kitchen table math, the sequel: 8/26/07 - 9/2/07

Saturday, September 1, 2007

remembering foreign language vocabulary

AbstractIn a 9-year longitudinal investigation, 4 subjects learned and relearned 300 English-foreign language word pairs. Either 13 or 26 relearning sessions were administered at intervals of 14, 28, or 56 days. Retention was tested for 1, 2, 3, or 5 years after training terminated. The longer intersession intervals slowed down acquisition slightly, but this disadvantage during training was offset by substantially higher retention. Thirteen retraining sessions spaced at 56 days yielded retention comparable to 26 sessions spaced at 14 days. The retention benefit due to additional sessions was independent of the benefit due to spacing, and both variables facilitated retention of words regardless of difficulty level and of the consistency of retrieval during training. The benefits of spaced retrieval practice to long-term maintenance of access to academic knowledge areas are discussed.
Maintenance of Foreign Language Vocabulary and the Spacing Effect
Harry P. Bahrick, Lorraine E. Bahrick, Audrey Bahrick, Phyllis E. Bahrick
Psychological Science, Vol 4, Issue 5, pp 316-321, September 1993

Assuming I'm reading this right (haven't looked at the article yet), you can swap repetition for spacing.

You can spend less time studying if you space that studying out over a substantial period of time -- and vice versa.


overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

same time, next year

I just noticed Ken's use of the phrase "how the spiral should work."

This is our problem.

A spiral curriculum is by definition a form of spaced repetition. You learn topic X in first grade; then you "revisit" topic X in second grade; then you take another return trip again in 3rd grade.

The notion that the space between repetitions will be 12 months' time is simply built in to the proposition. It's unexamined.

And, of course, no effort is made to ensure or assess whether students have reached mastery before the class peddles on.


overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

Ken's interval

re: study time and retention intervals

I would think it is even more efficient to systematically fade both the amount of problem and the interval between problem sets for each new problem type.

For example after a new problem type is learned via massed practice, subsequent practice sets might go something like this:

day 2: problems 10
day 3: problems 10
day 4: problems 8
day 5: problems 8
day 6: problems 6
day 7: problems 6
day 9: problems 6
day 11: problems 6
day 13: problems 6
day 16: problems 5
day 20: problems 5
day 25: problems 5
day 30: problems 5
day 40: problems 4
day 50: problems 4
day 60: problems 4
day 80: problems 3
day 100: problems 3

That's how the the spiral should work. As the student better learns the material, it gets refreshed at increasingly larger before the knowledge has a chance to fade from memory.

This is how Engelmann does it in all the DI programs. The only exception is when a subskill gets subsumed into a more complicated skill (once it has been mastered), then only the more complicated skill gets the distributed practice.

Saxon is interesting in this respect, because Saxon books give you practically no massed practice at all once you move to algebra. Prior to that kids to "Fast Facts" sheets every day.

In the high school books students do, at most, 4 or perhaps 5 problems in the new skill or concept covered in the lesson. Usually you do only 2.

I assume he does this because by the time you get to algebra you're constantly doing problems built out of well-learned embedded skills, but I don't know.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

Tex's political brain

Just saw a comment from Tex:

I will be examining my daughter’s new NSF math program, Think Math!, to see if it teaches to mastery and how long the interval is before it spirals back. In fact, this will be a good question on which to focus, instead of jumping on the school with all the other potential issues. (Other issues including the emphasis on group work, which I detest for my daughter’s math learning.)

I am (slightly) embarrassed to say that I had not quite come to this thought....

Tex is way right on this. These studies are gold. They're radically more useful politically (and practically, in terms of afterschooling) than the cognitive science studies discussed by Dan Willingham because they are concrete, specific, and extremely difficult to refute by recourse to the abstractions of ed schools.*

"Learning styles," "problem solving," "conceptual understanding" -- all of these shibboleths are off-topic, and will be instantly recognized as such by parents (I think). The fact is that all textbooks and all mathematics teaching, no matter what the philosophy, give students problems to solve. So the question is: do we organize these problem sets effectively, to maximize long-term retention, or do we not?

The answer in my district is, "I will not discuss curriculum and pedagogy with parents," (direct quote, math chair, 12/13/2007) which, in parent-speak, means 'no.'

The broader, constructivist, answer, of course, is, "We're not interested in long-term retention."

That's why Tex's insight is so important to those of us who value knowledge. Defining the conversation as being about long-term retention is akin to asking "when did you stop beating your wife" -- there's no good answer a constructivist can give, since in the constructivist realm learners are always and forever constructing knowledge (or meaning), not storing it. But no administrator or teacher can say directly, to a parent or a school board member, "Whether or not your child remembers algebra, vocabulary, or grammar (etc.) when it comes time to take the SAT isn't important."

An education professor can say such things in an education journal. No one working in a school district can do so.

long-term retention

Remember this phrase.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

* Don't get me wrong. Willingham's articles are life-altering; I am a devoted fan. But I haven't found a way, thus far, to use them effectively in school politics.

the shuffling of mathematics problems improves learning

Haven't read yet, but here's the abstract.

