kitchen table math, the sequel: 7/1/07 - 7/8/07

Saturday, July 7, 2007

Nuts and bolts of math

Some people here write about the nuts and bolts of math, but I think many of us use our time well to TEACH math to our own kids, and then we go to KTM to vent our frustrations over paying taxes only to end up doing this ourselves.

I'm one of the crazies that teaches math to their kids AND writes about it.

Clueless Mom blog

The Faulty Test

(Cross-posted from the Eclectic Educator)

Constance Kamii, author of Young Children Reinvent Arithmetic: Implications of Piaget's Theory offers her belief that children cannot be taught place value.

Her conclusion is based on a post-test she created in which most children, after direct instruction in place value, fail to demonstrate understanding of place value.

Here's the test from her book:

1. I put 16 chips out and asked the child to count them and to make a drawing o f"all these." The children made drawings either in a line or in a bunch.

2. The child was asked to write "sixteen with numbers" on the same sheet to show that there were 16 chips.

3. The child was asked what "this part" meant as I circled the "6" in "16" as shown. The answer was to be indicated by circling some of the chips drawn on the paper.

4. The child was asked what "this part" meant as I circled the "1" in "16." The answer was again to be indicated by circling some of the chips drawn on the paper.

5. Finally, the child was asked what "the whole thing" meant as I circled the "16" and probed into the relationships among "16," "6," and "1." For example, when a child circled [one chip instead of ten] I asked why "these" (the six chips circled) and "this" (the one circled) were circled but not "these" (the 9 leftover chips).

Kamii found that none of the children in her class said that the "1" in "16" stood for 10.

But there's a problem. I tried the test myself, and see that it's faulty. A loose collection of ten does not, indeed, represent the tens place in base ten.

The first two pictures depict Kamii's test. The third picture depicts a more appropriate test.

Tester circles digit "6" and asks "What does this part mean?"

Child selects the correct number of "ones"

Tester circles the digit "1" and asks "What does this part mean?"

Single chips don't represent the tens place, so the child becomes confused and circles one chip instead of all ten

Here's a better test

This test offers a place ten chip to circle. Perhaps the child can be given several, from which she should select just one to represent the "1" digit in the number "16."

From my preliminary research, it appears that Kamii's faulty test is the reason why reform math purists don't teach place value. Constance Kamii seems to be cited a lot, including by the publishers of TERC Investigations.

Teachers trained in this method won't use base ten blocks or any form of manipulatives to assist in understanding.

Instead, they offer example after example of equations using numbers with two places such as (20 books plus 15 books is 35 books) but without explanation of what the numbers mean. And they wait for a child to "understand" internally.

The reform math purists (of which some teacher are, while others are eclectic), tend to be very rigid about not teaching place value by any direct means at all.

As one pre-service teacher instructor put it:

"Have you seen the new TERC Investigations books? They have totally caved to some pressure and I wish I knew where it was coming from.

"They are now advocating the use of the base ten blocks to teach regrouping (long rods that represent 10, single cubes that represent 1, and flats that represent 100). I was shocked and deeply disappointed."

Disappointed? Shocked? To believe that teachers must not teach place value is a far cry from the conclusion that children don't understand place value when taught using base ten blocks.

And the basic premise, that children don't understand it when taught it, is based on a flawed test. But worst of all, to suppress all direct instruction in the belief that children can't learn place value by direct teaching is nothing less than cruel and myopic.

Good for TERC Investigations Version 2 for "caving." Still, I don't want TERC in my backyard.

special ed & NCLB

large version
source: Sysyphean Tasks

eduwonk links to an Ed Week story on NCLB & special ed, about a report out from the National Center for Learning Disabilities:

Almost fourteen percent – some 6.6 million – of this nation’s school-age children receive some level of additional support through special education. These children come from all race and ethnic groups and speak many different languages. Significant numbers are served by other school programs, such as Title I and English Language services, in addition to special education.

Many are indistinguishable from students who do not receive special education services.


It should be noted that vast differences exist across states regarding the percent of students receiving special education services. In the 2003-2004 school year, state rates ranged from a low of 10.5 percent in California to a high of 20.2 percent in Rhode Island.


Special education classification has too frequently been used to diminish the expectations for the students designated as eligible for such services and to minimize the responsibility of general education teachers and administrators
for their progress. Also, data suggests that special education classification is used to segregate minority students, particularly Black boys. Black students represent more than 20 percent of those receiving special education yet make up only 17 percent of public school enrollment.
Rewards & Roadblocks: How Special Education Students are Faring Under No Child Left Behind (pdf file)
National Center for Learning Disabilities

My good friend O. was in from Los Angeles last Christmas. Her son, now grown, is autistic - high-functioning - and attended LA Unified schools, where his SPED classmates were always black. In every school, she said, it would be her son in a classroom filled with black kids.

I had the same conversation with a SPED parent here when I asked if she knew how our disadvantaged kids are doing. She said that in fact she did know how they were doing, because they were all in the same class with her child, who has fairly significant disabilities though I'm guessing a normal IQ. (In other words, her child has real, significant, and fairly obvious problems, but isn't as challenged as my two autistic kids are.)

Another parent tells me that 25% of black students in Irvington are classified SPED.

update from Kathy Iggy:

I don't know what the exact statistics are here, but when Megan was in SPED preschool, one year she was the only Caucasian child and another year there were only 2 Caucasian kids in a class of 14. In self-contained last year, the class of 10 was half African-American. The overall percentage of African-American enrollment in our district is probably around 10%. Makes you wonder.

This is true everywhere.

At least, it's been true everywhere I've looked.

time to focus on results, not paperwork in SPED

Where are the cupcake recipes?

An anonymous commenter wrote:
Isn't it odd that none of you ever post anything about the nuts and bolts of teaching math? Maybe you should share cupcake recipes.

Of course, he's wrong about the nuts and bolts. Many of us are teaching math at home because we don’t think our schools are doing an adequate job. KTM has been the single most important source of “nuts and bolts” information for me. And, yes, I do my share of venting my frustration about this situation. That’s only natural, I think.

However, we have been remiss about sharing cupcake recipes. Therefore, I thought I would post a recipe for a traditional Latin American cake that has won raves for me. It would be appropriate to take to school for those Spanish language or “cultures around the world” events.

Warning: As you can tell, this cake is not low cal.

“Tres Leches” (Three Milk) Cake
1 yellow or white cake mix baked as directed in 13 by 9-inch pan
For topping mixture:
1 cup evaporated milk (or whole milk)
1 can sweetened condensed milk
1 pint heavy whipping cream
1/2 cup rum (optional)
To finish:
Whipped cream, to cover cake
5 ounces sweetened coconut or fresh strawberries or whatever else strikes your imagination to top the cake
Let cake cool 5 minutes. Poke holes in the cake with a fork or a skewer all over. Pour topping mixture over cake. Let mixture absorb completely in refrigerator.Cover with whipped cream or cool whip. Add coconut or strawberries on top.

