Does your student know 10/9 is a fraction?

American elementary and middle school mathematics programs are poor in myriad ways. They lack breadth and depth. They lack reasoning. They lack precision. An object lesson in all of the above is the common use of analogy to teach math, even in grades 5-8.

 Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analogous connection anyway, even if they can abstract away from the literal -- something most children cannot do. And if you are are ELL, it is probably entirely worthless. Plus, the analogy is by definition imprecise, so its correctness will break down with even the slightest scrutiny.

 You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "a fraction is part of a whole".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:

•  1. "there's no such thing as ten ninths." that's the majority response in classroom after classroom. Why? Because a fraction is PART of a whole. How can a part of a whole be bigger than the whole? What's the whole then?  
• 1b. therefore, they believe no fraction can be bigger than 1.
• 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6. 
• 3. "three thirds is A Whole." Not one. 
•  3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?" 

 Additionally they don't know decimals are fractions. how could 1.2 be a fraction? Twelve tenths isn't a fraction, remember?

 These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.

 The correct explanation is that a fraction is a number. What number? A number defined on the number line as follows:

1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.

4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.

 Yes, teachers will need to build up to this. They should do so.

Terri W is of two minds

re: Fill-in-the-blank has a really bad idea

I'm of two minds whenever I hear of particularly dumb ideas being floated in the school systems.

On the one hand, I'm thinking, "Cool, this gives my kids a leg up over the competition."

Then on the other hand, I realize that the overwhelming vast majority of the next generation of citizens are being educated with these cockamamie ideas and I think: "We are so hosed."

I'm on both sides of that fence with just one (typical) kid!

C. is now an official Math Victim of U.S. schools (including his Catholic high school, sad to say). And, at the same time, he's way out in front in the verbal realm compared to most h.s. seniors in America (also thanks to his Catholic high school, as well as to family background).

Speaking of 'official,' C. has just now reached the point of maturity at which he realizes, without being told by his mother: Holy ****, I don't know any math. 

He has two weeks left in high school. He came home the other day, said he'd had a talk with his math teacher (statistics!), who had convinced him he needed to take math in college. So he was wanting to know whether NYU has remedial math courses.

K-12 kids don't know, while they're K-12 kids, that they're going to be sorry they didn't learn math or writing or science or whatever else they didn't learn. I remember Glen, I think it was, once saying that we have to advocate for our children's future selves: for the selves they're going to be, eventually.

That's exactly right.

The Daily has a really bad idea

I love The Daily. Love, love, love. Read it every day.

That said, today's issue has an op-ed on the high cost of college that proposes, as a solution:

"Why not reduce undergraduate education to three years, while tacking an extra year onto high school?"^*
Josh Barro

Good God Almighty, as my father would have said.

Five years of public high school?

FIVE YEARS of Hunger Games in English and Core-Plus in math?

And here I was thinking everyone should skip senior year and go re-take algebra and English at their local community college instead.

(Is high school cheaper than community college? High school teachers here earn a great deal more than instructors in community colleges, but I don't know whether that's the case elsewhere.)

and see:
the founder, chairman, and CEO of Netflix has a really bad idea
Larry Summers has a really bad idea
David Brooks has a really bad idea
David Brooks has a really bad idea, part 2
All is forgiven.

^* That's the pull. Full passage: Why not adopt the Quebec model and reduce undergraduate education to three years, while tacking an extra year onto high school? 

Core-Plus Students at Michigan State

In summary, our data show a clear decline in the level of Michigan State University mathematics courses taken by Core-Plus graduates. The existence of that decline is statistically significant at any reasonable level. The decline in course level is accompanied by a decline in average grades for all but the very top students, as well as a decline in the percentages of those who eventually passed a technical calculus course. These trends occur in data that include students from a variety of communities. The data from individual high schools show that the timing of these declines corresponds precisely to the implementation of the Core-Plus program.
A Study of Core-Plus Students Attending Michigan State UniversityRichard O. Hill and Thomas H. Parker Thomas Parker

a cohesive paragraph about cohesive paragraphs

I spend a lot of time thinking about cohesion in writing.

student work

pretty good!

stop the multiverse

"We provide classroom teachers with lessons that allow them to teach standards-based math using topics students care about...Instead of teaching fractions and percent, teachers get to teach "Is The Wheel of Fortune Rigged?"
Mathalicious

I left a comment at eduwonk.

and see: stop the multiverse, I want to get off

talking isn't writing, part 3

The typical grammar of conversation is radically different from the typical grammar of informational writing.

