kitchen table math, the sequel: 4/1/07 - 4/8/07

Saturday, April 7, 2007

copying sunflowers

I have been reading up on Vincent Van Gogh to prepare to give a short Art Literacy lesson at my children's school. I will read from a script, but I do a better job if I have read more than the script, beforehand.

The constructivist ethos of our Art Literacy program is to give children as much freedom as possible in achieving the production goal that follows each lesson. In this case, the goal for each child will be to paint a sunflower and show its texture by using specially thickened paint and a palette knife. Vincent used a lot of paint on his canvases.

In art, as in learning to read words or to add numbers, constructivists do not want children copying someone else’s good idea, whether it's how to paint a sunflower or how to sound out a word or how to add two numbers by counting-on. The child who copies is the child of a lesser god. He has fallen short of the goal of constructing original aesthetic or literary or mathematical meanings.

So I will read the Van Gogh lesson script and assign the painting, but I'm not supposed to finish what I've started by leading the children step by step through one method of creating one copy of one of Vincent's sunflowers. I show them one sunflower, and then let them learn the rest by doing it themselves as best they can. Once. Because they won't get a second chance because there isn't time.

They must copy one of Vincent's sunflowers, but I can't teach them how to use the tools they are given to achieve an aesthetic meaning that would be mine and not their own. I would be putting words in their mouths, so to speak, if I led them.

I was amused to read letters that Vincent wrote to his brother Theo and find that Vincent was often quite skeptical of his fellow artists and their methods. Among other observations, here is one excerpt from a letter he wrote from St. Remy on the subject of copying another artist’s work:

It is a kind of study that I need, for I want to learn. Although copying may be the old system, that makes absolutely no difference to me. [I have copied the Pieta and] I am going to copy the Good Samaritan by Delacroix too. What I am seeking in it and why it seems good to me to copy them I will tell you - they are always asking we painters to compose ourselves and be nothing but composers. So be it - but it isn't like that in music - and if some person plays Beethoven, he adds his personal interpretation - in music and more especially in singing - the interpretation of a composer is something, and it is not a hard and fast rule that only the composer should play his own composition. Very good - and I, mostly because I am ill at present, I am trying to do something to console myself for my own pleasure. I put the black and white [Pieta] by Delacroix… in front of me as a subject - and then I improvise colour on it, not, you understand, altogether by myself, but searching for memories of their pictures - but the memory, the vague consonance of colours which are at least right in feeling - that is my own interpretation.

Many people do not copy, many others do - I started on it accidentally, and I find that it teaches me, and above all it sometimes consoles me.

I am sorry he was not consoled more, sooner.

During the Art Literacy lesson we are to tackle the subject of Vincent’s mental illness right out of the box because even the littlest kids already know that he cut off his ear, and we need to manage their interpretation of the event.

We can also say that Theo sent Vincent money to keep him alive and painting, and we can say everybody is sad that Vincent never knew how famous his paintings would become.

But we don't tell the children that he made copies. Or that he took art lessons in Paris and he read books on technique that Theo bought for him. We don’t explain that Vincent made preparatory sketches and studies before he painted. We skip right over his many careful revisions, as he frequently returned to favorite subjects in favorite colors like sunflowers. We don't tell the children that, in his correspondence with Theo (who was working for an art dealer in Paris) the two of them were always desperately trying to figure out how to sell more of Vincent's paintings.

Friday, April 6, 2007

Number Sentences

On the SAT 9 math test for the spring of fourth grade (or the fall of fifth), you’ll find items talking about “number sentences,” a remnant of New Math that has been resurrected in the New New Math. Go visit the mathematics department of a good university and ask a senior mathematician if “math is a language,” with “sentences” and the like. Chances are fair that someone will start screaming at you, or even might just beat you up.

Your State Test Was Not Divinely Inspired

Doomed By Careless Math Errors

My 5th grader brought home her recent pre and post-tests on fractions. I can't fault the schools, teachers, or the typically lousy EM curriculum on this one. The topic is appropriate -- fractions in 5th grade requiring her to find the common denominator, add or subtract or multiply, and putting fractions in order, least to greatest. She's had plenty of classroom instruction and practice (she's also had a lot of afterschooling on this as Singapore Math's 5A and 5B spend a lot of time mastering fractions). The tests themselves were not particularly challenging.