Friday, July 6, 2007

value added comes to Westchester

The Cohoes city school district, outside Albany, is considering a gifted program for elementary students and adding college-level courses after discovering that its top students improved less on standardized tests in the past two years than everyone else in the district.

In Ardsley, N.Y., a Westchester County suburb, administrators intend to place more special education students in regular classes after seeing their standardized test scores rise in the last year.

And as the New York City Department of Education begins grading each public school A to F for the first time this fall, more than half the evaluation will be based on how individual students progress on standardized tests.

All three changes resulted from an increasingly popular way of analyzing test scores, called a “growth model” because it tracks the progress of students as they move from grade to grade rather than comparing, say, this year’s fourth graders with last year’s, the traditional approach.

Concerned that the traditional way amounted to an apples-to-oranges comparison, schools in more than two dozen states have turned to growth models. Now a movement is mounting to amend the federal No Child Left Behind Act, which is up for reauthorization this year, to allow such alternative assessments of student progress.

money 'graph:

[M]any school officials in affluent suburbs favor growth models because they evaluate students at all levels rather than focusing on lifting those at the bottom, thereby helping to justify instruction costs to parents and school boards at a time of shrinking budgets.

Maybe, maybe not.

Some of these schools are going to be slide and glide. I think the Scarsdale school system is a candidate - though I'm not sure how the tutoring there factors in. (may have to hit refresh a couple of times)

In New York, education officials are developing a statewide growth model that will be in place by the 2008-9 school year, to be used as an additional way to measure student learning. Fifteen New York school districts, mainly in the Albany and Catskill regions, have experimented with growth models on their own through a voluntary program started by two regional support educational agencies in 2005. The districts typically pay these agencies from $1,000 to $6,000 to train administrators and staff, and an additional $2.50 a year for each student for the data analysis, which is partly reimbursed through state aid.

“There is absolutely a need for this kind of data,” said Timothy G. Kremer, executive director of the New York State School Boards Association, which represents about 700 districts. “It keeps the focus on student achievement, and not on whether you’re going to pave the parking lot or who’s going to get hired as next year’s coach.”


Even some supporters of growth models have expressed concerns that they could shift attention and resources away from the neediest students. Kati Haycock, director of the Education Trust, an advocacy group for disadvantaged children, said she was worried about growth models’ focusing too much attention on students at the top. “It risks so broadening the federal government’s involvement that its historical role will be dissipated,” she said.

Thanks, Kati.

money 'graph, part 2:

"When you look at achievement, every single wealthy suburb has high test scores,” noted Theodore Hershberg, a professor of public policy and history at the University of Pennsylvania. That’s a terrible way to measure the performance of a school or an individual teacher because what you’re really looking at is family background or family income.”

In the high-performing Ardsley schools, where more than 87 percent of the students passed state reading tests this spring, district officials have long mined scores on their own, compiling a thick data book for review and coining the saying: “In God we trust, everybody else bring data.” [Didn't Margaret Spellings coin that phrase? I certainly remember her using it.]

But this year, they employed a more sophisticated growth model, which showed, for instance, that seventh-grade special education students had benefited from learning in regular classes. So this fall the district will expand the mainstreaming to the elementary and high schools. “This gives us the ability to measure whether a program has any teeth or is all fluff,” said Richard Maurer, the superintendent. *


Ardsley is next door to Irvington.

Has a major autism program.

So.... naturally we moved to Irvington, not Ardsley.

On the other hand, Ardsley also has TERC.

So it's probably just as well we're here.

Which raises a question.

Value-added compares each child to himself. Does he achieve at least one year's growth each year in school, one year's growth as one year's growth is defined for him?

That's fine, but if a school has always had TERC isn't your child's typical "one year growth" going to be pretty poor?

A kid has TERC for 5 years, then every year thereafter the school will be basing "data-driven" decisions in the revelation that your kid has once again achieved one year's growth....

So I'm thinking value-added isn't going to end the math wars.

This is amazing:

Cohoes school officials have spent more than $1 million on programs for their most struggling students in the past five years, and wanted to find out how much they had progressed. They learned that the lowest-level students were doing fine, while their high achievers were starting to fall behind.

Charles S. Dedrick, superintendent of the 2,200-student district, said that parents had complained that their children were scoring too low on the Advanced Placement exams to receive college credit, but he thought there was just a problem with the A.P. coursework. Now, after examining over time the state test scores of students in advanced classes, he sees a more systemic problem. So the district has made top-level students a priority, too, and is considering starting a gifted program, expanding A.P. and college-level courses, and adding an International Baccalaureate program to keep them challenged. [I am hoping that in addition to "challenging" these kids the school will actually teach them. My experience of "challenge" to date has meant top-down instruction at school, direct instruction at home.]

“The fact is we serve all students, and not just the lower-end students,” said Mr. Dedrick, who travels across the state to speak about growth models to school superintendents. “If you’re just concentrating on one group of kids, it’s not fair because both sets of parents pay taxes.”

Schools Move Toward Following Students' Yearly Progress on Tests

I see many signs that our high-end kids are underachieving. First of all, I know for a fact that C. has not read a single book even close to his reading level this year. [scroll down]

Beyond this, though, the fact that the district is dedicated to keeping kids out of accelerated and Honors courses, as opposed to recruiting more kids into these courses tells me kids aren't reaching their potential.

I also suspect - haven't taken the time to look at this closely - that we're in the same boat as Cohoes when it comes to AP scores. I could be wrong, but I don't think our kids are getting the scores they could and should be getting.

I don't think our average kids are doing well, either (though I don't know about high-end SPED kids. From what a couple of parents have told me, they're in better shape than the regular, non-SPED, average-achieving kids). Still, the rhetoric around here is Kati Haycock. Last winter the middle school principal [scroll down to end] told a large group of parents at a school board meeting that he was going to implement the middle school model because it would help the low-end kids. paraphrasing: "I only care about the lower kids, because they're the ones who are struggling."

A parent was told the same thing by another administrator: "We don't care about the high-achieving kids. They're fine." Words to that effect.

These are direct statements made to the parents of high-achieving kids. "I don't care about your kids" - this is the sentiment expressed openly and unapologetically as taxes climb each and every year.

Our taxes when we moved here 9 years ago were around $12,000/year; we're up to $22,000 this year.

And the middle school principal, in his first year on the job, feels perfectly comfortable telling a group of parents that he "doesn't care about the high kids."

In general there is a constant undercurrent - sometimes not so "under" - of hostility towards pushy parents who think their kids are smarter than they are. "Don't push your kids" -- this is the advice given openly to parents of high school kids in this $21,000/per pupil district.

"Everyone has a place."

When I told the math chair, back in 6th grade, that "Christopher needs to be able to take math in college" she said, "he needs to take math to graduate from high school." Subject closed. She had zero interest in my concern that he be prepared for college math.

And then there was the middle school principal's Back to School night skit on school quality.

He'd just come to Irvington from Albany, where he was principal at a school attended by disadvantaged kids.

"I don't like NCLB," he said, "but it has one good idea, which is evaluating your school."