Douglas Biber and Susan Conrad

Katharine on awkward student sentences

Here is Katharine on students using "it" as the subject of their sentences:

In terms of awkward sentences written by students, what I'm seeing is an avoidance of modified nouns as subjects. Instead, the would-be modified-noun subject is "factored out" of the sentence into a modifier, and then replaced by "it":

In Happe’s article it is said that this deficit is due to an autistic children’s inability to infer a communicator’s intentions.

[As opposed to Happe's article says that... Notice, btw, that the final noun phrase, the object of "due to", is heavily modified]

Or:

By discovering which parts of communication are more challenging to develop, it can help speech researchers discover where people with other language and communication challenges stumble as well.

[Instead of: Discovering which parts of communication are more challenging can help...]

Actually, only the first example ("Happe's article") is a modified noun; the second one is a sentential subject ("Discovering which parts of communication are more challenging"). So more precisely what I'm seeing is an avoidance of any syntactically complex element in subject position.

Perhaps this goes for speech as well?

in the predicate: an autistic children’s inability to infer a communicator’s intentions

in the subject: it

Interesting.

Happy Passive Voice Day!

at Shaun's Blog

Geoff Nunberg on Our Friend the Passive Voice

Facebook Issues

Do any colleges ask for full Facebook access from applicants? I've heard this is starting to happen at some job interviews. In general, I would like feedback about how people set up accounts with their kids and what kinds of limitations are imposed. I find Facebook somewhat creepy in how I'm sent potential friends of friends as if everyone needs to know everything about everyone. I really don't want friend requests being sent to my sister's complete mailing list. I also suppose it's a good idea to set up an account using a new or separate email - one that isn't called "partyguy6972". I know that one can control access to information, but is there a model that seems to work? I assume that you can group information and you can group friends. Ideally, you would would want each piece of information in a separate group, and you would want each friend to have access to only specific groups. Unfortunately, you would have to tediously set this up for each new friend. Even if you have larger groups of information, you might not want Aunt Sarah to see everything that's in the "relatives" group. I've generally ignored Facebook, but now I would like some real world feedback - not so much the extreme stalker/bully sorts of examples, but the more subtle issues of privacy and access. My son's account might be proper, but his friends' accounts might not.

talking isn't writing, part 2

John McWhorter gives the timeline.

I think he may be wrong (or perhaps I mean misleading) about the simplicity of spoken English, however:

Thus spoken language is fundamental, while written language is an artifice. Not surprisingly, then, the earliest writing was based on the way people talk, and that meant short sentences with a direct logical throughline. Researchers have found that even educated people today speak in word packets of 7 to 10 words a pop.

Talking With Your Fingers

I'm not so sure about that "direct logic throughline" concept, I must say. As far as I can tell, the Longman Grammar corpus study found that conversational English is more grammatically complex than linguists have assumed, which may (or may not) mean that the logic of spoken English is less direct than the simple Subject-Verb-Object ordering we imagine is typical of speech. And it strikes me that transcripts of spoken English often show a certain meandering quality.

But I don't know.

[update 4/27/2012: As I think about it, I realize I have no idea whether transcripts do or do not show meandering...]

RELATED: The single most fascinating article I've read on the question of novice versus professional writing is Bill Robinson's Rhetorical and Grammatical Dependency in Adverb Clauses, which appeared in a 1995 edition Syntax in the Schools.

Robinson summarizes Kellogg Hunt's study comparing K-12 students to professional writers. Surprisingly, Hunt found that professional writers did not use more subordinate clauses than novice writers:

In short, the high school seniors were using coordination and subordination at almost the same rate as professional writers of superior ability.

The major difference between professionals and students was that professionals wrote much longer sentences, 40% longer to be exact. And what made the sentences of professionals longer wasn't the presence of more clauses per sentence, but the presence of longer clauses.

It seems that professionals do a great deal of "noun modification."

Which, upon reflection, I'm thinking is right. At the moment, if I had to say what I do that a student writer does not do, I would go with: noun modification and plenty of it!

EXCEPT: I'm not so sure that's true of blog writing.

How much noun modification is going on in this post, for instance?

Not too much. Assuming I know what noun modification actually is, of course, which I may not.