So what's the problem? On a test she could easily have gotten 100% she got a 78. Why? Careless errors. She added when she was supposed to subtract, she failed to read the question properly, one she skipped entirely -- just missed the question. The one she skipped she got right on the pre-test. The ones she got wrong on the post-test she got right on the pre-test and vice versa.

So what next? Pedagogically, instructionally, what is the best way of getting kids to take their time and be careful? Is this just one of those 5th grade things? I'm tempted to ask the teacher for a copy of the test and have her retake it. Is there anything gained by having a consequence/punishment for carelessness?

Attention to detail seems critical in math. Am I overreacting? Anyone have success with this particular aspect of teaching?

Monday, April 2, 2007

Math Study Guides - SparkNotes

Math Study Guides - SparkNotes

Another math site.

This one covers prealgebra through Calculus BC II. It covers all the formulas, has sample problems and has online tests for each class.

I am more certain than ever that my 3rd grader could probably complete Algebra by the end of 5th grade. He knows 50% of the stuff covered in the prealgebra test already.

against Saxon

Linda Seebach's column on Ron Aharoni, with a swipe at Saxon Math (not from Linda, but from mathematician Alexander Givental).

I will say that I've thought for awhile that if I were starting a homeschool program, I'd probably choose Primary Mathematics for grades K-6, then switch to Saxon.

Don't know exactly why I think that & can't justify it logically.

But that's where I seem to have ended up.

I would write more, but I have to pack and catch a plane for Evanston. However, you can find several posts about Aharoni's terrific work over at the old ktm (link to "Posts Search" on the sidebar); Linda's column tells you where to purchase Aharoni's book.

Aharoni's wonderful article in American Educator is here,

Back in a week!

(I'm taking Saxon Algebra 2 with me....)

Sunday, April 1, 2007

Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

I stumbled upon this recent metastudy on elementary math programs by Bob Slavin.

Here is the summary.

Here's the whole thing (DOC). (Link fixed)

There are lots of good observations in it,

  • How few math curricula perform well in research studies,
  • How many bogus post hoc studies are out there,
  • How Bob Slavin shouldn't be doing an objective study involving his own programs, and
  • I caught a mistake involving the CMC study I recently blogged. It was randomized, not post hoc.
I figured I'd release the KTM brain trust on it to pick it apart.

in case your child needs to select a poem from a book or anthology

The Harp and Laurel Wreath: Poetry and Dictation for the Classical Curriculum
by Laura M. Berquist

A Family of Poems: My Favorite Poetry for Children
by Caroline Kennedy

The Best-Loved Poems of Jacqueline Kennedy Onassis
by Caroline Kennedy

All 3 seem to be good (not that I've spent a lot of time reading them).

Laura Berquist's book is wonderful. Ed deliberately had Christopher choose The Charge of the Light Brigade in order to make a point about the importance of classical education.

He's really had it.

silver lining

The good news is: I now know that the name Alfred, Lord Tennyson includes a comma.

help desk
in case your child needs to select a poem from a book or anthology


This is hilarious.

non-school factors, math, and reading

More confirmation that advantaged kids get their reading comprehension at home & their math at school. (I'm adding bullets.)

I had been particularly influenced by Wesley Becker's famous Harvard Educational Review article (1977) noting that the impact of early DISTAR success with decoding was muted for reading comprehension in later elementary grades by vocabulary limitations. Becker argued that this was a matter of experience rather than general intelligence by observing that while his DISTAR students' reading comprehension fell relative to more advantaged students by grade 4, their mathematics performance remained high.

  • He suggested that the difference was that all the knowledge that is needed for math achievement is taught in school, whereas the vocabulary growth needed for successful reading comprehension is essentially left to the home.
  • Disadvantaged homes provide little support for vocabulary growth, as recently documented by Hart and Risley (1995).
  • I was further influenced by the finding of my doctoral student, Maria Cantalini (1987), that school instruction in kindergarten and grade 1 apparently had no impact on vocabulary development as assessed by the Peabody vocabulary test. Morrison, Williams, and Massetti (1998) have since replicated this finding.
  • This finding is particularly significant in view of Cunningham and Stanovich's (1997) recently reported finding that vocabulary as assessed in grade 1 predicts more than 30 percent of grade 11 reading comprehension, much more than reading mechanics as assessed in grade 1 do.
Teaching Vocabulary:
Early, Direct, Sequential
by Andrew Biemiller

Our schools here teach vocabulary K-8 (don't know about high school).