How does one evaluate one's school?

"There are four criteria," he said.

"Number one is the school. You can evaluate the quality of your school by evaluating your school. But if you're putting everything on the school, that's too much."

The next 3 criteria were:

  • quality of the students
  • quality of the parents
  • quality of the community

Irvington, he said, had high-quality students, high-quality parents, and a high-quality community (which he knew because he'd been driving around town admiring the nice lawns).

ergo: the school is high quality

This was the first time any of us had ever laid eyes on the man, and he performed a skit - he had 4 middle school girls holding up big posters with the 4 school-quality criteria written on them - about How to evaluate your school that said nothing about the school, the teachers, the curriculum, or the students' achievement.

slide and glide

update from concerned parent:

I was talking to a friend whose daughter will be going to the middle school. They are only allowing something like 3% of the 7th graders to take algebra. Everyone else will be taught using Connected Math Program.

If that's not institutionalizing underachievement, I don't really know what is.

I think it's time for me to read Jay Mathews' book Class Struggle. I have the book, and have read the first chapter, but it's time to sit down and concentrate.

Mathews wrote the book, as I understand it, because he had the compare and contrast experience of having written a book on Jaime Escalante and then moving to Scarsdale where there was a notable absence of Jaime Escalantes.

Cheri Pierson Yecke writes about a similar phenomenon in The War against Excellence, a terrific book every parent with a middle school-aged child should read.

The War Against Excellence
value added assessment PDK
value added assessment FAQ page
interview with William Sanders
A new way of judging how well schools are doing

Jay Mathews on the class struggle
Jay Mathews column on wealthy schools, AP courses, SAT scores
are wealthy schools worse?
value added comes to Westchester

* Can we apply value added to character ed?

KUMON & the components of long division

anonymous left this comment:

I'm running out of patience with Kumon. They keep making my daughter repeat the same workbooks. She already knows the stuff, she's not moving forward and it's not helping her. There's an opportunity cost to making her do all these Kumon books when she should be forging through Saxon Math.

On the other hand, if I let her drop Kumon, will she ever master long division?

I have a friend who's had the same experience. Her daughter is stuck in a Level and can't get out!

I'm guessing that the Morningside people wouldn't allow a child to be stuck on anything. I think they'd drop the child back to the component skills and practice those skills to mastery. Then you'd hope to see contingency adduction - you'd hope to see the child take a soaring leap up to the complex, composite skill.

So: what are the component skills of division?



or....subtraction, estimation, multiplication....?

contingency adduction


the sudden appearance of new learning without further instruction once component skills have been learned to fluency (or automaticity - pdf file)

Automaticity and Processing without Awareness
The Importance of Automaticity and Fluency for Reading Comprehension

Behavioral Fluency: Evolution of a New Paradigm (summary for laypeople)
wholes, not parts: Martin Brooks and The Constructivist Classroom
whole math

fluency defined

Fluency is functionally defined by skill retention after a period without practice; skill endurance over longer intervals than encountered during practice; skill stability in the face of distraction; a performance that can be effortlessly applied to new environments; and a skill that adduces easily with other skills to form new repertoires (RESAA). Precision Teachers have found that fluency can be promoted by building the frequency of an accurate response to high rates.

Behavioural Fluency in young children with autism
David John Bonser

fluency defined for laypeople:

learning a behavior until it becomes second nature

contingency adduction
Behavioral Fluency: Evolution of a New Paradigm (summary for laypeople)
wholes, not parts: Martin Brooks and The Constructivist Classroom
whole math

top down teaching

Top-down [teaching] means that students begin with complex problems to solve and then work out or discover (with the teacher's guidance) the basic skills required.
Educational Psychology, 7th edition
Robert Slavin, p. 259
wholes, not parts: Martin Brooks and The Constructivist Classroomwhole math

precision teachings articles from teach effectively blog

Professional Papers in Special Education

I'm going to start with the Binder, et al paper: "Fluency: Achieving True Mastery in the Learning Process." (pdf file)

Here are teach effectively's posts on precision teaching. (Teach effectively says sentient beings should give The Cambridge Center for Behavioral Studies money!)

And here, courtesy of teach effectively, is I Speak of Dreams who posts this passage:

Precision teaching is not so much a method of instruction as it is a precise and systematic method of evaluating instructional tactics and curricula.

Assuming the method is easily adaptible to one's home, this is something afterschoolers desperately need. The time I have to keep C. on task is extremely limited; every minute has to count.

And yet I have no real means of figuring out whether what I'm doing is worth the time or not.

For instance, after reading about contingency adduction yesterday, I started wondering whether I should have C. working on something simpler than what he's working on now.

Contingency adduction
, as I understand it, is the sudden appearance of new learning without further instruction.

Here's the discussion of contingency adduction that gave me pause:

Contingency adduction occurs when multiple repertoires or components combine with little, if any, instruction.... Johnson & Layng (1994) described how fraction word problems were consistently a problem for their students, some scoring as low as 3 correct in a minute (with up to 11 errors). Probes of the component skills revealed that none of these students were fluent with simple addition math facts. An instruction sequence was put into place that taught addition math facts to rates that predicted fluency (component). Following this, and with no extra instruction, a probe of fraction word problems revealed that all learners had jumped to 13 to 14 corrects per minute (composite). This example is consistent with most reports of adductions, with the common feature being that they often occur without any formal instruction (Binder, 1996).

Behavioural fluency for young children with autism
David John Bonser

and here's Binder, 1996:

Solsten & McManus, 1979) demonstrated that frequency building of components not only allows more rapid acquisition of composites but sometimes seems to produce composites with virtually no formal instruction — an effect that Johnson and Layng (1992) have called response adduction (Andronis, 1983; Epstein, 1985).

Behavioral Fluency: Evolution of a new paradigm
The Behavior Analyst 1996, 19, 163-197 No. 2 (Fall)

Saxon Math and contingency adduction

This is another synchronicity event. Just this morning I was emailing Barry G. saying that Saxon Algebra 2 had suddenly become much easier.

The reason it's suddenly much easier is that the lessons I'm doing now all involve combining components into composites. Since I've worked my way through 91 lessons, I'm fluent in the components. There are times when I turn to a new lesson and simply skim it, because I "already know" how to do whatever the lesson is teaching me to do.

Mind you, most of these lessons are on procedures and word problems I've never done before - procedures and word problems I've never even seen before.

I suspect there's no counterpart to contingency adduction for "top down teaching."

fluency and autism

Now that I've (finally) stumbled across precision teaching, I remember our ABA teachers starting to talk about it and work with it for autistic kids. The concept made perfect sense to me, because often an autistic child can do a particular skill - or come up with the proper word - if he works fast. (In fact, the three dissertations on fluency training I've tracked down thus far are on fluency teaching for autistic children.)

Fluency training struck me as a great idea then - and now, too.

fluency and KUMON

KUMON is all about fluency.

Speed and accuracy: the KUMON mantra.