I probably need a 1200-page corpus study to nail this down.

update 4/27/2012: Actually, there's a lot of noun modification going on in a subject as long as this one: "The single most fascinating article I've read on the question of novice versus professional writing..."

palisadesk on full inclusion and funding

re: Do general ed parents really want 'relief' from special education mandates?

palisadesk writes:

When they went for a full inclusion policy here, the senior bureaucrats were quite frank (at least in-house) that it was for financial reasons. While providing appropriate support to kids with exceptionalities in general ed classrooms is in fact very costly, providing pseudo-support and platitudes is relatively cheap.

The move from the top was, perhaps synchronistically, aligned with a lot of pressure from parents and groups representing children with disabilities for full inclusion. I can remember a time when many parents fought to get their kids into special classes for LD, language impairment, Total Communication (for deaf kids) and so forth. Those programs had a record of long-term success in getting students caught up and capable of doing challenging academic work.

Fast forward to now.....the district stopped investing in PD for special ed teachers, most of whom nowadays have learned nothing at all about effective practices, specialized teaching curricula or methods (Orton-Gillingham, DI, Lindamood-Bell, etc. etc.) so results in special classes are no better than in general ed. No surprise there. If you aren't doing something different, why expect a different result?

OTOH, we still do have special programs for very violent kids, for kids with severe or multiple disabilities, and increasingly, for autism. To comply with the laws around least restrictive environment *some* special ed classes remain, even for LD or slow learners, but are hard to get into. That means many children who really need a segregated program may spend a number of years floundering in the full inclusion environment before any other opportunity presents itself.

At the elementary level, teachers are used to a range of development and ability -- but there are limits. If one classroom has several extremely disruptive students, or students 4-7 YEARS below the class norm, it makes effective teaching of the whole class problematic. Much teacher time is diverted to preparing individual lessons and materials for the outlier students (there is NEVER a budget for special materials for them), and these students need much more teacher attention -- which is taken away from other students. Aides also have been cut back so that often they are shared between a number of classrooms.

There are some positive effects of inclusion but the absence of the necessary supports for the learning needs of the included kids is a serious equity issue, IMO. The exceptional students are not getting the teaching theyneed, and the other students are inevitably deprived of some of the instruction -- and much of the enrichment for the high achievers -- that THEY need.

I see this as a false economy.

the joy of corpus studies, part 2

more from Biber and Conrad:

[I]n fiction writing and newspaper writing, the verb say is much more frequent than any other lexical verb; in conversation, the verbs go and know are as frequent as say, and get is more frequent than any of those three verbs; while in academic writing, the only especially frequent verb is BE.
Corpus Linguistics and Grammar Teaching

answer key

Most common verb in spoken English: get

[T]he extremely high frequency of the verb get in conversation is more surprising for most people. This verb goes largely unnoticed, yet in conversation it is by far the single most common lexical verb. The main reason that get is so common is that it is extremely versatile, being used with a wide range of meanings. These include:

• Obtaining something: See if they can get some of that beer.
• Possession: They’ve got a big house.
• Moving to or away from something: Get in the car.
• Causing something to move or happen: It gets people talking again, right?
• Understanding something: Do you get it?
• Changing to a new state: So I’m getting that way now.

Corpus Linguistics and Grammar Teaching
Douglas Biber and Susan Conrad

I missed it by a mile. (I said 'be,' which I gather is right if you're talking about the most common verb used in writing.)

New Blogger post window is not easy, and not fun.

Right-side menu is now stuck open, covering up one or two words at the end of each line.

I wonder how hard it is to move to Wordpress?

I love Wordpress.

books vs 'visuals'

Just found this comment by an AP history teacher:

Videos (DVDs, films, whatever) are overrated. A full-length movie had better be virtual time travel to be worth the time showing it. Bits and pieces are okay. I used a lot of stuff off of YouTube - there are excerpts from everything there.

VERY IMPORTANT NOTE: Students are appallingly jaded when it comes to visuals, and, to be honest, they're so used to watching stuff, that they don't actually pay attention any more. If you show something, you're going to have to explain it much more than you think. What does haunt them is in the books. I always showed a Japan class some propaganda films with heavy atrocities - didn't faze them a bit - and then had them read "Hiroshima" - and they had nightmares from it.

off-topic: While I was in Illinois, Blogger completely reworked the post window.

I don't like it.

At least, not so far.