Abstract In most mathematics textbooks, each set of practice problems is comprised almost entirely of problems corresponding to the immediately previous lesson. By contrast, in a small number of textbooks, the practice problems are systematically shuffled so that each practice set includes a variety of problems drawn from many previous lessons. The standard and shuffled formats differ in two critical ways, and each was the focus of an experiment reported here. In Experiment 1, college students learned to solve one kind of problem, and subsequent practice problems were either massed in a single session (as in the standard format) or spaced across multiple sessions (as in the shuffled format). When tested 1 week later, performance was much greater after spaced practice. In Experiment 2, students first learned to solve multiple types of problems, and practice problems were either blocked by type (as in the standard format) or randomly mixed (as in the shuffled format). When tested 1 week later, performance was vastly superior after mixed practice. Thus, the results of both experiments favored the shuffled format over the standard format.

Instructional Science ($)
published online April 19, 2007

the Saxon shuffle
[A] very small number of mathematics textbooks use what we call a shuffled format (e.g., Saxon, 1997). A textbook with a shuffled format may have lessons identical to those in the standard format, and moreover, the two formats need not differ in either the number of practice sets within the text or the number of practice problems per practice set. But, with the shuffled format, the practice problems are systematically arranged so that practice problems are both distributed and mixed. For example, after a lesson on the quadratic formula, the immediately following practice set would include no more than a few quadratic formula problems, with other quadratic formula problems appearing in subsequent practice sets with decreasing frequency. Thus, the practice problems of a given type are systematically spaced throughout the textbook. This spacing intrinsically ensures that the problems within each practice set include a mixture of different types, as there are no more than one or two practice problems of each kind within each practice set. In order to achieve such variety in the early portion of the textbook, the first several practice sets can include problems relating to topics covered in previous years.


Perhaps the most well known example of the shuffled format is the Saxon line of mathematics textbooks (e.g., Saxon, 1997). In these textbooks, no more than two or three problems within each practice set are drawn from the immediately preceding lesson, and the remaining one or two dozen problems are drawn from many different lessons. We are not aware of any published, controlled experiments comparing a Saxon and non-Saxon textbook, but such an experiment may not be particularly informative because it would be confounded by the numerous differences between any two such texts. That is, regardless of the outcome of an experimental comparison of a shuffled textbook and a standard textbook, any observed differences in, say, final exam performance might reflect differences in the lessons rather than practice format.

Such confounds would be avoided, however, if two groups of students were presented with the same lessons and different practice sets. For example, each group of students could receive a packet that includes the lessons from a traditional textbook, and these lessons would appear in the same order for both groups. Both groups would also see the same practice problems, but the problems would be arranged in either a standard format or shuffled format. By way of disclosure, neither author has an affiliation with a publishing company or mathematics textbook, although the first author is a former mathematics teacher who has taught with textbooks from many different publishers, including Saxon.

There is no doubt in my mind -- none -- that shuffled problems produce better retention than massed problems.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

Friday, August 31, 2007

Saxon rules

Just caught the last paragraph of the long-term memory study Concerned posted:

All these experiments involved rote learning, but Rohrer and Pashler have also found similar effects with more abstract learning, like math. This is particularly troubling, the psychologists say, because most mathematics textbooks today are organized to encourage both overlearning and massing. So students end up working 20 problems on the same concept (which they learned earlier that day) when they should be working 20 problems drawn from different lessons learned since the beginning of the school year. In brief, students are wasting a lot of precious learning time.

That's Saxon.

30 problems a day, no more than 2 on the same skill or procedure.

Day in, day out.

Extreme mixed review.

Which reminded me to get out my mixed review books:

Middle-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
Doug Stoffel Creative Teaching Press 2000

Mixed Skills in Math
ISBN: 1-56822-861-9

These are both for middle school, but you can find mixed-review books for all grade levels.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

how long does new knowledge last?

Unfortunately, this isn't directly relevant to Concerned's post, but it's interesting:

Translated into practice, the spiral curriculum is a series of different, unrelated topics that parade past the kids year after year. Kids dabble in measurement for a while before moving on to the next unit, which may be geometry, which is followed by whole-number operations, which is followed by fractions, ... and so forth. Typically, about 60 school days pass before any topic is revisited. Stated differently, the spiral curriculum is exposure, not teaching. You don't "teach" something and then put it on the shelf for 60 days. It doesn't have a shelf-life of more than a few days. It would be outrageous enough to do that with one topic--let alone all of them.

Bruner's endorsement of the spiral curriculum suggests the extent to which cognitivists lack a comprehensive schema of a kid's brain. Don't they know that if something is just taught, it will atrophy the fast way if it is not reinforced, kindled, and used? Don't they know that the suggested "revisiting of topics" requires putting stuff that had been recently taught on the shelf where it will shrivel up? Don't they know that the constant "reteaching" and "relearning" of topics that have gone stale from three months of disuse is so inefficient and impractical that it will lead not to learning but to mere exposure? And don't they know that when the "teaching" becomes reduced to exposure, kids will understandably figure out that they are not expected to learn and that they'll develop adaptive attitudes like, "We're doing that ugly geometry again, but don't worry. It'll soon go away and we won't see it for a long time"? Apparently not, even though it would take very little time working in a classroom to document all of the above.