As with so many other subjects, vocabulary isn't taught systematically; the kids study word lists from whatever they're reading in class.

I imagine vocabulary instruction here would be more effective if they did some form of systematic instruction, too, but I don't know.

We're still struggling through Vocabulary Workshop, which I continue to believe is a dandy series. (Struggling, meaning struggling to find time to do it.)

I'm curious about the Greek and Latin roots approach, too.

non-school factors and math
non-school factors, math, and reading

The school that isn't

Local News No assignments. No tests. No grades. It's "no problem" for Bothell school. Seattle Times Newspaper:

I wasn't going to blog it. I wasn't really, but it's to easy.
Clearwater is one of about 30 schools that follow the philosophy of the Sudbury Valley School in Massachusetts. Such schools are sometimes called "free" and "democratic" schools, where students are responsible for their own learning and have a significant role in governing the school. They also have many parallels with "unschooling," a movement embraced by some homeschooling families who don't follow a set curriculum.
When asked what they like best about their school, Clearwater students say the freedom to do what they want, when they want. Even if that means they might not learn everything students at other schools do.

"I won't say I'm amazingly good at advanced calculus," said Josh Pidcock, 19, who's been at Clearwater since he was 12. "I'm not the most studied reader either. I'm OK with that. I figure I can learn that in a college atmosphere much better.(because college is the place to learn how to read)
Two of the graduates are now attending community college, and one is enrolled at Earlham College in Indiana. Two others are working. One student left without graduating, Sarantos said, and is in a job-training program.

Three more are set to graduate this year. So far, Campbell has been accepted at The Evergreen State College in Olympia. Before he applied, he organized a math class at school to prepare for the SAT and studied on his own.

But Clearwater doesn't measure success by college acceptances. (Thank God for that.)

help desk

Ed just spent 3 hours helping Christopher with his ELA assignment.

We are on "vacation."

The assignment is huge.

Select a poem (any poem) from a book or anthology; identify three elements of good poetry; interpret the poem in a "well-developed response"; cite sources according to MLA; type the poem and the assignment (school doesn't teach typing, so I guess that's my job); OPTIONAL: research poet, find out about poet, etc.

This is starting at the top.

So....recent assignments have been way over C's head, but the few books he's been required to read are below grade level.


Anyway, here's the question.

I am not an English major.

I have no idea how to analyze a poem.

I have no idea, even, how to read a work of 19th century literature intelligently.

I know nothing!

So I need books. Books with shortcuts. Books that will give me a superficial but useable map-of-the-world where poetry analysis is concerned.

I thank you in advance.

being your child's frontal lobes

We've just today managed to figure out that C., although terrifically responsible when it comes to homework he can do, is avoidant when it comes to homework he can't do.

Avoidant means:

  • didn't select a poem to analyze from the books at school & copy it down (good thing I spent hours researching and purchasing poetry anthologies last school year!)
  • avoids the assignment altogether; and fails to mention assignment to parents

I guess we can say he's not proactive when it comes to seeking extra help.

Ed awaits his B-.


Effective Mathematics Instruction The Importance of Curriculum

I found a nice little study comparing a fourth grade Direct Instruction math program with a well regarded fourth grade constructivist program. The results were surprising, to say the least. (Cross posted at D-Ed Reckoning.)

The study Effective Mathematics Instruction The Importance of Curriculum (2000), Crawford and Snider, Education & Treatment of Children compared the Direct Instruction 3rd grade math curriculum Connecting Math Concepts (CMC, level D) to the constructivist fourth grade math curriculum Invitation to Mathematics (SF) published by Scott Foresman.

Invitation to Mathematics (SF)

SF has a spiral design (but of course). and relies on discovery learning and problem solving strategies to "teach" concepts. The SF text included chapters on addition and subtraction facts, numbers and place value, addition and subtraction, measurement, multiplication facts, multiplication, geometry, division facts, division, decimals, fractions, and graphing. Each chapter in the SF text interspersed a few activities on using problem solving strategies. Teacher B taught the 4th grade control class. He was an experienced 4th grade math teacher and had taught using the SF text for 11 years.