KUMON is not about precision teaching, however. KUMON is almost the opposite of precision teaching. With KUMON everyone does all the worksheets, period.

There's no data.

There's no skipping ahead if you can already do the worksheets you've got super-fast.

There is patience and persistence. (you may need to refresh your browser a couple of times to access the page)

wholes, not parts: Martin Brooks and The Constructivist Classroom
whole math

Thursday, July 5, 2007

is nothing sacred?

So now it turns out women don't talk more than men.

What next?

yet another surprise from the animal world

precision teaching at home

My neighbor sent me a link to the precision teaching people last year, but I never followed up. It's probably time.

Apparently students at the Morningside Academy in Seattle can gain more than two grade levels each year while adults can "advance two grade levels per month." (pdf file - I gather the school offers a money-back guarantee.)

starter articles:
Precision Teaching: Discoveries and Effects
Effective FluencyBuilding Design and Coaching for Classroom Programs Binder Riha Associates
Using Precision Teaching in a Homeschool Setting by Cynthia Riha
Joe Layng Believes Education is a Science - University of Chicago Magazine

book: The Morningside Model of Generative Instruction: What It Means to Leave No Child Behind

wholes, not parts: Martin Brooks and The Constructivist Classroom
whole math

B.F. Skinner weighs in

This is interesting. I had no idea B.F. Skinner had been a foe of ed schools:

Skinner's (1983) classic paper "The Shame of American Education" was originally published in the American Psychologist and later appeared in a collection of Skinner's papers (1987). It was aimed directly at cognitive psychologists. Skinner blamed modern cognitive psychology, in part, for the United States being what the National Commission on Excellence in Education (NCEE, 1983) termed "A Nation at Risk" The NCEE report described the United States as being threatened by "a rising tide of mediocrity." Skinner (1987) also blamed colleges of education as the place where "psychological theories come into the hands of teachers" (p. 120).


The failure of colleges of education to teach science and subject- based pedagogy has a long history that cannot be attributed solely or even primarily to cognitive psychology.


Evers and Clopton (2003) suggest that progressive ideas in the early 20th century grew from an activist religious piety they termed "pietist millennialism." [I'm not sure what "pietist millennialism" means, but I like the phrase...]

Skinner on discovery

Skinner (1968) made his views of the method of discovery quite clear:

If the student can be taught to learn from the world of things, nothing else will ever have to be taught. This is the method of discovery. It is designed to absolve the teacher from a sense of failure by making instruction unnecessary. The teacher arranges the environment in which discovery is to take place, he suggests lines of inquiry, he keeps the students within bounds. The important thing is that he should tell him nothing.

The human organism does, of course, learn without being taught. It is a good thing that this is so, and it would no doubt be a good thing if more could be learned in that way. Students are naturally interested in what they learn by themselves because they would not learn if they were not, and for the same reason they are more likely to remember what they learn in that way. There are reinforcing elements of surprise and accomplishment in personal discovery which are welcome alternatives to traditional aversive consequences. But discovery is no solution to the problems of education. A culture is no stronger than its capacity to transmit itself....It is quite impossible for the student to discover for himself any substantial part of the wisdom of his culture, and no philosophy of education really proposes that he should. Great thinkers build upon the past, they do not waste time in rediscovering it. It is dangerous to suggest to the student that it is beneath his dignity to learn what others already know, that there is something ignoble (and even destructive of rational powers) in memorizing facts, codes, formulae, or passages from literary works and that to be admired he must think in original ways. It is equally dangerous to forgo teaching important facts and principles in order to give the student a chance to discover them for himself. Only a teacher who is unaware of his effects on his students can believe that children actually discover mathematics, that (as one teacher has written) 'in group discussion they can and do figure out all the relationships, facts, and procedures, that comprise a full program in math' (p. 110).


Skinner pointed out that each of the reform proposals that were being made in the period immediately preceding his paper had a curious omission: namely how teaching could be improved. He described it as a "conspiracy of silence." Skinner suggested that pedagogy had become a "dirty word." Soon after Skinner's paper was published, this silence would be broken and pedagogy would come to the forefront. From the mid 1980's until the present, there has been an open battle between the progressives and the instructivists.


During the 1990s the United States underwent an unprecedented curricular "reform" of the public schools. By 1990, the ideological legacy of Kilpatrick, Rugg and Dewey controlled virtually all colleges of education and their respective teacher organizations. The NCTM and the National Council of Teachers of English developed standards reflecting progressive values and pedagogy. These curriculum standards were then adopted by state after state as the frameworks for reading and math. The end result was the standardization of ineffective practices leading to inadequately trained teachers and low-achieving students in basic academic domains (Rumph, Ninness, & McCuller, 2001).

"foundational constituents and complex composites"

Teachers are asked not to teach but to facilitate learning. The teacher is not to tell or show the students but to allow them to discover their own individual knowledge. Paradoxically, it is permissible for an untrained peer to show and tell. The knowledge gained by students is said to be different for each student. Independent reality separate from the person's own knowledge is denied. Progressive pedagogy does not analyze subject matter into foundational constituents and more complex composites, nor does it sequence material in such a way that constituents are taught before composites. Intentionally, the progressives use top down teaching, in which the sequences are inverted such that the more complex skill is introduced first and the foundational skills for it are assumed to be acquired in the context of the more complex task. Slavin (2003) describes top-down teaching this way: "Top-down means that students begin with complex problems to solve and then work out or discover (with the teacher's guidance) the basic skills required" (p. 259). It is not that progressive ideas can never be put to good use; they can (Johnson & Street, 2004). However, using indirect pedagogy to establish foundational constituent skills is simply ineffective and inefficient. Inevitable difficulties will occur in obtaining the skill at all, and in the absence of practice, established skills remain fragile. Weakly established skills are subject to forgetting, hinder the acquisition of composite skills of which they are a component and are less likely to combine with other skills through contingency adduction (Johnson & Layng, 1992).* Behavior analysts Layng, Twyman, and Stikeleather (2004) have developed methods to engineer discovery after directly training constituent skills and using precisely designed instructional sequences. Designed instructional sequences are not a component of contemporary or historical progressive education pedagogy but are a component of most effective pedagogies.

the Shame of American Education redux
(pdf file)
By Rumph, Robin; Ninness, Chris; McCuller, Glen; Holland, James; et al
Author note: Portions of this paper were presented at the 31st annual convention of the Association for Behavior Analysis, Chicago, May, 2005. We gratefully acknowledge Angela Ford for contributions in developing this manuscript. Correspondence concerning this article should be directed to Robin Rumph, School & Behavioral Psychology Program, PO Box 13019 SFA Station, Stephen F. Austin State University, Nacogdoches, TX, 75962
Chris Ninness home page & publications

Robin Rumph and Chris Ninness are professors at Stephen F. Austin State University School & Behavioral Psychology.

spaced repetition: top-down teaching

"Top-down means that students begin with complex problems to solve and then work out or discover (with the teacher's guidance) the basic skills required."

