kitchen table math, the sequel: 7/22/07 - 7/29/07

Saturday, July 28, 2007

Education Consumers Clearinghouse - best practices

I think this is the report Brett mentioned in his comment:

Executive Summary:
Educational Practices of Six Effective Tennessee Schools (pdf file)

1. Principals receive frequent reports on individual student progress towards the Tennessee curriculum standards.

2. Teachers receive frequent reports on the progress of each of their students.

3. Schools keep parents regularly informed about their children’s progress and ask the parents for assistance when children are having difficulty achieving a particular state

4. In addition to Tennessee’s TCAP examinations, these schools use supplemental tests that assess the same knowledge and skill domains sampled by the state examination.

5. Schools use criterion-referenced progress tests.

6. Schools set mastery criteria higher than those required by the state curriculum standards.

7. Principals use student progress reports scores to evaluate each teacher’s effectiveness in bringing about student achievement gain.

8. Principals and teachers collaboratively group students to optimize student progress.

9. Teachers employ supplemental learning activities when one or more students are having difficulty achieving a particular objective, even though the rest of students have mastered
that skill.

10. Teachers who are experiencing difficulty in particular areas of teaching are mentored by the principal and other teachers.

11. Teachers and principals select activities that allow students to practice the knowledge and skills that will be tested.

12. Schools have adopted school-wide programs that reward positive social and academic student behaviors.

13. Principals acquire additional resources to help teachers whose students are making insufficient progress.

14. Schools survey parents annually to assess satisfaction with the school’s services.

Heaven on Earth.

how to talk about spirals

I am just now getting to the Singapore/Saxon thread (and I STILL have not read the 888 lesson - which is why I haven't weighed in - SORRY! I'm frantic about my book, stuck on a question about stereotypies & environment & the brain, etc.)

We're going away for the night, so I may not get to it now, either, but I've begun reading the Comments thread.

Like instructivist, I appreciate this parsing of the difference between the Singapore spiral and the US spiral:

Singapore's Framework is an Additive Spiral that Builds Topic Content Grade-by-Grade; NCTM's Framework Is a Repetitive Spiral Approach that Covers Similar Topics Across Grade Bands

Spiral approach by content strand (additive)
Specific for each grade (K-6)
Clear, specific topics
Mathematically logical sequence

Spiral approach by content strand (repetitive)
Within broad grade bands (K-2, 3-5, 6-8 )
Imprecise topics
Absence of boundaries and inclusive

The fluency idea has raised a question in my mind.

What's good about spiraling-with-mastery is the fact that after studying the same material 3 years in a row people remember it for the rest of their lives:

Studies show that if material is studied for one semester or one year, it will be retained adequately for perhaps a year after the last practice (Semb, Ellis, & Araujo, 1993), but most of it will be forgotten by the end of three or four years in the absence of further practice. If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.*(see below *) Exactly what knowledge will be retained over the long-term has not been examined in detail, but it is reasonable to suppose that it is the material that overlaps multiple courses of study: Students who study American history for four years will retain the facts and themes that came up again and again in their history courses.

Practice Makes Perfect -- But Only If You Practice Beyond the Point of Perfection

by Daniel Willingham
American Educator

(When I mentioned this finding to my sister-in-law, who is a federal prosecutor, she instantly said, "That must be why law school is 3 years.")

I'm wondering whether learning to the point of fluency, as opposed to mastery-defined-as-percent-correct, would alter this finding.

The article is about overlearning, which the fluency people suspect is the same thing as fluency. However, I don't know whether the subjects in the "50-year retention" studies studies had overlearned the same material in each of the three years they studied it.

Why I love math

Yesterday I was following a link mentioned here, etc. and ended up reading a great post written almost 10 years ago that captured why I love math. Here's the key quote:
That is why I have always liked math. Unlike false TV stereotypes, math is reasonable, honest, predictable and true. Girls need math, I always thought, because it is the only class that makes any sense, where results can be traced to their beginnings, and memorizing usually comes second to thinking. Other classes are OK, but only math gives you that payoff - and with programming and science it becomes a tool to make cool things that work.

CUNY Plans to Raise Its Admissions Standards - New York Times

CUNY Plans to Raise Its Admissions Standards - New York Times

The City University of New York is beginning a drive to raise admissions requirements at its senior colleges, its first broad revision since its trustees voted to bar students needing remedial instruction from its bachelor’s degree programs nine years ago.

In 2008, freshmen will have to show math SAT scores 20 to 30 points higher than they do now to enter the university’s top-tier colleges — Baruch, Brooklyn, City, Hunter and Queens — and its six other senior colleges.


Dr. Goldstein said that the English requirements for the senior colleges would be raised as well, but that the math cutoff would be raised first because that was where the students were “so woefully unprepared.”

In the fall of 2005, for example, more than 40 percent of students in introductory math courses — pre-calculus, college algebra and calculus — either failed or dropped out of the classes, numbers typical of many universities nationwide.

[emphasis mine]


Fourth grade question on Bergen Academies math competition

Bergen Academies, a magnet high school in New Jersey, offers math competitions for area math whiz kids from 4th to 8th grade.