Teacher B's math period was divided into three 15-minute parts. First, students checked their homework as B gave the answers. Then students told B their scores, which he recorded. Second, B lectured or demonstrated a concept, and some students volunteered to answer questions from time-to-time. The teacher presentation was extemporaneous and included explanations, demonstrations, and references to text objectives. Third, students were assigned textbook problems and given time for independent work.

The SF group completed 10 out of 12 chapters during the experiment.

Connecting Math Concepts (CMC)

CMC is a typical Direct Instruction program having a stranded design in which multiple skills/concepts are taught in each lesson, each skill/concepts is taught for about 5-10 minutes each lesson and are revisited day after day until the skill/concept has been mastered. Explicit instruction is used to teach each skill/concept. CMC included strands on multiplication and division facts, calculator skills, whole number operations, mental arithmetic, column multiplication, column subtraction, division, equations and relationships , place value, fractions, ratios and proportions, number families, word problems, geometry, functions, and probability. Teacher A had 14 years of experience teaching math. She had no previous experience with CMC or any other Direct Instruction programs. She received 4 hours of training at a workshop in August and about three hours of additional training from the experimenters.

Teacher A used the scripted presentation in the CMC teacher presentation book for her 45 minute class. She frequently asked questions to which the whole class responded, but she did not use a signal to elicit unison responding. If she got a weak response she would ask the question again to part of the class (e.g., to one row or to all the girls) or ask individuals to raise their hands if they knew the answer. There were high levels of teacher-pupil interaction, but not every student was academically engaged. Generally, one lesson was covered per day and the first 10 minutes were set aside to correct the previous day's homework. Then a structured, teacher-guided presentation followed, during which the students responded orally or by writing answers to the teacher's questions. Student answers received immediate feedback and errors were corrected immediately. If there was time, students began their homework during the remaining minutes.

The CMC group completed 90 out of 120 lessons during the experiment.

The Experiment

Despite the differences in content and organization, both programs covered math concepts generally considered to be important in 4th grade--addition and subtraction of multi-digit numbers, multiplication and division facts and procedures, fractions, and problem solving with whole numbers.

Students were randomly assigned to each 4th grade classroom. The classes were heterogeneous and included the full range of abilities including learning disabled and gifted students. There were no significant pretest differences between students in the two curriculum groups on the computation, concepts and problem solving subtests of the NAT nor on the total test scores. Nor did any significant pretest differences show up on any of the curriculum-based measures.

The Results

Students did not use calculators on any of the tests.

The CMC Curriculum Test

For the CMC measure the experimenters designed a test that consisted of 55 production items for which students computed answers to problems, including both computational and word problems. The CMC test was comprehensive as well as cumulative; problems were examples of the entire range of problems found in the last quarter of the CMC program. Problems were chosen from the last quarter of the program because the various preskills taught in the early part of the program are integrated in problem types seen in the last quarter of the program.

The results here were not surprising, although the magnitude of the difference between the two groups may be.

The SF class averaged 15 out 55 (27%) correct answers on the posttest up from 7 out of 15 correct on the pre-test. The CMC class averaged 41 (75%) correct on the posttest up from 6 out of 15 correct on the pretest. I calculated the effect size to be 3.25 standard deviations which is enormous, though biased in favor of the CMC students.

The SF Curriculum Test

The SF test was published by Scott, Foresman to go along with the Invitation to Mathematics text and was the complete Cumulative Test for Chapters 1-12. It was intended to be comprehensive as well as cumulative. The SF test consisted of 22 multiple-choice items (four choices) which assessed the range of concepts presented in the 4th grade SF textbook.

The SF class averaged 16 out 22 (72%) correct answers on the posttest up from 4 out of 22 correct on the pre-test. However, surprisingly the CMC class averaged 19 (86%) correct on the posttest up from 3 out of 15 correct on the pretest. I calculated the effect size to be 0.75 standard deviations which is large, even though the test was biased in favor of the SF students.

You read that right, the CMC students out performed to SF students on the SF posttest.

The NAT exam and math facts test

The CMC group also scored significantly higher on rapid recall of multiplication facts. Of 72 items, the mean correctly answered in 3 minutes for the CMC group was 66 compared to 48 for the SF group for the multiplication facts posttest. I calculated the effect size to be 1.5 sd.