Slavin, R. (2003). Educational psychology: Theory and practice (7th ed.) Boston,MA: Allyn and Bacon.

in a nutshell: behaviorist take on constructivism

  • teachers are facilitators
  • peers, but not teachers, may "show and tell"
  • each student's knowledge is different
  • "independent reality separate from the person's own knowledge is denied"
  • subjects are intentionally taught "top down"
  • "top down" means subject matter is not analyzed into foundational constituents and more complex components
  • material is not sequenced so that foundational constituents are taught before more complex components
  • Robert Slavin: "Top-down means that students begin with complex problems to solve and then work out or discover (with the teacher's guidance) the basic skills required."
  • top-down teaching is "ineffective and inefficient"
  • problems acquiring the skill will inevitably ensue
  • without practice "established skills remain fragile"
  • worse yet: "Weakly established skills are subject to forgetting, hinder the acquisition of composite skills of which they are a component and are less likely to combine with other skills through contingency adduction." [ Johnson, K. R. & Lanyng, T. V. (1992) Breaking the structuralist barrier: Literacy and numeracy with fluency. American Psychologist, 47, 1475-1490.]
  • it is possible to "engineer discovery after directly training constituent skills and using precisely designed instructional sequences" [Layng, J., Twyman, J., & Stikeleather, G. (2004). Engineering discovery learning: The contingency adduction of some precursors of textual responding in a beginning program. Analysis of Verbal Behavior, 20, 99-110.] - but constructivist pedagogy does not do this
  • "Designed instructional sequences are not a component of contemporary or historical progressive education pedagogy but are a component of most effective pedagogies."

* "Contingency adduction occurs when multiple repertoires or components combine with little, if any, instruction (Andronis, Goldiamond & Layng, 1983; Johnson & Layng, 1996; Binder, 1996). "The problem of skill generalisation" (introduction to an unpublished dissertation, I think)

wholes, not parts: Martin Brooks and The Constructivist Classroom
whole math

My school has adopted Think Math! for grades k-5

In response to my expressions of concern about adopting this latest NSF program, my school assured me that they “believe it offers the balance we are trying to achieve, while teaching children to be the best in math... and, most importantly, to like math.”

SO, THE MOST IMPORTANT THING IS FOR THE STUDENTS TO LIKE MATH. It is more important to like math than to learn math, according to my school. This sounds like a familiar theme among reformers. When you take this approach to education, it seems inevitable that standards will drop.

I’m skeptical about Think Math!, to say the least. They insist that they provide abundant practice, including “lots of computational practice embedded in what may look to someone like a free exploration”. Hmm, that should be interesting.

Overall, the rhetoric on their website alerts me to be on the lookout for the usual suspects of reform math - excessive unguided discovery, group work, spiraling, skipping around and success for all. And, I’ve already been assured forewarned that all lessons will include letters sent home to parents.

I will also be looking out for the good parts of Think Math! I hope I find them.

Are you a "micro-managing" parent?

I caught this over at Instructivist in the comments to Spreading the Blame and was itching to get your take on a response to the statement "...when it comes to being involved in decisions of substance I'm on the other side of the door."

And that is where micro-managers should be kept. Put on your realism cap and take off your self-interest cap for a moment. If you require a "say" then all parents should have the same "right". Put 25 kids in a classroom and multiply by two. When the fifty adults come to a decision, then tell me what I am going to teach today, tomorrow, and next week. We will coloring while we wait for your collective wisdom. What next, parent created assessments?

If you are all that, then skip off to ed school and come and fix the problem. Don't forget, you'll have summers off!

I don't believe that being involved (or trying to be involved) in decisions that have a direct bearing on the future of my child is micro-managing. That's why I'm a parent, for pete's sake. So I guess either parents aren't involved enough (don't bake enough cupcakes) or they are too involved (ask questions and express concerns about the new reform math curriculum). I happen to be a cupcake baking question asker. Does that make me a micro-manager too?

help desk - books on handling disruptive behavior

Lessons two comments from anonymous:

Ordered the book IMMEDIATELY! I'm in search of scripted responses to disruptive situations. I'm currently in love with Engelmann's Direct Instruction. I'm teaching the 4 year old to read with his 100 lessons! [ISBN-10: 0671631985 ISBN-13: 978-0671631987]

It's great! There have been a few tough days...learning to rhyme and learning to sound out the word silently and then just saying the word were tough. But after the struggle (literally) the next day was like epiphany for her! She is reading better everyday!

If you know of any great books like Techniqes for managing...

Send your recommendations NOW!


I'm a teacher also. I'm not using the book for my own kid. I just wanted to point out that I enjoy the scripts and I need some scripts to handle the disruptive students.

Also, I'm a high school teacher. It's my first year. I taught younger kids before this.

Thanks for any input.

This is a big topic for me, because whenever I think about maybe teaching middle school some day I instantly wonder whether I would be able to keep order in a classroom. I know I couldn't do it given what I (don't) know now - would there be anyone who could teach me?

So I'm interested in books on this subject.

The one book I have, which looks terrific though I've read only a couple of chapters, is: Reluctant Disciplinarian: Advice on Classroom Management from a Softy who Became (Eventually) a Successful Classroom Teacher by Gary Rubinstein.

Here's one of my favorite passages:

As part of the training process, future teachers are often sent to observe dynamic teachers. They scribble notes furiously, as these phenomenons silence chatter by merely asking, "Are you respecting y our classmates right now?"

After observing one teacher with excellent classrom control, I asked, "What do you do if they throw paper airplanes?" She answered, immediately, "I don't put up with that kind of nonsense!" In my notebook I jotted, "Don't put up with that kind of nonsense!"

That would be me, I fear.

Here's another terrific passage:

In-service topics range from the utterly useless to the totally useless. One year my colleagues and I spent three hours learning the subtleties of new grade sheets, with advice like, "When you bubble, use a number two pencil, and be sure to erase any stray marks." At my friend's school, teachers recently received "risk management" training. This program could have been called "How to avoid hurting yourself on campus so we don't have to pay your disability income," with advic like, "You can prevent slipping on rainy days by thoroughly drying off your shoes when going indoors."

If a television is posted near the podium, teachers can be sure they are about to endure the least effective in-service imaginable — the video. I resent this medium because it just encourages those teachers who too often elect to "make it a Blockbuster lesson." The video usually depicts a round-table informational meeting with a group of teachers asking the moderator about the in-service topic. The video, with its bad acting and unnatural dialogue, takes the tone of a late-night infomercial.

Sometimes teachers are given an information packet to supplement the video. Once, while watching a video describing the latest standardized tes, I flipped through the booklet and discovered a section titled "Commonly Asked Questions." The questions and answers seemed very familiar. I soon discovered that they had given us the very script from which the teachers on the video were reading. I quickly pointed this out to some of the more obnoxious members of our staff, and they began reading the answers, loudly, along with the video.

ISBN-10: 1877673366

Wednesday, July 4, 2007

Happy July Fourth 2007

original here

a real world problem

C. just came downstairs & told us that the New World Champion of hot dog eating consumed 66 hot dogs in 12 minutes, breaking the previous world record, held by Takeru Kobayashi, of 54 3/4 hot dogs consumed in 12 minutes.