Here's a question on the 4th grade test from 2005:

Mary bakes a pie that starts off at 300 degrees, but it cools at a rate of 4 degrees per second. The room she is in is initially at 50 degrees, but heats up at a rate of 0.5 degrees per second. How long will it take for the room to reach half the temperature of the pie?

Friday, July 27, 2007

CMT Scores Released

The results of the annual Connecticut Mastery Test have been posted online. These scores are the most recent ones from 2007.

I haven't had a chance to comb through. I'm not sure if poking around in the data is all that revealing now that I've learned how incredibly low the bar has been set. From the few items that have been released, this 4th generation of the CMT looks to be the easiest to pass and given far later in the year. A mere 60% correct qualifies a student for "goal" and if you get 70% right, you are considered "advanced."

There is nothing higher than advanced on this barely grade-level exam. If my daughter ever got 70% correct on a grade level test at the end of the year, I doubt I'd applaud her as an advanced student. So what will this test tell us?

But I like the interactive data format the state is using to put the information out there.

home writing program in place, for now

The precision teaching folks have shown me the way....

the books I'm using:

I may shuffle things around once I'm finished reading William Kerrigan's Writing to the Point. The book's sole review on Amazon explains why:

Kerrigan and Metcalf provide a comprehensive and clear method for writing expository papers. Although geared for a college freshman writing class, using this text at the junior high and high school level could preclude its use in college. I have used this text with my gifted intermediate elementary school students with great success. The students love the straight forward approach to writing, and do not feel that their creativity has been hampered in the least. If you are having difficulty teaching written expression, check out this novel approach!

Buried treasure. The book is out of print; I would never have found it if not for my precision teaching quest, though the precision teaching folks don't seem to have heard of the book, either. I think I found it on a homeschooling listmania Amazon posted to the Why Johnny Can't Write page.

Speaking of which, Why Johnny Can't Write is (almost) an ur-text for writing the same way Knowing and Teaching Elementary Mathematics is for Math.


help for the struggling writer
sentence combining exercise
we're starting a copybook
man-eaters of Kumaon - text reconstruction
expert advice on teaching writing from Joanne Jacobs
more from Joanne Jacobs
doctor pion on writing a precis and critical reading
first crack at editing exercise
home writing program in place, for now
why kids should do text reconstruction
results of sentence combining exercise
Arthur Whimbey obit
BGF Performance Systems (carries Whimbey's books)
Tips for Teaching Grammar from the Writing Next Report (pdf file)
Writing Next: Effective Strategies to Improve the Writing of Adolescents in Middle and High Schools (pdf file)

Writing to the Point Fourth Edition Table of Contents
Amazon review Kerrigan & home program
Writing to the Point, first installment
William J. Kerrigan and the sentence
writing and swimming: pp 1 & 2 Kerrigan
writing and swimming: pp 1 & 2 Kerrigan
To the Instructor

college success & math & same-subject preparation

new study to appear in Science:

Researchers at Harvard University and the University of Virginia have found that high school coursework in one of the sciences generally does not predict better college performance in other scientific disciplines. But there's one notable exception: Students with the most rigorous high school preparation in mathematics perform significantly better in college courses in biology, chemistry, and physics.


Authors Philip M. Sadler of Harvard and Robert H. Tai of Virginia say the findings run counter to the claims of an educational movement called "Physics First," which argues that physics underlies biology and chemistry, and therefore the traditional order of high school science education -- biology, chemistry, physics -- should be reversed.


"Many arguments have been made for chemistry and physics preparation to benefit the learning of biology," says Tai, an assistant professor in Virginia's Curry School of Education. "On the scale of single cells, many processes are physical, such as neurons 'firing' electrically. Also, the complex molecules at the root of life obey chemical laws that are manifested in macroscopic processes. Yet our analysis provides no support for the argument that physics and chemistry principles are inherently beneficial to the study of biology at the introductory level."


[T]he controlled data indicated that high school preparation in any of the scientific disciplines -- biology, chemistry, or physics -- boosted college performance in the same subject. Also, students with the most coursework in high school mathematics performed strikingly better in their introductory biology and chemistry courses in college; introductory college-level physics performance also benefited. Conversely, little correlation was seen between the amount of high school coursework in biology, chemistry, or physics and college performance in any of the other disciplines in this trio.

"The link between math and biology is not exactly an intuitive one, but biology has become an increasingly quantitative discipline," Sadler says. "Many high school students are now performing statistical analysis of genetic outcomes in addition to dissecting frogs and studying cells under a microscope."

The current order of high school science education was established in the 1890s, in an attempt to standardize what was then a system of wildly disparate science education in high schools across the U.S. Biology was given primacy in that ordering in part because the late 19th century experienced a flowering of interest in the natural world, and also because it was perceived to be less daunting intellectually than either chemistry or physics.

This has been my operating assumption throughout the past 3 years off reteaching & preteaching math to C.

Math is key.