War Against the Schools' Academic Child Abuse by Siegfried Engelmann, p. 108

Early in the book he has this to say about practice:

In addition to having a good program, we have a great deal of knowledge about how kids learn and how to teach well. We know how much practice it takes for the kids to master the various details of our sequence. Oddly enough, the amount of practice that we've had to provide to meet our goal is possibly five times the amount provided in other published programs that teach the same subject. We have also learned thast kids tend to "lose" information if we don't keep it "alive" in the program." This observation had led us to activities that require kids to use all the important skills and concepts they've been taught.

War Against the Schools' Academic Child Abuse by Siegfried Engelmann, p. 17

Saxon Math is built on this principle. At the end of the year students are still practicing skills and procedures they learned at the beginning of the year.

Saxon gets content into long-term memory.

I've decided to use a variant of the Saxon "recursive" structure with C. & Singapore Math.*

We're going straight through the Primary Mathematics textbook & workbook starting with 3A. But I'm going to assign daily problems from the Intensive Practice book out of sequence, making them last throughout the entire year.

* The principal in Linda Perlstein's new book Tested describes Saxon this way (will post the passage at some point). I like it.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

"Overlearning" Overrated?

Science Daily had an interesting article out this week discussing the value of overlearning. Recent research has found that studying material beyond mastery may be a waste of time in the long run.

University of South Florida psychologist Doug Rohrer decided to explore this question scientifically. Working with Hal Pashler of the University of California, San Diego, he had two groups of students study new vocabulary in different ways. One group ran through the list five times; these students got a perfect score no more than once. The others kept drilling, for a total of ten trials; with this extra effort, the students had at least three perfect run-throughs. Then the psychologists tested all the students, some one week later and others four weeks later.

The results were interesting. For students who took the test a week later, those who had done the extra drilling performed better. But this benefit of overlearning completely disappeared by four weeks. In other words, if students are interested in learning that lasts, that extra effort is really a waste. They should instead spend this time looking at material from last week or last month or even last year.

Researchers concluded that once mastery was achieved it was better to leave that subject alone for a while and return to it later. They found that an optimal "study break" of about a month resulted in long-term learning-- something they refer to as the "spacing effect".

Is this "spacing effect" an argument for the spiral approach? Perhaps so, yet it does seem to be a well executed spiral in which the content is first studied to mastery and then revisted for reinforcement later. This is certainly not the haphazard "spiral" I've witnessed my children being subjected to with Everyday Math and seems to be more in keeping with Saxon or Singapore Math's idea of a spiral curriclum.

I hope they keep looking into this subject. Children have such precious little time to learn so many important things. Imagine all that could be accomplished if we started implementing teaching and study skills that were actually efficient.

Source: Back to School: Cramming Doesn't Work In The Long Term

ABSTRACT—Because people forget much of what they learn, students could benefit from learning strategies that yield long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions—the spacing effect—depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.

Increasing Retention Without Increasing Study Time

Catherine here, diving into Concerned's post.

What a find! I've just pulled the article; will read shortly.

In the meantime, here are the Willingham articles that discuss overlearning:

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

Thursday, August 30, 2007

more stats from 2007 SAT

No idea whether this is real or noise:

Among the surprises in the results: The test takers reported as being among the wealthiest were one of the groups that saw the biggest declines. Students with family incomes of more than $100,000 saw declines in all three sections combined of 19 points from the previous year's scores. By contrast, the only income group that actually saw increases were those with reported incomes between $10,000 and $20,000. In that group, there were increases in all three sections totaling 21 points.

College Board officials say the data aren't very reliable because students report their perceptions of what their parents make. And over one-third of students didn't answer the question.

Class of 2007 Logs Slide In SAT Scores

another reason to teach to mastery
why SATs predict college success

more stats from 2007 SAT

why SATs predict predict college success

Speaking of SATs, here is John Allen Paulos' explanation for why SATs do predict college success, nothwithstanding ritual assertions to the contrary from university presidents, admissions officers, and a chorus of lookers-on:

School's getting out, but the dreaded Scholastic Assessment Test, better known as the SAT, looms just a summer away for next year's high school seniors.

Given this, many might be inclined to agree with the president of the University of California, who announced several months ago that he would like to abolish the test as a requirement for admission to the school. (He would retain the SAT II, which measures achievement within particular disciplines.)


How predictive of success in college are SAT scores? More precisely, what is the correlation between high school SAT scores and first-year college grade point average?


Most studies find that the correlation between SAT scores and first-year college grades is not overwhelming, and that only 10 percent to 20 percent of the variation in first-year GPA is explained by SAT scores.

This association appears weaker than it is, however, for an interesting, but seldom noted statistical reason: Colleges usually accept students from a fairly narrow swath of the SAT spectrum.

The SAT scores of students at elite schools, say, are considerably higher, on average, than those of students at community colleges, yet both sets of students probably have similar college grade distributions at their respective institutions.

If both sets of students were admitted to elite schools or both sets attended community colleges, there would be a considerably stronger correlation between SATs and college grades at these schools.