Naturally this inspired a unit multiplier problem AND a bonus percent change problem.

Which is great, because I was planning to teach percent change this summer!


A new world record for eating hot dogs has been set: 66 hot dogs in 12 minutes.
The previous record was 54 ¾ hot dogs consumed in 12 minutes.

Use unit multipliers to determine:

How many hot dogs did the new world champion eat per minute?

How many seconds did the new world champion require to consume one hot dog?

How many hot dogs did the former world champion eat per minute?

How many seconds did the new world champion require to consume one hot dog?

What is the percent increase in number of hot dogs consumed in 12 years, from the former world champion (59 hot dogs in 12 minutes) to the new world champion (66 hot dogs in 12 minutes)?

World Hot-Dog Eating Champion Crippled by Jaw Injury
how dogs eat hot dogs (scroll down)

are meerkats constructivists?

So yesterday I mentioned that a good 50% of the peer-reviewed articles on animal behavior I read open with some variant on the observation that we thought only people did this stuff.

I may have to revise that estimate upwards.

Last night I finally read "Teachers in the Wild" by Gergely Csibra (TRENDS in Cognitive Science, Vol. 11 No. 3).

Here's the abstract:

Three recent studies challenge the apparent consensus about the absence of teaching in non-human animals by providing evidence that certain behaviours of ants, birds and mammals satisfy a strict definition of teaching. However, these behaviours, although capable of facilitating information or skill acquisition in youngsters, could not support the transmission of cultural knowledge across individuals, which human teaching arguably serves.

translation: We thought only people had teachers; turns out South African pied babblers, ants, and meerkats do, too.

ergo: South African pied babbler, ant, and meerkat teaching isn't real teaching; South African pied babbler, ant, and meerkat teaching can't transmit cultural knowledge.


teaching is transmitting!

According to Trends in Cognitive Science.

scorpian school for meerkats

Scorpian school (teachers are "mature members of the meerkat community") has three grades:*

1st grade: the youngest meerkat pups are given dead scorpions to practice killing

2nd grade: older meerkat pups are given live but disabled scorpions (stinger has been removed) to practice killing

3rd grade: the oldest meerkat pups are given live, non-disabled scorpions (stinger intact) to practice killing

No word on what gift the pups get for graduation.

discovery isn't teaching

Human versus non-human teaching

These three studies describe phenomena that fulfill the accepted criteria for teaching [8], and there is no doubt that they serve the function of facilitating knowledge and skill acquisition of the young at the teachers’ expense. Still, they do not seem to be particularly good examples of the activity that, as humans, we would call ‘teaching’. What the ants and the babblers do could be considered as charitable information donation [13], whereas the meerkat helpers’ behaviour seems to be a good example of what is called ‘scaffolding’ (modifying the environment to support individual learning) in developmental psychology [14]. However, the prototypical human teaching is neither pure expression of episodic information nor just environmental scaffolding but a type of social learning that transmits generalizable (semantic) knowledge from the teacher to the pupil through (not necessarily linguistic) communication [15].


Conversely, the opportunity teaching that has been demonstrated in meerkats does result in the acquisition of a generalizable skill: it not only provides youngsters with food but also ‘teaches’ them how to kill scorpions. However, the transmission of this skill is achieved by individual, not social, learning. If the helpers demonstrated (i.e. communicated in their behaviour) how to disable or kill scorpions, the pups could learn this skill from them, rather than through individual learning in the supportive environment that is provided by their teachers. Thus, none of these studies has shown us a behaviour that is equivalent to typical human teaching because they .... facilitate skill learning in a way that the information does not directly come from the source (as with meerkats). As such, these types of teaching are unable to support the transmission and maintenance of cultural forms with unrestricted content because the content of the transmitted information is .... left to be discovered by the pupil (as in opportunity teaching).

So there you have it:

1. scaffolding means arranging the environment so as to produce learning in a student

2. scaffolding forces the student to discover knowledge

3. forcing a student to discover knowledge is not true teaching

4. discovery teaching can't transmit and maintain "cultural forms"

fyi, number 4 is the goalpost moving passage, the point in the study where scientists struggle to explain why what animals (and insects!) are doing can't possibly be what we're doing.

I discount these statements. Where is the evidence that discovery teaching can't transmit and maintain "cultural forms"?

Still, I was amused to find TRENDS simply asserting that of course scaffolding, discovery, etc. aren't real human teaching. Scaffolding, discovery, etc. are the kind of thing a meerkat does, not a person.

A person does direct instruction.

I wonder if Dan Willingham has seen this article. He's the one who told to subscribe to TRENDS.

update from Lynn:

Catherine, is there any practical reason for being able to remove a stinger from a live scorpion other than as a teaching technique?

What kind of professional development is going on for the teacher meerkats?

I mean, how do they all know that in 2nd grade, this live scorpion/no stinger step is necessary? It's nonobvious as meerkats are not farmers, are they? So it seems that the real level of teaching going on is not between the adult and youngster, but between the teachers.

Good catch.

I'm not sure whether meerkats remove the stinger before killing a scorpion....but I assume from the article that they don't.


Also, see Mr. Person's comment.

FORAGER U. As it gets older, a young
meerkat begging for prey is more likely to
receive a live meal that it can practice subduing.
Image courtesy of Andrew Radford/Sophie Lanfear/
Alex Thornton/Katherine McAuliffe

NEWBIES. Pups about 30 days old are
just ready to go out foraging.
A. Radford, S. Lanfear, Thornton, McAuliffe

Live Prey for Dummies: Meerkats coach pups on hunting
by Susan Milius
Science News


Radford, A.N. and Ridley, A.R. (2006) Recruitment calling: anovel form of extended parental care in an altricial species. Curr. Biol. 16, 1700–1704

Franks, N.R. and Richardson, T. (2006) Teaching in tandem-running ants. Nature 439, 153

Thornton, A. and McAuliffe, K. (2006) Teaching in wild meerkats. Science 313, 227–229

Pied babbler research project
Andrew Radford web page

yet another surprise from the animal world
are meerkats constructivists?

* It's not this systematic: "For the youngest pups, 65 percent of those servings were alive, but for the oldest pups, almost 90 percent were still living."

Any math-heads want to join?

Okay I don't necessarily mean you PhD's, although you're certainly welcome. I really mean you moms trying to get your brains wrapped around math again.

I'm challenging myself by taking last year's seventh grade Bergen Academies Math Competition test. It's a timed test that runs 90 minutes, but later for the timed part. I just want to get through the test.

Here's the blog entry, which also links to the test that you can print and take. Anybody game?

Clueless Mom Challenges Herself

Tuesday, July 3, 2007


The largest collection of Warriner's ISBN numbers in captivity.

Don't say I never gave you anything.

can parents be held accountable?