I assume this new publication is drawn from the same Sandler/Tai survey finding that only solid math achievement in high school, not AP science courses, predicts success in college science:

Mathematical fluency is the single best predictor of college performance in biology, chemistry, and physics, giving a strong advantage to students whose high school science courses integrate mathematics. "Draining the math out of high school coursework does students a disservice," Sadler says. "Much of college biology, chemistry, and physics are taught using the language of math, so students without fluency quickly become lost."

Last but not least, here's what the famous Toolbox study has to say about math and college completion rates:

The highest level of mathematics reached in high school continues to be a key marker in precollegiate momentum, with the tipping point of momentum toward a bachelor's degree now firmly above Algebra 2. But in order for that momentum to pay off, earning credits in truly college-level mathematics on the postsecondary side is de rigeur. The world has gone quantitative: business, geography, criminal justice, history, allied health fields—a full range of disciplines and job tasks tells students why math requirements are not just some abstract school exercise. By the end of the second calendar year of enrollment, the gap in credit generation in college-level mathematics between those who eventually earned bachelor's degrees and those who didn't is 71 to 38 percent (table 21). In a previous study, the author found the same magnitude of disparity among community college students in relation to earning a terminal associate degree (Adelman 2005a).

My goal: C. needs to be able to take college courses in math after he graduates high school.

The math department doesn't seem to share this goal, judging by the chair's reaction the one and only time I raised it with her.

Me: "Christopher needs to be able to take math in college. That's our family goal."

Math chair: "He needs to take math to graduate high school."

End of discussion.

behavioral studies of creativity

John Wills Lloyd, of teach effectively, left links to 4 studies of creatitivy.

math dad writes textbook

Some parents pitch in with homework when kids get bad grades in math. Nicholas Aggor literally wrote the book.

The Riverview engineer was so distressed when sons Samuel, 14, and Joshua, 13, brought home bad marks, he took it upon himself to rewrite their textbooks chapter by chapter.

Four years later, they are in advanced classes and the Ghana native's pet project has become a passion that's produced a math curriculum for grades kindergarten to nine -- 14 books in all. And soon, it may not be just his kids whose grades are improving.


[C]urriculum director Paula Daniels said she wants copies by September for parents as a tutoring guide.


"There's step-by-step instruction -- if the kids don't get the concept from the teacher, they can just about teach themselves," said Shelley Zulewski, a math teacher at Riverview's Seitz Middle School, where Aggor's books will be the sole text for 10th-grade geometry and a supplement to other grades.


The key to his texts, Holloway said, is that children can understand them. Aggor uses marbles, board games, sports and other examples kids understand to explain math concepts.

And he brings together the best of old-fashioned arithmetic and the "new math" concepts that baffle many parents, according to Holloway.


"It's all broken down to where it's not all a bunch of mumbo jumbo that kids can't figure out ... It's a meeting of both thoughts of education, the old school of 'just do this and you can balance your checkbook,' and the new way of exploring and understanding it."


John Bruwer of Brownstown Township said he gave a copy of Aggor's book to his 13-year-old, Darron, out of desperation last spring. Even though Bruwer is an engineer, he had trouble helping his son with his eighth-grade math homework from Patrick Henry Middle School in the Woodhaven-Brownstown School District.

"Darron would come home from school, just having gone through a chapter, and would really struggle," said Bruwer, a statistical problem-solving coach with Chrysler Corp. "I'd have to really read through it to make sure I could explain it to him.

"(Aggor's book) took the fear away of understanding math," Bruwer said. "He became more relaxed and more self-reliant -- he'll try the examples on his own. Basically, I was cut out of the equation."

Dad's math book makes the grade
via Gadfly

I'd love to take a look at these books. I bet they're good.

Thursday, July 26, 2007

creativity, knowledge, and the project method

For reasons that escape me now, I came across three separate articles on the subject of creativity yesterday, each one quite good, or at least interesting.

The contradictions amongst the three (and within the text of one) are interesting as well.

The one tenet upon which all 3 agree: high levels of knowledge are essential to creativity - knowledge meaning knowledge stored in longterm memory, not on the internet.


    Carson had long been struck by the idea that creativity is marked by a bringing together of seemingly unrelated ideas, memories, images, and thoughts. Cubism is creative, for instance, because it juxtaposes images of living things with the hard edges and sharp angles of geometry. Might it be the case, she wondered, that creative people have a slightly leaky mental filter? If so, then perhaps they do not dismiss as easily as the rest of us “irrelevant” ideas that pop into their heads, but instead entertain them long enough for one of them to connect with another thought that is kicking around – giving birth to a novel, creative idea.

    There is little doubt that screening from conscious awareness that which is irrelevant to your immediate needs helps focus concentration. It may also be good for mental health, since paying attention to every little sight, sound, and thought can drive you batty. Indeed, reductions in this filtering mechanism, called latent inhibition, have long been linked with a tendency to psychosis. But Carson wondered whether that “failure” might also spur original thinking. To find out, she and colleagues had 182 Harvard students undergo tests in which they listened to repeated strings of nonsense syllables, heard background noise, and saw yellow lights on a video screen. The students also filled out questionnaires about their creative achievements (which is how Carson identified all those composers, scientists, and the rest), and took standard intelligence tests.