Those schools that attract students with a wide range of SAT scores generally have higher correlations between the scores and first-year grades.

This is a general phenomenon; the degree of correlation between two variables depends on the range of the variables considered.


Soccer Assessment Test

Assume there were an SAT (Soccer Assessment Test) that measured the speed, coordination, strength, and soccer experience of students in a certain city. Assume further that the students roughly divided themselves into five leagues depending on their scores on this SAT, players in the top leagues having higher SAT scores on average than those in the lower leagues.

One wouldn't expect that a measure of success in the sport, say number of goals scored, to vary much among the leagues. There would be good scorers and bad scorers in every league and, just as grade point distributions are similar in most colleges, the distribution of goals scored would probably be similar in the five leagues.

In each league the better scorers would probably have only slightly higher SATs on average. In other words, there wouldn't be a high correlation between SAT scores and success in soccer within any league.

There would, however, be a much higher correlation between SAT scores and soccer success were the students randomly assigned to the teams in the five leagues.


The analogy between soccer and scholastics is not perfect, of course, but the point remains. Like the soccer "SAT," the scholastic SAT provides incomplete, but useful information to students and colleges. A rough measure of intellectual preparedness, the SAT shouldn't be made into a fetish, but neither should it be ignored.

Without it, colleges would undoubtedly place more emphasis on high school grades and extracurricular activities, measures that also have serious shortcomings grade inflation and meaningless resume-puffing being the main ones.

The SAT is a flawed predictor, but it is also relatively objective and, among other virtues, sometimes provides a way for the bright, yet socially inept student to be recognized.

I was happy to read this.

I always knew the "SAT scores don't predict college success" meme was wrong, but I couldn't work out why.

For what it's worth, Ed says GREs absolutely predict success, and do so in a fashion that's entirely commonsensical.

Applicants with verbal scores in the 700s are a lock. (for newbies: I'm talking about a graduate program in French studies) Professors know these candidates are intellectually equipped to succeed in a masters or doctoral program. The GRE doesn't tell an admissions committee anything about a candidate's emotional stability or potential for suffering Ph.D.-killing life crises, but admissions committees don't expect it to.

Applicants with scores in the 500s likely will not be able to handle the program.

Applicants in the 600s are a mystery. Some will wash out; some will be in the middle along with their scores; and some will be flat-out brilliant, best in show. These are the folks who come bearing surprises.

I like that.

another reason to teach to mastery
why SATs predict college success
more stats from 2007 SAT

another reason to teach to mastery

With results that will hardly put to rest the controversy over the merits of the SAT, scores for the high-school class of 2007 are in -- and they're down.

The point drop in the average math score for the latest year was the biggest in more than three decades, reported the College Board, the New York nonprofit that administers the college-admissions test. Math scores declined three points, to 515, from the previous year's 518 out of a possible 800. Critical reading -- formerly called "verbal" -- also dropped, to 502 from 503, marking the lowest score since 1994. Scores for the writing section slipped to 494 from 497 the previous year -- the first time the new section's scores were released. Taken together, the scores are the lowest of the decade so far.

Class of 2007 Logs Slide In SAT Scores
by Anne Marie Chaker WSJ 8-29-2007; Page D1

I'm not sure anyone knows what's going on, if anything.

However, it does sound as if the test has gotten harder, and we know for a fact that it's longer:

"The new test was designed to be more challenging," says Seppy Basili, senior vice president at Kaplan Inc., the test prep concern owned by Washington Post Co. The inclusion of material from advanced algebra is "a serious concern for some students," he says, some of whom don't reach the material until their junior year.

Level of Attention

The revisions to the test also added 45 minutes to the length. "It is a problem," says Scott White, director of guidance at Montclair High School, Montclair, N.J. "There is no way on earth a kid can test for nearly four hours and have the same level of attention and acuity as with a three-hour test."

Andrew Bennett-Jackson, a senior at Montclair, says he found his "focus really go down" toward the end of the exam, which he took over the summer. He scored a 1610 out of a possible 2400. He plans on trying the ACT, a rival admissions test accepted by a growing number of schools, in the fall. One big plus: The ACT's writing section is optional, which makes it shorter for students who just want to retake the other two sections. "It just makes the test a little bit easier," Mr. Bennett-Jackson says.

I don't need a statistical analysis to tell me that your best shot at a high score on a nearly-four-hour test is going to be knowing algebra & geometry to automaticity. The whole point of automaticity (well, one of the points) is that your brain uses far less energy to work with "automatic" material than it does with material you're shaky on. The reason a well-learned subject seems "easier" is probably that it is easier. You use fewer resources.

I still remember how sad I felt the day I saw research showing that mentally retarded children's brains are far more "active" on brain scans than normal children's. I didn't know as much about scans & brain metabolism then, and the sudden realization that mentally retarded children were working very hard to understand and remember things typical children could do while barely lighting up a CT-scan was heartbreaking.

The brain is an expensive organ:

The calculations presented in this paper show that the energy metabolism of the brain may account for about 50 percent of the BMR in the new-born. As indicated by our calculations the proportion of BMR accounted for by the energy metabolism of the brain decreases with age, but it still accounts for about 30 percent of the BMR at age 12 years.