Instructivist has a post on "spreading the blame" in which he writes:

Home life has the greatest influence on a child's success or failure in school. It shapes the behavior of the child. It is where values and attitudes are communicated. Home life can be intellectually stimulating or impoverished. Attempts to remedy educational disparities also need to focus on this neglected aspect.

While these observations are certainly true (I always base my case against character education in this exact premise!), I'm not absolutely sure they have to be true.

We have a friend who is on the board of one of the successful charter schools in NYC (I've never been able to remember the name, unfortunately).

He told us that tiny little kids - in Kindergarten & 1st grade - come in with "attitude." They are hard and defended. Apparently they talk and sound like the're 18.

It takes a couple of months for the school to get through the shell, but get through it they do. The kids regain their sweetness, and start to focus and learn.

I don't think this school has any kind of remedy for the kids' home life. In fact, I'm sure of it, because I remember our friend expressing surprise that parents never come to school events, contact teachers, etc.

I think Siegfried Engelmann's philosophy is that you have to teach the kid you get - right?

If the kids you get have parents who aren't able to support their children's education - or are unwilling to do so - then you have to work within that reality.

For me that would mean, first and foremost: NO EXPECTATION OF HELP WITH HOMEWORK. When we talked to the new assistant superintendent she said this principle is an absolute for "turnaround schools." If you're trying to turn a low-performing school around you assume zero parental help with homework.

However, if I had a school full of kids who didn't get "easy" homework done, I'd take this a step further and stop assigning homework at all - or, rather, I would assign homework that would be completed under supervision at school, as "La Salle High School" does:

4. In their freshman year, all students are required to attend study halls during the hour or two when they are not in class. The study halls are also resource centers, which contain many of the books and references needed by freshmen in their courses. In such study centers, emphasis is placed on assisting students with their work. The teacher aide who runs the center is familiar with the assignments that freshmen receive. (This stands in contrast to study halls in other schools, which provide only custodial care during study periods.) Some freshmen who are deemed to be academically deficient are required to attend a separate study skills center adjacent to the freshman study hall. There, the intention is to provide more intense help than is available in the freshman study hall. Together, the freshman study hall and the study skills center serve to initiate freshmen into the academic culture of the school.

5. During their sophomore, junior, and senior years students who are experiencing academic or truancy problems are assigned to study halls during times when they are not in class. Again, emphasis is on providing academic assistance.

6. A few students exhibiting extraordinary behavioral problems are assigned to “supervised study.” In this room, custodial care is supplemented with a strong emphasis on interaction between the aide and the students. The room has only 12 desks, indicating that supervised study is necessary for only a tiny portion of the student population.

7. The school places a premium on student attendance in classes and has designed an effective monitoring system whereby parents are notified by the classroom teacher of class cuts on the same day that they take place.

I don't know how many teachers would be able to require students to do their homework under supervision without benefit of a well-thought out and well-staffed system like La Salle's.

Carol Gambill does, but she's working with kids who, by and large, are doing their homework successfully and are highly motivated.

I have no idea what I'd do if I were teaching in a school in which parents didn't supervise homework, the kids didn't do it, and there was no supervised homework option in the building. I'd probably try to spend as little class time as possible on instruction and as much class time as possible on homework.... ?

are there ways to hold students and parents accountable?

You could do things like fine parents whose kids don't show up for school - I think some communities have tried this, right?

I'm not sure what you can do to hold kids accountable apart from detention.

Kids should definitely have detention for not doing homework; rule should be enforced and consequences should be real.

This is another of my beefs with my own middle school. One of the teachers sends a student around to "check" to see whether the kids have done their homework. Naturally some of the kids simply write that day's date on an old homework assignment, and the student checker marks down that the homework is completed. This shouldn't be happening. The teacher shouldn't entrust homework checking to a student.

update from instructivist

Judging by most of the responses to my post, the thesis is widely misunderstood. I'm concerned with the environment parents create simply by being, i.e.having or lacking certain attributes. These attributes can be any number of things, e.g. providing a loving and nurturing environment conducive to the healthy emotional development of the child; valuing respect for others and teaching good manners; attaching value to education; providing an intellectually stimulating environment even in incidental ways. Contrast this with dysfunction and psycho- and sociopathology as is so often the case, a pathology that poses nearly insurmountable obstacles to education and perpetuates stratification. The thesis does not concern itself with minutiae like school board relations.

In this respect the thesis seems unremarkable.


(So, can anyone tell I'm a little burned-out?)

here's Joanne Jacobs

My book, "Our School," is about Downtown College Prep, a San Jose charter high that targets low-achieving Mexican-American students. Many parents had an elementary education in Mexico and speak English poorly or not at all. The school assumes parents can't help with homework but asks them to check off a homework log showing that they saw their child doing something. If a student misses two or more homework assignments, a teacher-counselor calls the parents.

Students behind on homework must attend special study sessions during what otherwise would be free time and/or Saturday school. Getting kids to do the homework is a big job in ninth grade. If they do it poorly, that's not a huge problem. Once the work habits are in place, they will improve.

Ninth, tenth and 11th graders have a 75-minute study session at the end of the school day. Tutors are available on request with ninth graders getting priority.

Niki Hayes told me something interesting.

She said you have to establish homework habits in K-5, because the middle school years really are "hormonal," and that's the worst time to try to do it!

I had never thought of that, but it makes sense.

mixed review workbooks

Middle Grades Math Minutes is the one math workbook Christopher doesn't mind doing.

Creative Teaching Press (mixed review workbooks – popular w/parents)
First-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Angela Higgs
ISBN-10: 1574718126
ISBN-13: 978-1574718126

Second-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Angela Higgs
ISBN-10: 1574718134
ISBN-13: 978-1574718133

Third Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Alaska Hults
ISBN-10: 1574718142
ISBN-13: 978-1574718140

Fourth-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Alaska Hults
ISBN-10: 1574718150
ISBN-13: 978-1574718157

Fifth-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Alaska Hults
ISBN-10: 1574718169
ISBN-13: 978-1574718164

Middle-Grade Math Minutes: One Hundred Minutes to Better Basic Skills
by Doug Stoffel
ISBN-10: 1574717235
ISBN-13: 978-1574717235

more mixed review:

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

and, from Susan S & instructivist:


Costs approximately $25/yr ($40 if you subscribe to high school material as well).

Site will print out mixed review or single-problem worksheets with answers. All subjects.

A lifesaver.

Instructional Fair Workbooks
Skill Builders Word Problems Workbooks (fantastic & cheap $3.95/book)
Skill Builders Workbooks (all)

mixed review workbooks
Spectrum Math workbooks

Spectrum Math workbooks

I'm pretty sure instructivist recommended the Spectrum books. I haven't used them yet, but probably will. Spectrum books have pre- and post-tests for each topic along with lots of word problems. (Other subjects, too.)

Spectrum Math Kindergarten
ISBN-10: 076963690X
ISBN-13: 978-0769636900
Revised updated edition (I’ve never seen the updated editions):
ISBN-10: 0769677916
ISBN-13: 978-0769677910

Spectrum Math Grade 1
ISBN-10: 1561899011
ISBN-13: 978-1561899012

Spectrum Math Grade 2
ISBN-10: 156189902X
ISBN-13: 978-1561899029

Spectrum Math Grade 3
ISBN-10: 1561899038
ISBN-13: 978-1561899036

Spectrum Math Grade 4
ISBN-10: 1561899046
ISBN-13: 978-1561899043

Spectrum Math Grade 5
ISBN-10: 1561899054
ISBN-13: 978-1561899050

Spectrum Math Grade 6
ISBN-10: 1561899062
ISBN-13: 978-1561899067

Spectrum Math Grade 7 by Thomas J. Richards
ISBN-10: 0769636977
ISBN-13: 978-0769636979

Spectrum Math Grade 8, by Thomas J. Richards
ISBN: 1-56189-908-9
ISBN-10: 1561899089
ISBN-13: 978-1561899081
Revised (haven’t looked at this):
ISBN-10: 0769637086
ISBN-13: 978-0769637082

Instructional Fair Workbooks
Skill Builders Word Problems Workbooks (fantastic & cheap $3.95/book)
Skill Builders Workbooks (all)

mixed review workbooks
Spectrum Math workbooks

Skill Builders workbooks - $3.95/book

Fantastic series! Christopher is doing the Skill Builders Algebra 1 along with Instructional Fair's Algebra 1.

I've been threatening to make him use the Skill Builders Spanish series, too - and I will! (Skill Builders has books for every subject under the sun...I should probably look to see if they've got one for Earth Science...)

Skill Builders Series
hard to pull up on Amazon, but lots of places sell them
ABC Teacher’s Outlet is a good source

Rainbow Bridge Publishing:

Division: Grades 4-6
ISBN-10: 1594412685
ISBN-13: 978-1594412684

Fractions, Decimals, & Percents: Grades 5 & 6
ISBN: 1594412693

Math grade 6
ISBN : 1-932210-05-9

Word Problems: Grade 5
ISBN-10: 1932210776
ISBN-13: 978-1932210774

Word Problems: Grade 6
ISBN-10: 1932210784
ISBN-13: 978-1932210781

ISBN: 1-93221-009-1
ISBN-10: 1932210091
ISBN-13: 978-1932210095

Algebra 1
ISBN-10: 1932210105
ISBN-13: 978-1932210101

Algebra 2
ISBN-10: 1932210830
ISBN-13: 978-1932210835

Geometry Grades 4-5
ISBN: 1-932210-79-2

Skill Builders word problem books:

Skill Builders World Problems Grade 2
ISBN-10: 1594412790
ISBN-13: 978-1594412790

Skill Builders World Problems Grade 3
ISBN-10: 1932210709
ISBN-13: 978-1932210705

Skill Builders World Problems Grade 4
ISBN-10: 1932210717
ISBN-13: 978-1932210712

Skill Builders World Problems Grade 5
ISBN-10: 1932210776
ISBN-13: 978-1932210774

Skill Builders World Problems Grade 6
ISBN-10: 1932210784
ISBN-13: 978-1932210781

Instructional Fair Workbooks
Skill Builders Word Problems Workbooks (fantastic & cheap $3.95/book)
Skill Builders Workbooks (all)
mixed review workbooks
Spectrum Math workbooks

Instructional Fair Workbooks

Given the fact that I'm having a nervous breakdown over stereotypies, dopamine, opioids, serotonin, novelty and a bunch of other things I can't remember all at the same time, today seemed like a good day to assemble a list of workbooks for my friend P., who just got her kids' final math assessments in the mail.

Their assessments may have been even worse than the one Christopher got in June 2006, e.g.: one of her kids has mastered integers, and that's it.

That seemed odd.

"Integers are hard!" I said.

"My kids are twisted," my friend said.

She's buying workbooks.


I love the Instructional Fair workbooks; we're using Algebra 1 this summer.

Instructional Fair series

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

ISBN: 0-7424-1787-5

ISBN-10: 0742417883
ISBN-13: 978-0742417885

Algebra II 5-8
ISBN: 0-7424-1789-1

Intro to Geometry
ISBN: 0742417778

Precalculus by Wayne J. Bentley
ISBN: 1-56822-488-5

Instructional Fair Workbooks
Skill Builders Word Problems Workbooks (fantastic & cheap $3.95/book)
Skill Builders Workbooks (all)
mixed review workbooks
Spectrum Math workbooks

yet another surprise from the animal world

Probably at least half the animal studies I read open with some variant of we thought only people did this stuff.

As a matter of fact, the entire history of research on animal behavior could be summed up as a protracted exercise in goalpost moving.

(I'm joking.)

btw, I now believe animals have language.

I can't prove it. I can't even argue the case competently.

But I believe it. I guess I believe it thanks to my Bayesian brain (and a glimmer of an insight that Jeff Hawkins' argument will lead to the conclusion that animals have language....) I've read SOOOO many articles that start out with "We scientists thought only humans do this" that if someone forced me to bet money on whether animals do or do not have language, I would have to bet yes. I just don't see enough, "We scientists always thought animals couldn't do X and, sure enough, animals can't do X" articles to make me bet "no."

In other words, I'm seeing a trend.

(or a pattern!)

Thinking that animals have language is heresy, in case you didn't know.* Ed, as it turns out, did not. I dropped my animals-have-language bomb the other morning at breakfast and he said, "Didn't I read an article saying animals have language?"

answer: no

You did not "read an article" saying animals have language. You may have read a book coauthored by your wife saying that animals (might) have language, or something close to it.

It's funny how one person's heresy is another person's misconception.

Anthropologists have long recognised altruism as a vital element in allowing complex social groups to form. This raised the question of when it first evolved. True altruism has always been seen as a uniquely human trait in that only people were thought capable of deliberately helping others, knowing there would be a cost to themselves.

Warneken and his colleague Brian Hare carried out the study on 36 chimpanzees at the Ngamba Island Chimpanzee Sanctuary, in Uganda.

The animals were not allowed to interact with any humans they knew or had received food from.

In the first experiment, the chimpanzee saw a person unsuccessfully reach through the cage bars for a stick on the other side, too far away for the person but within reach of the ape.

The chimpanzees spontaneously helped the reaching person regardless of whether this yielded a reward or not. When the chimpanzees could see the person making no effort to reach the object, they did not help.

A second experiment was designed to make it much harder to offer help, with the chimps forced to climb 6ft to get the stick. There were no rewards but, again, the animals still helped.

A third piece of research looked at the apes’ willingness to help each other. One chimp was made to watch as a second animal tried to get into a closed room containing food.

The only way it could get in was if the watching animal removed a chain on the door. In each case they did so.

Elsewhere in the animal world there are many examples of apparent altruism. Dolphins, for example, will support sick or injured animals, swimming under them for hours at a time and pushing them to the surface so they can breathe.

Chimps beat us to that human touch

yet another surprise from the animal world
are meerkats constructivists?

* It may not be heresy in the UK. Not sure.