    Comparing the measures of the students’ latent inhibition (how many of those noises and lights they noticed) with their IQ scores and creativity, the scientists found that the more creative had significantly lower scores for latent inhibition than the less creative.


    “Getting swamped by new information that you have difficulty handling may predispose you to a mental disorder,” Carson says. “But if you have high intelligence and a good working memory, you are more likely to be able to combine bits of new information in creative ways.”

    The “high intelligence” part of Carson’s statement is key. Studies going back to the 1980s have shown that an inability to filter out extraneous perceptions and thoughts is linked to mental illness, in particular schizophrenia. Says Carson, “Highly creative people in our studies showed the same latent-inhibition patterns found in other studies of schizophrenics.” But if creative individuals are often characterized by an eerie ability to see connections and make associations that are beyond the rest of us, how do they manage to do so while staying sane?


    Carson’s answer: high IQ. There is no dearth of intelligent people who lack any spark of creativity, or of creative people who are not, by conventional measures, particularly brilliant. But some minimal level of intelligence is required for creativity. The reason is that in order to generate novel combinations, it helps to have a wealth of mental elements to work with. Without a sufficient supply of elements that can be combined in an original way, creativity is impossible. Since intelligence is generally associated with a large store of such knowledge, the greater the intelligence the larger the potential supply of elements that can be combined in an original way.

    “We saw creativity increase as IQs climb to 130 and even up to 150,” Carson says. But in those with average IQs, around 100, reductions in latent inhibition did not boost creativity. High intelligence, she adds, “should help you to better process the increasing information that goes along with low latent inhibition. To be creative, you can be bright and crazy, but not stupid.”

    Why Mad Scientists Are Mad: What's Behind the Creative Mind?
    by Sharon Begley
    In Character

    In Character: How would you define creativity?

    David Gelernter: The ability to see a relation between two seemingly unrelated ideas, and to draw conclusions. This is the standard definition in cognitive psychology and philosophy of mind; it’s the definition I used in my book about thought and creativity (The Muse in the Machine). But we also need to include visual and musical creativity – the ability to invent and manipulate images or musical phrases. These seem to well up by themselves out of mental freshwater springs, so to speak. But I suspect that if we studied them minutely, we’d discover that new visual or musical ideas are suggested by something or other – they do have something to do with unsuspected relations between thoughts.
    The mind is more supple when you’re young; you’re more apt to hit on novel combinations. But there’s a countervailing process: as you get older, you know more and you’ve experienced more. You don’t move through the branches as nimbly as you once did, but the forest is larger, which makes the possibilities greater.


    We can encourage creativity indirectly by stocking the human brain (like a trout pool) with the information it needs in order to exercise whatever inborn creative faculty it was born with. I’m not sure I’m in favor of encouraging creativity, but I’m definitely in favor of education.


    Creative children are born with the ability to swing like chimpanzees through the branches of the “memory forest”; they’re able to travel far and fast through the canopy. But they need a large enough forest to travel through. . If they live in a puny forest, their remarkable gift for branch-swinging won’t ever come out. A good education creates a large, lofty forest.

    David Gelernter, interview
    In Character

    Around the same time, psychologist Joy Paul Guilford of the University of Southern California noted that intelligence did not mirror the totality of a person's cognitive capacity. In the late 1940s Guilford developed a model of human intellect that formed the basis for modern research into creativity. A crucial variable is the difference between "convergent" and "divergent" thinking. Convergent thinking aims for a single, correct solution to a problem. When presented with a situation, we use logic to find an orthodox solution and to determine if it is unambiguously right or wrong. IQ tests primarily involve convergent thinking. But creative people can free themselves from conventional thought patterns and follow new pathways to unusual or distantly associated answers. This ability is known as divergent thinking, which generates many possible solutions.


    Convergent thinking is also required for a creative breakthrough. Inspirational thunderbolts do not appear out of the blue. They are grounded in solid knowledge. Creative people are generally very knowledgeable about a given discipline. Coming up with a grand idea without ever having been closely involved with an area of study is not impossible, but it is very improbable. Albert Einstein worked for years on rigorous physics problems, mathematics and even philosophy before he hit on the central equation of relativity theory

    Unleashing Creativity
    Scientific American Mind
    by Ulrich Kraft

    another 21st century skill that isn't

    Working collaboratively in groups is the last thing you should be doing if you want to solve problems creatively.

    In many cases, creativity seems to emerge unconsciously, often when you are thinking of something else. That may explain the responses people gave to a survey about where and when they are most creative. Nearly 20 percent of American adults say they think most creatively in their cars.... Respondents also said the ideal conditions for their creative thinking were solitude and quiet. When asked to complete the sentence, “My most creative ideas come when ... ,” 66 percent chose “I am alone,” with 47 percent opting for the closely akin “it’s quiet and there are no disruptions.” Interestingly, given the culture’s infatuation with brainstorming, only 24 percent chose “I’m working with others.”