Energy metabolism of the brain in children

This is cool. Numbers Guy says there probably is a change in test population. That's the Sun's take (I'm not going to link; no chance the editorial will be free). The Sun says that the decline in NYC SATs happened because more black students took the SAT, which is good. More black students are in the running.

another reason to teach to mastery
why SATs predict college success
more stats from 2007 SAT

gut feelings

My copy of Gerd Gigerenzer's Gut Feelings arrived yesterday, and today I've found an interview with Gigerenzer in the TIMES science section.

Interesting passage on school quality:

In the 1990s, I was living in Chicago, where there are high dropout rates from the high schools. People often asked, “Is there a way to know which school has the lowest dropout rate?” There existed data measuring different cues of school performance: the pay of teachers, the number of English-speaking students in a class, things like that.

I wondered: could one feed these into a computer, analyze them and obtain a prediction on which high school produced the fewest dropouts? We did that. And we were astonished to find that computer-based versions of Franklin’s bookkeeping method — a program that weighed 18 different cues — proved less accurate than going with the rule of thumb of “get one good reason and ignore the rest of the information.”

Q: What was the “one good reason” that got you the right answer?

A: Knowing which school had high daily attendance rates. If two schools had the same attendance levels, you needed one more cue — good writing scores — and then you could ignore the rest.

Through Analysis, Gut Reaction Gains Credibility
by Claudia Dreifus


Writing instruction is probably the single most labor-intensive form of teaching, so..... high writing scores as a measure of school quality may in fact be a measure of teacher (and student?) effort.

Our high school apparently has fantastically high attendance rates, which the principal once offered as evidence that they were doing something right. I remember thinking that made sense.

Low writing scores, though. At least, the scores on the first year of the SAT writing section looked low to me. Ineffective writing instruction has been a perennial parent complaint.

I suspect that with the loss of the middle school math chair and the adoption of Trailblazers we'll see math scores decline, too, unless parents and students can find a way to make up lost ground in high school. I don't know much about the math teachers there, but the word on them ranges from "pretty good" to "very good." Still, I don't see how a high school teacher, no matter how good she is, is going to teach fractions, decimals, percents, and algebra 1, algebra 2, & geometry.

Wednesday, August 29, 2007

teacher turnover redux

I've just stumbled across what appears to be an interesting blog, which -- synchronicity alert -- turns out to have a link to a Quick and the Ed post on teacher turnover summarizing a massive NCES report (enough with the embedded phrases -- blech).

The salient points:

1. Teachers stay put more than we think. Teachers have relatively low attrition rates and are actually leaving the profession at lower rates than their peers in other professions. So caution the characterization of teachers as a bunch of fickle ship-jumpers. Or at least no more so than the rest of us.

2. When they leave, it's mostly for family reasons or to go into an entirely different field. Not surprisingly, it's the ladies who leave for family and the men who leave for business and engineering, often for pay reasons. See #s 4 & 5.

Of course, given the bang-up job the NCES did on the charter school report last year, I'm not sure I trust them any more than I do the Times, but still.

Any time I see a news article citing scare figures without comparison data -- how many people quit other jobs? -- I figure I probably don't know any more now, having read the story, than I did before.

Singapore Math, day one

I've just walked C. through his first day of Primary Mathematics 3A, including the 3A workbook.*

Then I had him do one "Challenge Problem" from Intensive Practice 3A:

1. I am a three-digit number. All the three digits add up to 9. My tens digit is twice my hundreds digit and my ones digit is three times my tens digit. I have no zeros. Who am I?
(page 10)

He took a look at it and said, "That's hard!"

I thought so, too. As a matter of fact, I am currently doing word problems only slightly more advanced than this in Saxon Algebra 2, at the end of the book. [update: RJ and Cheng noted that the sum of the digits of their two-digit counting number was 9. If the digits were reversed, they found that the new number was 27 less than the original number. What was the original number? Saxon Algebra 2, Second Edition, p. 432, no. 1.]

I handed him the workbook to look at while I wrote the problem on separate sheets of graph paper for both of us to do. I figured it might be a good idea to get some practice drawing bar models.

A couple of seconds later C. said, "I have the answer."

He did. answer: 126

I pushed him to tell me how he got it, and finally he said (paraphrasing), "I asked what is 2 times, and then I realized it was 2, and then that was 6."

I believe this is called "making children's thinking processes visible," right? And, yes, this is the way kids talk when they're describing a mental process. Mental processes are slippery. When you get an answer fast, you don't necessarily remember -- or even know -- how you did it

What I take Christopher's report to mean is that he intuitively -- unconsciously -- understood immediately that the hundreds digit would have to be 1, then consciously asked himself what two times 1 was, had a eureka moment (I could tell this from his tone of voice) when he realized that 2 was put him in the running to end up with a grand total of 9, and from there semi-consciously realized that the ones digit was 6.

At least, that's the way I piece it together. Obviously I don't know how he came up with the answer five seconds after he read the problem.

This gives me hope. I was saying yesterday that Ed and I were hoping the fact that C. appears to be correctly guessing answers to fraction word problems means he's developed some number (& fraction) sense; today's Challenge problem seems to be evidence that he has.

I think I've mentioned before that "Math Dad," the parent here who is a veteran NY state math teacher, has told me, and I'm pretty close to quoting here, "It's too late for these kids. They are lost." He was speaking of Christopher's group, the kids who need effective instruction and a sound curriculum in order to succeed in accelerated math.

When I heard that I thought: over my dead body.

Maybe, just maybe, these past 3 years of slave labor and protracted Math War have produced some "growth" after all.


These are the three questions for which C. correctly guessed the answer. I'd forgotten that one of them is extremely simple algebra. Sigh.

Robin and Jim took cherries from a basket. Robin took 1/3 of the cherries and Jim took 1/6 of the cherries. What fraction of the cherries remain in the basket?
answer: 1/2

___ represents the number of magazines that Lina reads each week. Which of these represents the total number of magazines that Lina reads in 6 weeks?

6 + ____

6 x ____

____ + 6

( ____ + ____ ) x 6

[yes, I am horrified that I have a "rising" 8th grader who had to guess the answer to this question - though this does serve the purpose of illustrating why "Math Dad" goes around saying things like "It's too late for these kids."]

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then left one-fourth of the remaining marbles to Jim. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with?
answer: 48

* For newbies, C. is entering 8th grade and began taking Math A -- integrated algebra and geometry -- in the second semester of 7th.

Tuesday, August 28, 2007

How We Train New Teachers

Lynn’s post about a teacher shortage generated some great commentary about teacher training, and as of this afternoon I have some first-hand information I think I can use to add to people’s awareness of how we often train our teachers.

Yesterday, the first day of school, our English AP sent a desperate email asking if anyone would volunteer to take on a student teacher this year. I said sure. I’m in the ideal position to do this: I've sponsored student teachers before; I have all English III, 3 of them inclusion classes; this year, I’m the English III Team Leader in our department, and the inclusion classes will give the student teacher plenty of experience in that as well. I am currently mentoring a new-to-our-school 3rd year teacher and a brand-new first year teacher. In addition to that, the inclusion teacher I have 2 classes out of 3 is a brand new teacher as well - to inclusion and teaching. For those who don’t know me, I went through an alternative program after working for a number of years, and completed no student teaching at all. This is my 9th year as a teacher.

After I agreed to take on the student teacher, I received a general email which was supposed to contain an attachment with the teacher’s name and info, as well as when this teacher would show up. That attachment was not included. I emailed to get the info. I received no reply throughout the day today.

The teacher himself showed up at my classroom door at 2:30 this afternoon, just in time for my last period. I had no idea who he was, but he introduced himself and said he’d just finished training that morning and was told to come to the school to meet me. No one had bothered to escort him - he’d had to find my room on his own. I was greeting students at the door and getting them settled in and taking notes (I’m now officially the MEANEST English teacher, according to other teachers, because I made them do work the first day and gave homework), so I had him sit down and start doing the same work the students would do that day.

Since I teach bell to bell, I didn’t have any time to interact with him until afterward. Here’s what I know about him so far:

He is a former marine, and worked in the prison system for about two years. This bodes well for him in my opinion, and I’m not saying that to make any sort of joke about student behavior. He decided to become a teacher after working in the prison system began to weigh heavily on him. He has been taking online courses since September 2006. He has had no face-to-face contact with any instructors, nor any classroom observation at all. I have no idea what this program is that he’s in.

He has not read any books about education for his courses. He has not received any instruction in lesson planning. “Just kind of outlining ideas,” is how he described it.

I asked him where he felt he was in terms of preparedness. He said, “To be honest, I feel like I’m just starting.” In my mind, the person who is reporting for student teaching should NOT feel as if he’s just starting. I asked him if anyone had told him what our schedule would be, when he should get to school in the mornings, or what he would be doing. The answer was no.

The expectation is that he will take over 2-3 of my classes by the second 6 weeks of school, this young man who has not been taught to write a lesson plan yet and hasn’t even observed a high school classroom.

I hustled him down to the library and got him a copy of Henry Wong’s First Days Of School, just to start him off with something that is easy to read and has some pretty solid techniques. Sent him home with the assignment to read five pages or so, and come back tomorrow morning at 8:15.

more, more, more at Mindless Math Mutterings

Everyone who hasn't already done so should go read all of Mindless Math Mutterings right this minute.

How to Teach to the Test - brilliant!

And I need to find my copy of Teach Like Your Hair's on Fire.

Rafe Esquith at Mindless Math Mutterings

it's always worse than you think

A teacher pal in the Midwest tells me that the single worst thing about her district (where she's happy & has taught for many years) is the chronic implementing of new programs without aforethought.

Every five years the district implements new stuff. Without fail. (Last summer they bought Lucy Calkins, but then didn't use the curriculum; they also piloted TERC, but decided against it - whew.)

Administrators come in, implement stuff, and move on to better jobs where they implement more stuff.

It's the same everywhere, it seems. If I remember correctly, and I think I do, the material sent out to parents last year, introducing our new middle school principal, said he'd implemented character education in his school in Albany.

And that was it, pretty much.

He implemented character education in an urban school attended by disadvantaged kids. So, apparently on the strength of that accomplishment, he was hired to implement stuff here.*

Of course, what do I know? If you listen to the New York Times, character education is what disadvantaged kids need.


The fact that the star teachers in our schools -- and most schools are going to have a star or two -- have to sit around listening to professional developers instead of creating and providing the professional development themselves and being paid to do it really cheeses me off, and always has.


churning the curriculum
Paul Hill, an education researcher at the University of Washington, sees parallels between education practices and medicine as practiced in the Colonial period. "There was not a lot of science behind medicine then," says Hill. "It was swept by fads." Each patient's suffering was thought of as unique, a special case. There was no broad research that demanded standardized responses to standard ills. Each patient was in need of individualized care. Ever hear one of your child's teachers use that word? We provide individualized instruction. Each class is unique. "Physicians used to say the same thing," says Hill. "There was a natural tendency to think of everything as much too unique to want to generalize your practice."

But while medicine has moved into an era of standardized practice driven by medical research, education remains in the era of leeches and bleeding. Where was the research that demanded teachers drop phonics, teach in "open" classrooms or try "new" math? Where, for example, was the research that led so many elementary school teachers to tinker with "heterogeneous grouping," where children of mixed abilities were put into groups with the hope that the faster learners would tow along the slower learners? Like so many educational fads, it sounded like a good idea, but it rarely worked in crowded classrooms lacking the talented aides who could pull it off.

At Harvard, former teacher Tom Loveless teaches a subject on this very topic: "Controversies in Education Reform." The syllabus, which consists entirely of required readings on school failures, resembles an indictment of a Mafia chief. Loveless has a personal feel for the problem that dates back to his nine years of teaching in Sacramento public schools. "This is an industry with tremendous turnover at school sites," says Loveless. "Half the principals change schools every six or seven years and superintendents even more frequently. I went through three or four principals, all of them saying: "I have some new ideas and we're going to change things." In come the big changes, out goes the principal within a few years, in come more big changes. "Nobody would stay in one place long enough to be responsible for outcome," says Loveless. "By the time everyone figures out what they're doing doesn't work the principal is gone and you're off to a new approach."

source: Neglected Evidence

The beauty of this system is that taxpayers get to pay for all this stuff. If you're a taxpayer and a parent, you get to pay for the tutors and the time-off-work to (try to) fix the problems, too.

* Middle school model in our case. Our middle school is apparently destined to become exemplary no matter what the parents think.

TIMSS test for kids (and grownups)

You can take sample 10-item 4th & 8th grade TIMSS tests here. They've also got some 12th grade tests, though none in math.

I had C. take the 8th grade math test yesterday.

The news wasn't quite as bad as I expected.

He scored 100% correct. I was thrilled ecstatic until I read the questions, 3 or 4 of which seemed over his head. Sure enough, he'd guessed 3 of his answers, all on fraction questions.

On the other hand, he got the percent problem correct, and I know for a fact he couldn't have answered that question 2 months ago. Score another win for Singapore Math, it seems.

Ms. Thierry and 3 friends ate dinner at a restaurant. The bill was $67. In addition, they left a $13 tip. Approximately what percent of the total bill did they leave as a tip?
correct answer: 20%

When I asked how he got this answer he said, "I took 10% of $67 and multiplied by 2." That was great to hear, since both Ed and I have been hounding him all summer long about the fact that you can find 20% of a number simply by moving the decimal point over 1 place (WHICH MEANS DIVIDING BY TEN, I INVARIABLY POINT OUT) and then multiplying that number by 2. Apparently it sank in.

I was semi-glum over the missed answers, but Ed thinks the fact that he is "guessing" correctly is good, and I tend to agree. Consistent correct guesses on fraction questions probably indicate that C. has developed some number sense for fractions.

Fraction sense.

(Does that seem wrong?)

He'll do his final page of problems on percent in Primary Mathematics 5B today. (We've done all the corresponding workbook problems, as well.)

I was thinking I'd have him also do the problems in 5B Intensive Practice, but it occurs to me as I write that I should probably distribute those through the next weeks a la John Saxon.

I think we'll finish up Percent in 5B and head on back to Primary Mathematics 3A.

First topic: Numbers to 10,000.

recent comments widget

Well, in theory I am going to add a recent comments widget to ktm-2.

Unfortunately, I've lost the post from hackosphere that explained how to do it.... so I'm thinking about using this guy's advice.

Is there any reason why I shouldn't?

I ask, because I'm going to be a very sad panda if I blow up the blog.

Miss Teen on U.S. Education

A 2006 National Geographic poll determined that young Americans are geographically illiterate and I do believe this response qualifies. The survey found that young adults in the United States fail to understand the world and their place in it. This is what we end up with when that happens.

Monday, August 27, 2007

watch all of this

I'm serious.

Watch until you get past the slide show to the part where the guy starts talking.

on the other hand

re: vacation from the vacation

On the other hand, I got to see Susan S' new house - beautiful!

And spend 5 minutes on the phone with Karen H.

And the Lincoln Presidential Museum is fantastic! We didn't really see the museum, unfortunately, because Andrew melted down the minute we set foot inside the "White House" and things got worse from there. Some guy accused Andrew of trying to "boost" his camera; then, in the "war room," a group of touring Asians kept swiveling around and craning their necks to get a look at Andrew and me in our hiding place off the main room, where I'd taken him to try to minimize the disruption to everyone else's Presidential Museum visit.* Andrew was by then slapping himself violently in the face while I tried to hold his hands and figure out what a behavior analyst would do.

Finally I gave up and did what a parent would do, which was send Andrew outside with his father so I could purchase tourist merchandise from the gift shop in peace.**

So we didn't see the museum, but we did see the documentary about Lincoln's life. It's a Sensurround-type deal, with flashing lights, rumbling seats, and extremely loud, sudden, explosion noises that seemed designed to overwhelm anyone with the slightest degree of sensory overload.***

Andrew was rapt.

He sat perfectly still in his seat, gazing at the screen, his hands clasped together as if in prayer. Jimmy had his eyes closed and his head down; he looked like a tortoise without a shell. I was jumping out of my chair & shrieking at every crack of fake cannon fire and every clap of fake thunder.****

Andrew was transfixed.

Afterwards Ed and I realized that for Andrew the documentary must have been like PBS with sound effects.


* I specify "Asians" because we have Japanese friends here who tell us they can't return to Japan because of the stigma attached to having an autistic child there, and because another Japanese family here in town stopped speaking to us and canceled a play date invitation to Christopher when Ed revealed that C's brothers have autism, and I've generalized from those experiences. The fact is I have no idea why the Lincoln Museum tourists were staring at us, but I felt Andrew and I had to go. I apologize to any foreign nationals who find this account insulting or hurtful. On the other hand, the camera-booster guy was in the same room with the tourists when we came in, and he scrammed out of there as soon as he saw us. Good.

** This sounds snarky, so I need an asterisk. The movie is wonderful, and the sounds effects - and hologram! - work. The whole museum is wonderful; Ed (for newbies: Ed is a historian) thought so, too. He liked it so much he wanted to go back alone.

*** two Presidential Museum mugs and a matching Presidential Museum t-shirt & baseball cap

** My startle response is getting way worse with age, and no one seems to have the slightest idea how to fix it.

vacation from the vacation

We're back.

We're back and in vacation from the vacation mode.

Why are we in vacation from the vacation mode?

Because tornadoes, elderly parents, and autistic children are a bracing mix, that's why.

And that's all I've got to say about that.

Is there a Teacher Shortage?

I got home from vacation last night and was greeted this morning with a front page article in the New York Times bemoaning teacher "turnover." I'm not sure if the article means to imply that there is a shortage of teachers for the schools, a shortage of qualified teachers for the schools, or just a lot of teachers swapping jobs.

The article, "Schools Scramble for Teachers Because of Spreading Turnover" starts by mentioning the high rate of teacher retirements and the stress of working in low performing schools, then says that this combination of events is "fueling a crisis in teacher turnover that is costing school districts substantial amounts of money as they scramble to fill their ranks for the fall term."


But I've been hearing dire warnings for at least a decade about this teacher shortage. So I'm a little skeptical. There seem to be plenty of classroom teachers to go around, the last time I looked, but I would have to agree that qualified teachers in math and science are harder to find.

Back to the article, Guilford, NC apparently has hired new teachers for every class every term.

This is different from a shortage, and the Times is careful not to call it a shortage of teachers. But teachers do seem to switch schools frequently.

I'm not sure if this is a problem or not. It might well be the way a market has to correct itself. If we allow teachers to move around freely, no one is likely to stay where they hate their job, if they can get a better one somewhere else. Perhaps Guilford, NC should look at what's going on in the classrooms that no one wants to stick around for very long. Maybe the pay is too low, maybe the curriculum is too stupid to teach, maybe the administrators are horrible people to work for. I don't know, but if there's a lot of teachers moving around, it might be a good idea to ask why.

The Times article is not all that interested in asking why. I find this disappointing. The article talks about the efforts schools make to recruit and retain, and pay is at the top of the list, not surprisingly.

If you read deep into the article -- page 13 -- Thomas G. Carroll, president of the National Commission on Teaching and America's Future is quoted as saying,
"Our teacher preparation system can accomodate the retirement rate. The problem is that our schools are like a bucket with holes in the bottom, and we keep pouring in teachers."
I find this weird. If the ed schools are pumping out enough teachers to compensate for retirements, why is there a problem? And there does seem to be a problem. Carroll's Commission calculates that 1/3 of all new teachers leave the profession after just three years, and that after five years almost half are gone.

This is astounding. After spending years of going through ed school, getting certified, and surviving practice teaching, 1/2 of all teachers are gone in 5 years?

Here's a question worth pondering, if people are giving up teaching after devoting years of their time preparing for it, maybe they weren't very well prepared for the reality of a classroom? Maybe ed schools are the problem? If teachers were better prepared to succeed in the classroom, i.e., if they knew how to help kids learn, maybe they'd stick around longer.