Posttest comparisons on the computation subtest of the NAT indicated a significant difference in favor of the CMC group. Effect size = 0.86. On the other hand, neither the scores for the concepts and problem-solving portion of the NAT nor the total NAT showed any significant group differences. The total NAT scores put the CMC group at the 51st percentile and the SF group at the 46th percentile, but this difference was not statistically significant.


The CMC implementation was less than optimal, yet it still achieved significantly better performance gains compared to the constructivist curriculum. The experimenters noted:

We believe this implementation of CMC was less than optimal because (a) students began the program in fourth grade rather than in first grade and (b) students could not be placed in homogeneous instructional groups. A unique feature of the CMC program is that it's designed around integrated strands rather than in a spiraling fashion. Each concept is introduced, developed, extended, and systematically reviewed beginning in Level A and culminating in Level F (6th grade). This design sequence means that students who enter the program at the later levels may lack the necessary preskills developed in previous levels of CMC. This study with fourth graders indicated that even when students enter Level D, without the benefit of instruction at previous levels, they could reach higher levels of achievement in certain domains. However, more students could have reached mastery if instruction were begun in the primary grades.

Another drawback in this implementation had to do with heterogeneous ability levels of the groups. Heterogeneity was an issue for both curricula. However, the emphasis on mastery in CMC created a special challenge for teachers using CMC. To monitor progress CMC tests are given every ten lessons and mastery criteria for each skill tested are provided. Because of the integrated nature of the strands, students who do not master an early skill will have trouble later on. Unlike traditional basals, concepts do not "go away," forcing teachers to continue to reteach until all students master the skills. This emphasis on mastery created a challenge for teachers that was exacerbated in this case by the fact that students had not gone through the previous three levels of CMC.

Why didn't the CMC gains show up on the NAT problem solving subtest and total math measure? The experimenters opine:

Our guess is that a more optimal implementation of CMC would have increased achievement in the CMC group, which may have shown up on the NAT. In general, the tighter focus of curriculum-based measures such as those used in this study makes them more sensitive to the effects of instruction than any published, norm-referenced test. Standardized tests have limited usefulness for program evaluation when the sample is small, as it was in this study (Carver, 1974; Marston, Fuchs, & Deno, 1985). Nevertheless, we included the NAT as a dependent measure because it is curriculum-neutral. The differences all favored the CMC program.

That no significant differences occurred either between teachers or across years on the NAT should be interpreted in the light of several other factors. One, the results do not indicate that the SF curriculum outperformed CMC, only that the NAT did not detect a difference between the groups, despite the differences found in the curriculum-based measures. Two, performance on published norm-referenced tests such as the NAT are more highly correlated to reading comprehension scores than with computation scores (Carver, 1974; Tindal & Marston, 1990). Three, the NAT concepts and problem solving items were not well-aligned with either curriculum. The types of problems on the NAT were complex, unique, non-algorithmic problems for which neither program could provide instruction. Performance on such problems has less to do with instruction than with raw ability. Four, significant differences on the calculation subtest of the NAT favored the CMC program during year 1 (see Snider and Crawford, 1996 for a detailed discussion of those results). Because less instructional time is devoted to computation skills after 4th grade, the strong calculation skills displayed by the CMC group would seem to be a worthy outcome. Five, although the NAT showed no differences in problem solving skills between curriculum groups or between program years, another source of data suggests otherwise. During year 1, on the eight word problems on the curriculum-based test, the CMC group outscored the SF group with an overall mean of 56% correct compared to 32%. An analysis of variance found this difference to be significant...

And, here's the kicker. The high-performing kids liked the highly-structured Direct Instruction program better than the loosey goosey constructivist curriculum:

Both teachers reported anecdotally that the high-performing students seemed to respond most positively to the CMC curricula. One of Teacher A's highest performing students, when asked about the program, wrote, "I wish we'd have math books like this every year.... it's easier to learn in this book because they have that part of a page that explains and that's easier than just having to pick up on whatever."

It may be somewhat counter-intuitive that an explicit, structured program would be well received by more able students. We often assume that more capable students benefit most from a less structured approach that gives them the freedom to discover and explore, whereas more didactic approaches ought to be reserved for low-performing students. It could be that high-performing students do well and respond well to highly-structured approaches when they are sufficiently challenging. These reports are interesting enough to bear further investigation after collection of objective data.