    ...[C]reative people ... more likely to be introverted than extroverted, independent, enthusiastic, and hard-working.... Introverts are more likely to tolerate the long hours of solitary thought necessary for creativity. (Despite the popularity of brainstorming and group problem solving in corporate America, study after study has shown that these techniques produce fewer workable creative ideas than does solitary problem solving. In fact, people working alone generally hit upon better ideas than do the same people working together.) People who are independent or even iconoclastic are less likely to reject a novel idea without first giving it some thought.

    Sharon Begley

    When was the last time you heard anyone say a camel is a horse designed by a committee?


    Gelernter says conversation is essential to creativity.

    Certainly true for me.

    Conversation on a blog - even better!

    the sun, the moon, and the stars at instructivist

    Incredibly cool post at Instructivist!

    C. & Ed are going to love it.

    While you're there, be sure to read his post on the NY Regents History exam, too. The suffragette question reminds me of Celebrity Jeopardy.

    algebra in jobs

    from instructivist:

    Geometry and algebra can crop up in all sorts of situations.

    I was once teaching ESL to a Mexican worker in a limestone factory that made stones for construction, and unexpectedly found an opportunity to teach some math. Part of his job was to make templates for arches. All he was given by the customers was the chord and rise. He needed to know the radius to make the template. He was desperate to learn how to calculate the radius. After some research, I was able to teach him the formula and how to use it. He was beaming. At other times he needed to make templates for ellipses. I taught him how to calculate the distance between foci and how to construct an ellipse with nails, a string and pencil. He was beaming once again.

    All this mixes geometry and algebra in the practical world.

    from Googlemaster:

    I write software for a living, so you could say that I use algebra every day. I mean, what is:

    double avgRevenue = totalRevenue / numSold;

    if it isn't algebra? Okay, that's a simplistic example that isn't representative of what I really do, so maybe that's just arithmetic with symbols... but wait, isn't "arithmetic with symbols" just a long-winded phrase for "algebra"?

    The software that I write deals with statistics, econometrics, linear programming, etc. It's not your average shopping cart application. But even your average shopping cart application contains code with symbolic representations of mathematical equations.

    IMHO, this goes back to the idea that, sure, many people will never use anything higher than basic arithmetic in their careers, but if you stop your math learning with basic arithmetic, you've effectively cut yourself off from being a doctor, a pharmacist, a computer programmer, a veterinarian, any sort of engineer, an economist, a patent attorney, and a host of other enjoyable white-collar careers.

    It's always nice to have choices.

    I knew I forgot an important one. Architects use both algebra and geometry.

    from Steve H

    I suppose that a high school teacher could argue that (after all of the damage has been done and all of the doors have been closed) there are other things that would help these students for their daily living more than algebra.

    However, it would be nice to see these teachers still trying to open doors no matter how closed they seem. With a little more effort on their part, and a willingness of the student to try one more time, the teacher might completely change their world.

    This is so much more important than learning to balance a checkbook. By the way, a little algebra will greatly help kids understand how a mortgage (the time value of money) works.

    from susan j:

    instructivist and googlemaster both did a great job. I would extend the first answer to any building trade or any activity where you use formulas.

    Many jobs require setting up and using spreadsheets. Even secretaries need this skill nowadays.

    People with cash flow problems may need to use algebra in deciding how to juggle their various accounts.

    Tom Loveless on automaticity

    Question from Alison Corner, Principal, Pawtucketville Memorial, Lowell MA:

    Their is alot of emphasis in K-4 on teaching math as a problem solving exercise. How would you recommend incorporating teaching computation skills for automaticity?

    Tom Loveless:

    Some lessons have to be devoted to computation alone, including why procedures work. Automaticity involves both accuracy and speed so I would stress both in memorizing basic facts in addition, subtraction, multiplication, and division and in the use of algorithms.

    The State of Math Standards - transcript

    I don't know how I managed to remember automaticity (which I have in theory been trying to achieve with C.), but forget speed.

    Actually, I do know how: my years of ABA, when Jimmy and Andrew worked to a 90% criterion for mastery, dominated my memory of what mastery meant.

    One year in KUMON ("speed and accuracy") wasn't enough to overwrite 4 years of ABA, or however many it was....

    how to determine fluency for an individual child

    An adult-to-child proportional formula can also be helpful in setting performance aims. Measures are first taken of the student’s tool skills rate and the rates at which a competent adult performs both the tool skill and the target math skill.* The tool skill for answering math facts is writing random numbers without solving any problems. This fast-as-you-can number writing rate provides a ceiling for the fastest rate at which answers to math facts can be produced. The proportion obtained by dividing the adult’s performance rate on the target math skill by his or her tool skill rate is then multiplied by the student’s tool skill rate. The resulting figure is the fluency aim for the student. For example, if the adult solves math facts at a rate of 60 correct answers per minute and can write 120 random numbers per minute, his or her target skill to tool skill proportion is .5 (60 divided by 120). Based on the adult-child proportional formula, the fluency aim for a student whose tool skill rate is 80 would be set at 40 correct answers per minute (80 times 0.5). Providing direct and repeated practice on the relevant tool skills may be an effective way of improving the overall fluency of some children. Alternative modes of response, such as answering orally, should also be considered for children who exhibit very slow writing or poor fine motor control.

    Do Your Students Really Know Their Math Facts: Using Daily Time Trials to Build Fluency (pdf file)
    by April D. Miller and William L. Heward
    p 134
    Intervention in School and Clinic, Volume 28, Number 2, November 1992

    Effects of sequential 1-minute trials with and without inter-trial feedback and self-correction on general and special education students' fluency with math facts

    * tool skills; component skills; composite skills - see speed test

    Dude just did that in his head!


    Wednesday, July 25, 2007

    algebra 1 & geometry, too

    This exchange from the Ed Week chat is interesting:

    Question from Jon Joseph Madison WI School System:

    The most important mathematics I teach high school learners is the math of daily living - doing taxes, computing percents, understanding a mortgage. One of the reasons that parents can't help the students with the type of math we currently teach is because they have never used it since they graduated from high school. Granted, some students need algebra, geometry and calculus but for the vast majority is this really required?

    Tom Loveless:

    Students must have a thorough grounding in arithmetic, that's for sure. But an awful lot of jobs in the new economy require knowledge of algebra and geometry as well, and there is no reason why the vast majority of students can't master those topics. Other nations do it, and so can we. By the way, Teaching the New Basic Skills by Richard Murnane and Frank Levy is a great book to read on this question.

    I've always wondered about this - how much math do regular people actually use beyond arithmetic?

    Geometry makes sense, but how do regular folk use algebra on the job? (I don't object to state standards requiring mastery of algebra 1; quite the opposite. But I haven't been able to see where many people use algebra in the workplace.)

    I had no idea Murnane addresses this; I have the book upstairs.

    The State of Math Standards

    newsflash: I've just learned that Ed Week has scheduled an online chat re: math standards for July 25.

    Which, I see, turns out to have been today.

    Who knew?

    So I've missed the Ed Week chat. Lucky for me, they've posted the transcript.

    Good to be reading Tom Loveless again:

    Question from Leslie Skantz-Hodgson, Director of Curriculum and Media Instruction, Smith Vocational and Agricultural High School:

    Just what exactly are other countries doing, that we are not, to get high performance in math?

    Tom Loveless:

    No one knows for sure, but I'll give you my best guess. High achieving nations value mathematics, stress its importance starting even before a child enters school, have high expectations for mathematical learning in the elementary grades, and put most students through a rigorous set of high school math courses. Elementary and middle school teachers have taken mathematics courses offered by college math departments. In addition, the curriculum is focused on only essential topics; textbooks are lean and concentrate on a few key ideas.

    Lots of K-12 defenders of the faith:

    Question from Jeanne Cerniglia, mathematics teacher, JRL Middle School, Southeast Local District, Wayne County, Ohio:

    There seems to be a move to go "back to the basics" (skills). While skill mastery may be lagging, my concern is in the area of critical applications - especially in the area of data analysis as it applies to functioning as a responsible citizen nationally and globally. Where can I find research that explores how to connect mathematics and general critical thinking strategies?

    I know the answer to that!

    There is no research to be found, because general critical thinking strategies are a myth. General critical thinking strategies are an urban legend; general critical thinking strategies do not exist. [see e.g.: Inflexible Knowledge: The First Step to Expertise; Critical Thinking: Why Is It So Hard to Teach?; The Mind's Journey from Novice to Expert]

    That is, general critical thinking strategies do not exist unless you happen to have a Ph.D. like me:

    Although this is not highly relevant for K-12 teachers, it is important to note that for people with extensive training, such as Ph.D.-level scientists, critical thinking does have some skill-like characteristics. In particular, they are better able to deploy critical reasoning with a wide variety of content, even that with which they are not very familiar. But, of course, this does not mean that they will never make mistakes.
    Critical Thinking: Why Is It So Hard to Teach?
    by Daniel Willingham

    I am praying I do possess a general critical thinking skill or two, because otherwise I am not going to be able to wrestle CHAPTER ONE to the ground.



    I've just discovered that the first chapter of Linda Persltein's new book Tested is posted online.

    I read this chapter over breakfast one morning last week; I couldn't put the book down 'til I'd finished. It's the opening scene in the book, the scene in which the principal of Tyler Heights Elementary School is waiting for the state scores to arrive. The tension is ferocious.

    When the envelope finally comes and the principal opens it and sees the scores, I cried.

    That doesn't happen too often to me when I'm reading books about education policy.

    Maybe it should happen, but it doesn't!


    There's a very nice passage re: Saxon Math later on in the book.

    Will post shortly.

    cortical homunculus

    image posted at Mind Hacks

    This model shows what a man's body would look like if each part grew in proportion to the area of the cortex of the brain concerned with its sensory perception.
    Natural History Museum

    I'm reading Rodolfo Llinas' I of the Vortex.

    The book is pretty spectacular, possibly life-altering.

    First time I've ever felt queasy reading about neuroscience, though.

    update from Mr. Person:


    I'm feeling faint.

    developing intelligence summary/review
    Chris Chatham home page
    interview with Rodolfo Llinas
    recent publications
    I of the Vortex (MIT Press)

    The Future of the Brain by Steven Rose
    On Intelligence by Jeff Hawkins
    Jeff Hawkins posts

    Tuesday, July 24, 2007

    Looking for Data on Math Curricula

    I'm collecting data about the efficacy of math curricula. What I'd like is to get some statistical evidence that shows the effect of various mathematics curricula on student outcomes. Any other data--I'm thinking covariates that might affect the delivery of the mathematics instruction--would be appreciated.

    So, for an example, how many students score proficient (or whatever) on the NAEP who use Saxon math? And the same for Everyday Math, Connected Math, etc.


    NCLB 3.0: Dancing and Sculpture?

    No Child Left Behind means a lot of things to a lot of different people. Who knew that one of NCLB's "failings" was that it didn't spend enough time or money on ensuring our struggling schools had enough music, dancing, and sculpture classes? Apparently, that was a top-of-mind concern for Democratic presidential candidates at last night's CNN/YouTube debate.

    Gov. Richardson, and to a lesser degree Sen. Biden, demonstrated that too many people either don't understand NCLB or choose not to believe NCLB successes. And that's the real shame. As a community, we should be rallying around the effective teaching, research-based instruction, strong assessments, and student achievement mandated under NCLB. Instead, we are embarrassed by our past support of it and quick to denounce it in public forums.

    If you've read my past ramblings, you can guess that Eduflack has a lot to say about last night's debate. Check it out at

    NCLB works, and there's the data to prove it and students, teachers, and classrooms to exemplify it. It's time for those success stories -- and not the negative crowing of the status quo -- to define what NCLB really stands for.

    Monday, July 23, 2007

    Saxon bar models

    I was all set to scan in a page from Saxon Math 7/6 when I realized I should search the old site first.

    et voila!

    Saxon bar models.

    large image here

    The sad news is that the ktm-1 post gives the exact same problem I'm going to be having C. do tomorrow.

    Two years after we did it the first time.

    Anyone here think two years might be a tad long between exposures?

    This time around we are going to keep at it. We're going to be doing Saxon bar models for fractions, decimals, and percent, and we're going to carry on doing Saxon bar models for fractions decimals, and percent until C. reaches fluency.

    Plus we're going to be setting up equations the Saxon way and solving them, and we're going to be doing that, too, until C. becomes fluent at setting up and solving simple equations.

    And we're going to do many, many, many story problems.

    starter variables in Saxon Math

    The Saxon way of setting up equations, fyi, involves using variables that are abbreviations of the unknown you're looking for.


    What number is 1/4 of 100?
    WN = 1/4 · 100


    25 is what fraction of 100?
    25 = WF · 100

    Saxon also uses WP for what percent. (I'll probably throw in some WDs for what decimal to boot.)

    These Saxon "starter variables" are an ingenious way to spare working memory, IMO. Later on, in Algebra 2, he transitions students to x and y when he explains, in a lesson functions & function notation, that x is typically used to represent the independent variable & y the dependent variable.

    I did that lesson this weekend; it was a revelation. All these years using x & y -- alongside the concepts of independent and dependent variables -- while having no idea that the two were connected.

    Good grief.

    Saxon equations with starter variables

    "relative effectiveness of the primary senses"

    Some of you may remember this anecdote from the old site:

    My sister-in-law is a federal prosecutor in Philadelphia. One day we were talking about learning styles. Pace Dan Willingham, I don't believe in learning styles, but since everyone else does I don't automatically launch into a cognitive science lecture when the subject comes up.

    So we were talking about learning styles, and I said something about visual learning styles, and my sister-in-law said, "Everyone has a visual learning style."

    "That's the first thing they tell you about presenting evidence to juries. If you want the jury to remember what you've said, you have to give them a visual."

    I know she's right about this because..... because I just know. Visual memory was a topic Temple and I never managed to nail down while writing Animals in Translation. Memory for things visual is strong; that much we knew. But we couldn't figure out the research basis for this belief, or how exactly it related to the book's thesis or to the visual thinking of autistic people and animals.

    A loose thread.

    Ken Spencer's books posted online

    Sunday I came across the books Temple and I needed: Media & Technology in Education (1966) and The Psychology of Educational and Instructional Media (199). Both look terrific, and both are posted in full.

    This chart appears on page 1 of Chapter 5: Media and Technology in Education: Theory and Practice:

    Figure 5-1. The Relative Effectiveness of the Primary Senses



    Media and Technology in Education: Theory & Practice
    by Ken Spencer
    Chapter 5: Human Information Processing and the Audiovisual Approach to Education Educational
    (pdf file)

    Assuming this is right, and I think it is, it explains a lot.

    For one thing, it accounts for some of the effectiveness of peer tutoring and collaborative learning.

    What's going on in peer tutoring and collaborative learning? Talking out loud!*

    When I gave the list to Ed he said, "That's why you learn so much from teaching. I never forget anything I've taught."

    True for me, too.

    I always thought that was because having to teach a subject forced you to organize it in your own mind, which I'm sure is true. But part of the effect probably stems from the simple fact that you're talking out loud.

    Remember the Commenter who suggested I have C. teach math to me?

    We're starting tomorrow.

    * The book Why Johnny Can't Write has fascinating material on talking-out-loud as a study technique. Will get to that at some point.

    Anchorage School District Middle School Math Textbook Adoption

    According to their website, the Anchorage School District has adopted MathScape as their 6-8 grade math text book.

    Of course there was never any doubt about whether they were going to adopt "reform" math. The four choices they had to decide upon were, Connected Math, Math in Context, MathScape, and Math Thematics. I suppose it was just a matter of figuring out which one was the fuzziest.

    According to the School District memorandum, here are some of the strengths of the program (emphasis mine)

    The program provides the following for the needs/rights of students:

    • know the purpose of learning, including objectives, standards, goals, criteria and evaluation rubrics
    • choose from a variety of strategies to explore, solve, and communicate math concepts
    • engagement through a variety of activities, which may include independent projects, cooperative learning, manipulatives, technology, collaborative work, etc.
    feel connected and free to take risks
    • a belief that math can be learned
    • opportunities for self-monitoring and self-reflection
    • make connections to real life applications
    • support at individual learning levels

    • Teacher makes meaningful connections between math and real-life.
    • Teacher has high expectations for success and achievement for all students.
    • Teacher uses a variety of instructional and assessment strategies (differentiated instruction, cooperative learning, exploration & learning extensions, use of manipulatives and technology, and other best teaching practices).
    • Teacher clearly states classroom expectations, and content and language objectives. • Teacher provides time for student reflection & meta-cognition.
    • Teacher communicates with and is available to parents and students.
    • Teacher receives appropriate and ongoing professional development & training (knowledgeable of pedagogy, content, and vertical alignment of curriculum).
    • Teacher is provided adequate time and opportunity for grade-level and vertical collegial collaboration and support.
    You notice there is nothing about fluency, mastery, clear examples, computation, standard algorithms, etc...

    What do you expect from a school district that uses Everyday Mathematics?

    Cross-posted at Parentalcation


    inventory: lost items, Monday morning, 7-23-07:
    • my fanny pack (empty, thank God, but no longer occupying its designated spot inside basket on top of office bookshelf)
    • my watch (apparently the watch isn't replaceable, either, especially seeing as how I can't remember the company that makes it)

    Also, Andrew broke his bed frame this weekend.

    On the bright side, Andrew’s Care Trak is working great. Terrific news video here (CareTrak & autistic kids).

    Forget jetpacks & robots.

    In my own personal futureworld, everything I own will have its own chip with its own frequency, so it can tell me where it is after the borrowers have stuffed it down inside the sofa cushions. A couple of years ago I actually bought the "Electronic Locator" Sharper Image sells, but it stopped working after about five seconds & I didn't have the patience to try to get it running again.

    missing Andrew

    calculus textbook recommendation

    from anonymous:

    You might want to try the Larson 8th ed or the Finney books. Much more "user friendly" [than Foerster] so to speak. I use Larson in my AP Calc (both AB and BC) but supplement from several others. Foerster has a supplement that goes with it called calculus explorations (and one for precal also) that are excellent.

    Pre-calculus and Calculus at Key Curriculum Press


    Sunday, July 22, 2007

    percent troubles

    Students who can work with fractions and decimals still have an inordinate amount of trouble with percents. They can answer the questions, “What decimal is 3/4?” and, “What percent is .75?” and yet not be able to write 3/4 as a percent. They can know that .2=20%. They can work with the problem, “80% of what number is 40?” and can know that 100-20=80 and yet not have any idea of how to solve the problem, “Jackson paid $40 for a jacket which was on sale for 20% off. What was the regular price of the jacket?” (How many students would solve that problem by taking 20% of $0=$8, and concluding erroneously that the regular price must be $0+$8=$48?!)

    Math Word Problems Decimals & Percents Level B, p. iv
    by Anita Harnadek
    ISBN 0-89455-821-8

    hyperspecificity in autism
    hyperspecificity in autism and animals
    hyperspecificty in the rest of my life
    hyperspecificity redux: Robert Slavin on transfer of knowledge

    Inflexible Knowledge: The First Step to Expertise
    Devlin on Lave
    rightwingprof on what college students don't know
    percent troubles
    what is 10 percent?
    birthday and a vacation

    Information about Foerster's Calculus book

    I'd appreciate any information about Paul A. Foerster's Calculus Concepts and Applications book.


    Catherine here - dropping into ec's post

    Here's a link to calculus text recommendations at ktm-1. (This may be all of them, but if not I'll try to post the other links later.)

    bonus: calculus advice from Rudbeckia Hirta.

    Barry's calculus text recommendation.