeling you get when your mother withholds such privileges as answering your questions about ceiling fans, has become intrinsically aversive. But can we take this a step further? Can a moral compass be acquired through operant conditioning?

However sad being bad makes you, it's hard to suppress the temptation--a temptation that, just a few days ago, he put into words for the very first time:

"Sometimes I do bad things because I like to get a reaction."

"But why don't you like getting a good reaction?"

[Short pause.] 

"Getting a good reaction is harder than getting a bad reaction."

Unfortunately, he's probably right. When he does something wrong, he can rely on us to take predictable measures to undo it; for positive behaviors like putting things away (as opposed to throwing them around) or fixing them (as opposed to breaking them) there's nothing to undo. Furthermore, people are quicker to notice things when they're out of place or messed up than when they're restored to their proper circumstances.

Beyond this, how often are our displays of delight and gratitude anywhere near as as salient (in the rapidity of our response, the volume of our voice, and the explicitness of our facial expressions and body language) as our displays of fear, anger, or dismay? And how often is eliciting someone's pleasure as obvious, simple, and straight forward as eliciting their disappointment?

Math problems of the week: traditional vs. IMP trigonometry

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.399-400 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 10-11 [click to enlarge]:

Homeschooling update

It’s nearing the end of the school year and high time for another homeschooling update. In the course of the year, my daughter has finished the Lord of the Rings trilogy, the Little House books, the Narnia books, the Jungle Book I and II, The Island of the Blue Dolphins, The Secret Garden, The Little Princess, and the 6th Harry Potter book (all but the last of which were part of my own childhood cannon). Just recently she’s finished Classic Myths, Story of the World Volume II, and Singapore Math 6B, and has moved on to the Arabian Nights, Story of the World III, and Singapore Math’s Challenging Word Problems for 6th grade (which I purchased several years ago along with the 3rd -5th grade Challenging Word Problems when I heard they were mysteriously going out of print).

Besides this, she’s summarized readings, constructed and diagrammed sentences, cycled weekly through self-made vocabulary flash cards (derived from unfamiliar words in her readings), mapped the world’s continents, rivers, mountain ranges and empires, read about molds, fungi, and insects, and watched David Attenborough on the oceans, mammals, birds, bugs, and dinosaurs. She’s continuing to work through a traditional French ALM grammar book and to follow Mireille and Robert in the very non-traditional French in Action (Pierre Capretz is decidedly not a fan of ALM).

For extracurriculars and social activities there’s music (piano, organ, violin, and trio practice), gym (consisting of several weekly 1-3 mile bike rides back and forth to music lessons, roller skating, ping pong, and hiking), art (Draw 50 Animals; pottery at the neighborhood art school; independent projects like cartooning and dioramas, which she loves doing when they aren’t assigned to her by other people), creative writing (at the neighborhood art school), girl scouts, and formal and informal play dates.

We certainly have our moments of frustration and getting on each other’s nerves. But this continues to be a huge improvement over regular school, and I continue to feel very lucky to have a flexible enough schedule. I probably spend about the same amount of time working with her now that I did back when she had Math Investigations assignments and monstrous projects that she lacked the motivation and organizational skills to do (most recently her classmates had to write 30 poems, 5 in each of 6 genres--the PSSA state test writers being obsessed with genres), but instead of prompting her step by step through uninspiring, low-ratio-of-effort-to-learning tasks, I’m relearning history and mythology, engaging with challenging word problems (which, as they get harder, are starting to feel like the kind of mental workout that could provide the same long-term cognitive benefits as crossword puzzles), and brushing up on my French.

I was at a party this weekend talking with someone who specializes in the theme of abandonment. His focus is on the trauma of abandonment and how it informs religion, so I mentioned the various babies left in baskets in rivers or raised by wolves who became future leaders, and all those heroes of children's literature whose adventures depend on the absence of parents.

"How do you know all this stuff?" he asked me.

"I'm home schooling my daughter," I replied.

21st century skills: look not to the future, but to the past

If one listens to 21st century employers rather than to 21st century education experts, the biggest skills deficits in this high-tech century of ours aren't in "collaborative skills," "meta-cognitive skills" and "creativity" (especially as defined by education experts), but, more than ever before, in technical skills (especially those relating to engineering, programming, and basic numeracy).

Why are there deficits in technical skills? People tend to assume it's because schools haven't kept up with technology. The solution? More computers in the classrooms, more Internet access, more Smart Boards--in short, more technology. But the real problem is that, thanks to a combination of Reform Math "math" reform, and declining math proficiency among elementary school teachers, fewer and fewer students master the foundational math skills through Algebra II (at the bare minimum) that technical skills require.

When people talk about the need for schools to adjust to the 21st century, they tend to assume that schools need to be more forward thinking than they currently are. I haven't heard a single high-profile "expert" state that schools should instead should be looking backwards. But, for all the ways in which the 21st century is different from all preceding ones, the foundational math and programming skills needed for 21st century technology haven't changed. The frontiers of math are, of course, always advancing, but not its foundations. The same is true of computer programming.

For programming skills, schools should look backwards to the 1980s when electives and after-school clubs included basic programming and computers were better suited to such programming. (For basic programming skills, schools are arguably better off using old TRS 80s than 21st century PCs). For math skills, schools should look backwards even further, to the 1950s, the last time most American math textbooks covered foundational math in sufficient depth and rigor and taught basic skills to mastery--and looked anything like the kind of math text still used by most developed countries around the world whose students are increasingly outperforming ours not just in math, but in all those technical fields that depend on math.

Ironically, so-called "21st century skills" curricula, like that which Pearson Publishing has foisted on the Montgomery County, Maryland, Public Schools, water down technical skills in favor of the kind of "collaborative skills," "meta-cognitive skills" and "creativity" that no employer is asking schools to teach.  Montgomery County's implementation of Pearson's curriculum appears to go even further, supplanting the earlier Math Pathways program that allowed capable students to attend above-grade-level math classes.

In practice, more lap tops, more ipads, more Internet access, and more Smart Boards also water down the technical aspects of the curriculum, distracting students and teachers away from teaching and learning math skills to mastery, and from rigorous, focused mathematical and computational problem solving. Technology in the classroom may help create a generation of 21st century consumers, but not of gainfully employed 21st century producers.

Math problems of the week: traditional vs. IMP trigonometry

The third in a series of posts comparing the introduction of trigonometry in traditional high school math vs. the Reform Math program Interactive Math Program.

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.397-398 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 8-9 [click to enlarge]: