kitchen table math, the sequel: 3/16/08 - 3/23/08

## Thursday, March 20, 2008

### For those who wanted specifics from NMAP

This document has it for algebra.

The exec summary says this:
What is usually called Algebra I would, in most cases, cover the topics in Symbols and Expressions, and Linear Equations, and at least the first two topics in Quadratic Equations. The typical Algebra II course would cover the other topics, although the last topic in Functions (Fitting Simple Mathematical Models to Data), the last two topics in Algebra of Polynomials (Binomial Coefficients and the Binomial Theorem), and Combinatorics and Finite Probability are sometimes left out and then included in a pre-calculus course. It should be stressed that this list of topics reflects professional judgment as well as a review of other sources.
Symbols and Expressions
• Polynomial expressions
• Rational expressions
• Arithmetic and finite geometric series

Linear Equations
• Real numbers as points on the number line
• Linear equations and their graphs
• Solving problems with linear equations
• Linear inequalities and their graphs
• Graphing and solving systems of simultaneous linear equations

• Factors and factoring of quadratic polynomials with integer coefficients
• Completing the square in quadratic expressions
• Using the quadratic formula to solve equations

Functions
• Linear functions
• Graphs of quadratic functions and completing the square
• Polynomial functions (including graphs of basic functions)
• Simple nonlinear functions (e.g., square and cube root functions; absolute value; rational functions; step functions)
• Rational exponents, radical expressions, and exponential functions
• Logarithmic functions
• Trigonometric functions
• Fitting simple mathematical models to data
Algebra of Polynomials
• Roots and factorization of polynomials
• Complex numbers and operations
• Fundamental theorem of algebra
• Binomial coefficients (and Pascal’s Triangle)
• Mathematical induction and the binomial theorem

Combinatorics and Finite Probability
• Combinations and permutations, as applications of the binomial theorem and Pascal’s Triangle

The doc itself is much more specific, as you'd guess. The tables and figures are quite insightful.

APPENDIX A: Comparison of the Major Algebra Topics in Five Sets of Algebra I and Algebra II Textbooks with the List of Major Topics of School Algebra
APPENDIX B: Errors in Algebra Textbooks

List of Figures
Figure 1: Percent of Students At or Above Proficient in Mathematics Achievement on the Main NAEP Test: 1990, 2003, and 2007
Figure 2: States with Standards for Algebra I and II Courses
Figure 3: Topics in a 1913 High School Algebra Textbook
Figure 4: Singapore 2007 Algebra Standards for Grades 7-10
Figure 5: Algebra Objectives for NAEP’s Grade 12 Mathematics Assessment
Figure 6: Topics to Be Assessed in the ADP Algebra II End-of-Course Test
Figure 7: Mathematics Topics Intended from Grade 1 to Grade 8 by a Majority of TIMSS 1995 Top-Achieving Countries
Figure 8: Mathematics Topics Intended from Grade 1 to Grade 8 in the 2006 NCTM Focal Points Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries

List of Tables
Table 1: Frequency Counts for Broad Topics in 22 States’ Standards for Algebra I and II Courses
Table 2: Algebra Elements Covered by State Algebra or Integrated Mathematics Frameworks, by State and Two-Thirds Composite
Table 3: Comparison of the Major Topics of School Algebra with Singapore’s Secondary Curriculum
Table 4: Comparison of the Major Topics of School Algebra with the 2005 NAEP Grade 12 Algebra Topics
Table 5: Comparison of the Major Topics of School Algebra with ADP’s High School Algebra Benchmarks, Core Topics in its Algebra II Test, and the Topics in the Optional Modules for its Algebra II Test
Table 6: K-8 Grade Level Expectations in the Six Highest-Rated State Curriculum Frameworks in Mathematics Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries

### "100+ years to fire all the bad teachers"

Steve Barr has organized a parents' union, too.

### Barry Garelick on the post-NMP world

David Geary, a cognitive developmental psychologist at University of Missouri, said that the reason a panel such as NMP was formed was because of the failure of schools of education to do what the country wants: Train teachers using research-based techniques, rather than running a playground for untested methods. Schools of education should be held accountable for their work, he said.

Vern Williams noted the current state of affairs in math education in which correct answers have been deemed over-rated and algebra has been redefined to include statistics and pattern recognition. He expressed his hopes that as a result of the NMP report teachers will feel it is once again crucial to consider content - and correct answers.

During a break in the meeting, however, an event occurred which to my mind simultaneously underscored and transcended the importance of NMP's report. Williams' 8th grade algebra class which had assembled at the back of the gym gathered, in rock fan fashion, around Hung-Hsi Wu - a panelist and math professor from Berkeley - to get his autograph and take pictures.

"I guess this shows that kids can get excited about math without sitting in groups doing projects and using math textbooks that look like video games," Williams said.

I hope for the best in this post-NMP world.

Living in a Post National Math Panel World

That's beautiful.

### number sense in action

I have it!

Number sense means you can look at an item on SAT-Math and instantly eliminate two of the choices.

..............................

Actually, I have no idea whether being able instantly to eliminate two of the four choices on an SAT math item is or is not an example of number sense.

I do think I-can-eliminate-two-answers is a distinct stage in the novice to experienced to proficient sequence. Last summer, visiting Susan S, I discovered that I could correctly pick the answer to the quadratic function problems in her ACT prep book more often than not. Just a year before I hadn't been able to do any of these problems on a practice SAT test, nor could I eliminate any of the 4 alternatives.

I've seen the same thing with C. at various points. Suddenly he'll be guessing right. He still doesn't know what he's doing, but he knows the answer -- knows it, or has a pretty good idea.

### Dan K on number sense

I don’t have a good definition of number sense and have not researched any references to attempt to find one. Where I would be tempted to apply the term is in the context of error checking or “sanity checks.” In other words, there are errors that students make that you would think would not slip by them if they had more “number sense.”

For example, if you multiply a number by an improper fraction, your product should have a magnitude greater than the original number. So, if I make an error in multiplying 419 by 5/2, I might get a number like 168. Now I make errors all the time, and I don’t catch them all. This one, however, I would catch because I have enough “number sense” to know that I should get an answer greater than 419. You could argue that this is knowledge of some rule about proper and improper fractions rather than some “sense,” but then I would counter by saying that it was my number sense that led me to reach for the rule to apply.

Similarly, a story problem might tell me that a company with revenue of 1.84 billion dollars invested 4% of revenue in research and ask how much was invested in research. I might make an error and multiply the revenue number by 4 instead of .04. Seeing that my result was greater than the total revenue number would trigger my number sense to tell me something didn’t look right. In geometry, when computing the length of a hypotenuse, one should get a number larger than either leg of the triangle. In trigonometry, number sense would recognize that something was wrong if you applied the tangent function when you should have used the inverse tangent.

I guess the antithesis of number sense, to me, is when a student computes the third side of a triangle with sides of length 20 and 30—by the Pythagorean Theorem or the Law of Sines or however—and his calculator tells him that the third side is something nutty like 0.561 long. If he had any number sense, he’d know to not trust that answer and re-check his work. I guess a more specific case might be computing a hypotenuse with the Pythagorean Theorem, but failing to take the square root as the last step. A right triangle with sides 3, 4, and 25 (instead of the correct 5) makes no sense.

That’s my two cents’ (sense?) worth.

I see this kind of number sense as a side effect of good instruction: a natural result of good teaching but not the goal. It's an emergent property. That being the case, I imagine that the kind of Everyday Math estimation homework Concerned Parent's son brings home is most likely a FWOT.

Because number sense develops with proficiency, it's useful as an informal assessment. For instance, if I ask C. what 10% off a sale item is and he has no clue, I know we're in trouble.

These days C's number sense for percent is pretty good, but his number sense for the root or roots of quadratic equations is (probably) nonexistent.

That doesn't tell me we need to rustle up some worksheets on number sense.

It tells me we need more practice solving quadratic equations.

### average SAT scores by family income

Higher Ed has a list here. (scroll down) Other interesting factoids:

Median math scores --
Asians: 578 (equivalent score prior to 1995: 565)
Whites: 534 (equivalent score prior to 1995: 510)

Asians: 514 (prior to 1995: 435)
whites: 547 (prior to 1995: 470)
source for score equivalents: SAT I Mean Score Equivalents

Both differences are statistically significant, assuming I'm interpreting the chart below correctly.

source:
Comparing Group Scores on the SAT - for Colleges and Universities (pdf file)

the 1995 recentering

When the College Board recentered SAT scores, back in 1995, Verbal scores went up 70 points and Math scores about 25. So, if recentering had not occurred, the current Verbal average of 506 would have been about 437 and the current Math average of 514 would have been about 494.
source:

Comparing Group Scores on the SAT - for Colleges and Universities (pdf file)
Comparing Group Scores on the SAT (pdf file)
SAT Data Tables

### NYT article

I'm on semi-hiaitus from blogging, but I had to come out of my hole to highlight this article in the New York Times:

Bill Would Bar Linking Class Test Scores to Tenure
By JENNIFER MEDINA
Published: March 18, 2008
While the state was consumed by the downfall of Eliot Spitzer last week, the Assembly passed a bill that would pre-emptively bar New York City and other school districts from linking teacher tenure to students’ test scores.

There's not much more to say, other than how appalling this is.

## Wednesday, March 19, 2008

### the number sense, redux

Unbeknownst to the NCTM, it seems, people have been looking into the question of what the number sense is or is not:

This led to Dehaene’s first encounter with what he came to characterize as “the number sense.” Dehaene’s work centered on an apparently simple question: How do we know whether numbers are bigger or smaller than one another? If you are asked to choose which of a pair of Arabic numerals—4 and 7, say—stands for the bigger number, you respond “seven” in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene’s experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the Ã‰cole Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number.

Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. “It is a basic structural property of how our brains represent number, not just a lack of facility,” he told me.

[snip]

These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.

Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first. Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: “two, three is one, four is two, five is three, six is four, six.” But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.) The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 x 6, the reflex activation of 7 + 6 and 7 x 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 x 8, the error rate can exceed twenty-five per cent.
The Numbers Guy
by Jim Holt
The New Yorker

in a nutshell:
• number sense--right now! is bunk. number sense--5 or 10 years from now! might be more like it, depending on what kind of numbers we're talking about. For instance, number sense for exponents--not in this lifetime! would capture a normal human being's ability to grasp intuitively the nature of exponential growth. [Is an exponent a "kind" of number? Probably not.]
• the fact that I spent 30 years of my life believing that 7x6=43 is perfectly normal and nothing to be ashamed of.

Which reminds me: I never wrote part 2 of my post on cumulative practice.

### Favorite Sentence from Wu's paper

"It is an intriguing question how to judge students’ learning processes if they are fed extremely defective information."

He goes on for quite a bit about how in ed research, how we don't decouple the student from the teacher. We measure students' performance, but not don't control the input variables: teacher knowledge/methodology/pedagogy/etc/etc/etc.

Thinking of it this way, it's worth going back to even the most well known research on the effects of teaching on IQ and student performance and asking how much of those outcomes are simply transference: maybe they really are merely measuring teacher IQ, in a sense . Certainly, without decoupling, we can't tell which pieces matter. Catherine posted a few weeks ago about Miss Apple Daisy, who actually changed the outcome later in life for nearly every single student she had. Was she a necessary or a sufficient outcome for them?

Decoupling would require controlling the variables. That means controlling the teachers and the curriculum. It means Direct Instruction is a Necessary, but Not Sufficient condition. First, you have to create the means by which the teachers are controlled enough that we could even tell what script works and what doesn't--DI controls its teachers. But then we still have to test what makes all kids understand. Funny idea, that we would believe all kids could understand, isn't it?

### Critical Concepts for Understanding Fractions

Hung Hsi Wu has written a document for the NMAP about fractions. This document "contains a detailed description of the most essential concepts and skills together with comments about the pitfalls in teaching them. What may distinguish this report from others of a similar nature is the careful attention given to the logical underpinning and inter-connections among these concepts and skills."

Basically, this is a document about what you need to know about fractions and how they work. This is how they should be understood. It is NOT a teacher's manual, and certainly not a student's textbook, but it is the mathematical basis of one, and it's a darn good start.

It is dense. Like a good math paper, there are no extra words to confuse. There is exactly as much precision and description but no more. That makes it slow reading if you are unfamiliar with the material.

Over the next few days, I'll write more posts about this document, expanding on some of his writing. Perhaps we can make a KTM parent manual from this document with enough feedback and examples.

### Number Sense—Right Now!

Number Sense—Right Now!
NCTM News Bulletin, March 2008

"Is 4 × 12 closer to 40 or 50? How many paper clips can you hold in your hand? If the restaurant bill is \$119.23, how much should you leave for a tip? How long will it take to make the 50-mile drive to Washington? If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"
OK. I've generally ignored it when people talk about "number sense", but here it is from the president of NCTM. He says: "Number sense is important and needed—right now." What is it, exactly? Is it estimation? No, it seems to be more than that. Does Mr. Fennell define it? No. He just gives the examples above. Let's look at each one.

"Is 4 × 12 closer to 40 or 50"
Do it exactly in your head. It's part of the times table.

"How many paper clips can you hold in your hand?"
How accurately do you have to make this estimate to show number sense? He doesn't say, but I don't think he is talking about plus or minus 25% accuracy. I don't think I could guess that closely.

"If the restaurant bill is \$119.23, how much should you leave for a tip?"
Does he think that those who have mastered the traditional method of multiplication are stuck doing this calculation right-to-left on paper? This is a straight estimation problem, and my traditionally-taught wife takes pride in calculating these things down to the penny in her head.

Is number sense more or less than estimation? It seems to be both more and less. Number sense means more than just estimation, but it doesn't require you to provide accurate estimations.

"How long will it take to make the 50-mile drive to Washington?"
About an hour? What are the assumptions? Is number sense equal to estimation with common sense added in? Apparently the common sense level is not very high. How about the number sense to determine how long it will take to drive 400 miles on the highway, accounting for stops for gas, eating, and traffic? What if I gave you the exact times for stops and the lower speed in the traffic? Mastery of the basics leads to number sense, not the other way around.

He seems to be making the case that there is no linkage between mastery of the basics and number sense. But then he really isn't talking about mastery of estimation. Schools could hand out "Arithmetricks" and practice, practice, practice. No, he seems to be talking about some sort educational number sense osmosis. Low expectations.

"If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"
He goes on to say:

"Students who have a good sense of number are able to provide a reasonable response to the examples above, including the driving example. And they know that there is no proportion-driven response for the final example."
Sure there is. If a 10 year old child is 5 feet tall, then proportion tells you that there is some other effect going on when they get to 20. If you were talking about some unusual species of tree, then proportion or number sense is not going to help. You need content knowledge, and we all know what schools think of content. I think I'll coin a new term: Content Sense. That's what we need-right now! Just like with fractions on the real number line, kids need to be able to place major historic events on a timeline.

After all of this, I still don't know what he means by number sense or what performance level is required. Whatever it is, it seems pretty low. Mr Fennell can't define it, but he wants it fixed "right now".

There is another example:

"A sense of number emerges that is built on the foundations discussed above, which
yield responses such as, “I knew 3/4 was more than 3/5 because the pieces were bigger in fourths.” This is what all math teachers want. Such “aha!” classroom moments remind us about the importance of understanding."
"the pieces were bigger in fourths"?

So number sense is something other than math; something other than mathematically knowing why 3/4 is greater than 3/5. And "understanding", according to him, is something other than mathematical understanding. What if the student said that 3/4 = .75 and 3/5 = .6? Is that number sense? Does that show understanding or is that just rote knowledge?

Math is all about tools and methods that you can rely on to give you correct results in spite of the fact that what you might be doing defies common sense. That's the power of math. As I've said before, let the math provide you with the understanding. If you're worried about estimation, teach it directly. Number sense, whatever it is, will take care of itself.

## Tuesday, March 18, 2008

Catherine’s Visual Learning Post got me thinking about my “picture overshadowing” theory of sight words.

I have a lot of theories about sight words (theories are at the end of the post.) All of my remedial students had problems with wildly guessing at words. From my survey of hundreds of children in schools which taught with varying emphasis on sight words, I found that the more sight words used and the longer the student was exposed to them, the harder it was to break out of these guessing habits and focus on sounding out words from left to right. I also found that more sight words resulted in more reading problems. In my sight word case study post, I examine the case of a girl who after learning the 220 Dolch sight words (this girl was not one of my students)
The skill of sounding out simple words, that she had been able to do shortly after she turned three, had been completely lost. If she didn't know a word by sight, she was stuck. [snip] ; even if a word was in her spoken vocabulary, she couldn't recognize it on the page if she hadn't seen it before in print, even if it was totally phonetically regular, with all short-vowel sounds. And when she came to these words she didn't recognize, she would try to guess…
While most of my students had not completely lost their ability to sound out words, it was like swimming through molasses to get them to sound out words. And, the more sight words taught, the thicker the molasses.

I had a theory about sight words and picture overshadowing already, but did not have a good explanation for how spelling fit in. I knew that spelling was helpful for my students, but spelling, like sight words, seemed to be dealing with wholes as well, at least on the surface. While I intuitively believed that spelling was different, I was not able to explicitly explain the difference.

Providentially, the article Catherine linked to, Words Get in the Way, provided the missing spelling explanation:
For instance, in a 1995 study, Schooler reported that verbal descriptions disrupted white volunteers' memories for the faces of white but not black individuals. He proposed that thanks to their extensive experience in looking at white faces, white volunteers used rapid, nonverbal perception to evaluate each such face as a unified entity. In contrast, volunteers spent more time studying individual features of the less-familiar black faces. Subsequent written descriptions were more consistent with the features that white participants remembered about the black faces than with the unified images they had stored for the white ones, Schooler concludes.
This led me to see how spelling could fit nicely into the picture overshadowing theory of sight words. While you do examine the whole word to learn it for spelling, you are also studying the individual features (letters) of the word.

Charles Perfetti’s article The role of discourse context in developing word form representations: A paradoxical relation between reading and learning states,
In our experiments, children attempted to read words they could not previously read, during a self-teaching period, either in context or in isolation. Later they were tested on how well they learned the words as a function of self teaching condition (isolation or context). Consistent with previous research, children read more words accurately in context than in isolation during self-teaching; however, children had better retention for words learned in isolation.
In my remedial work, I’ve found that students learn better when taught words in isolation. I try not to introduce any outside reading material until all phonics skills have been over-learned. When teaching my daughter to read, I found that she also did better when taught words in isolation, just like my remedial students. Moreover, she did even better when we switched to Webster’s Speller, learning spelling and syllables in isolation. Don Potter has found that his students (both beginning and remedial students) learn better when taught words in isolation as well. He is sharing a method for teaching phonics words in isolation with his nationwide campaign to get a free copy of Blend Phonics to every elementary teacher in America.

The article Words get in the Way states,
In one study, conducted by Kim Finger of Claremont (Calif.) Graduate University, participants who wrote a description of a man's face after studying the face for 5 minutes suffered no memory loss if they were then nudged back into a perceptual frame of mind. To do this, Finger asked them either to solve a printed maze or to listen to 5 minutes of instrumental music. Both strategies yielded face memory equal to that of volunteers who didn't provide a written description.
I’ve found that my students also do better when I get them switched back from “guessing mode” (visual) to “sounding mode” (verbal.) To switch them out of “guessing habits,” and into “sounding out habits,” I found the use of nonsense words helpful, especially if I announced up front that the upcoming words were nonsense words. Some of my students who had been exposed to sight words for years were very hard to break of their guessing habits. They would even try to guess at nonsense words—unless warned that the word was a nonsense word and that there was no way they would ever be able to guess it because it was not a real word. Repeated nonsense words would usually switch them from “guessing mode” to “sounding mode” and allow me to begin phonetic teaching work on regular words.

So, I now have a more complete theory of sight words and “picture overshadowing.” Sight words are processed on the visual side of the brain (pictures). Words taught with phonics are processed on the verbal side of the brain (sounds.) People with dyslexia (organic or induced by sight words) have been shown to improve their reading abilities and have changes in brain activity consistent with this picture overshadowing theory. [See note 2 below] This "picture overshadowing" explains the molasses effect I saw with sight words and my students' impaired ability to sound out words.

I believe that guessing at words from pictures or context can also switch students into this visual “guessing mode,” while reading words in isolation forces students to focus on the letters and sounds of the word, the verbal “sounding mode.”

In my informal survey of hundreds of children, those who read the best were those taught with phonics methods that used very few sight words.

You can determine if someone is suffering "picture overshadowing" from too many sight words by giving the Miller Word Identification Assessment, or MWIA, available for free download from Don Potter. It measures the speed at which a student reads sight words verses less frequent phonetic words. Anyone reading the phonetic words more than 10% slower than the holistic words should consider a good phonics program with no sight words.

Blend Phonics is a good program that uses no sight words, and does not teach words in context, which I also believe leads students to switch over from the verbal to the visual mode. Don Potter also has developed a Blend Phonics Reader which has words of similar configuration (bed, bid, bat, bit, etc.) next to each other to help the student learn to see and overcome visual configuration guessing habits.

Webster’s Speller is another very good phonics program that uses no sight words. Its use of spelling may also help prevent dyslexia by teaching students to sound out and spell words and syllables before they read them in context, making sure that they have a firm letter by letter mental image of the word before they attempt to read them in context. It also teaches using syllables via a syllabary, which may be helpful for preventing dyslexia. Syllables were also a very powerful reading method for my students and for Catherine’s son, resulting in rapid reading grade level improvements.

My free online phonics lessons also use syllables and teach no sight words. The first several lessons have now been switched to all uppercase to prevent guessing from visual configuration.

Richard G. Parker warned of the dangers of reading words you hadn’t yet learned to spell (and sound out, see note 3) in 1851:
I have little doubt will be found true, and that is, that it is scarcely possible to devote too much time to the spelling book. Teachers who are impatient of the slow progress of their pupils are too apt to lay it aside too soon. I have frequently seen the melancholy effects of this impatience. Among the many pupils that I have had under my charge, I have noticed that they who have made the most rapid progress in reading were invariably those who had been most faithfully drilled in the spelling book.
When I taught my daughter to read using a variety of phonics programs and only 2 sight words, I found that she would occasionally guess at words when reading stories. After learning to spell and sound out syllables and words using Webster’s Speller, she no longer guesses at words when reading them in context.

Note 1: all but 2 of the most commonly taught 220 sight words can be taught phonetically.

Note 2: I disagree with Flowers’ statement that this confirms that dyslexia is biologically based. While some forms of dyslexia probably have a genetic component, sight word teaching could be the cause of many of these brain differences, and parents who do not know phonics cannot teach their children to sound out words at home, which could account for the seeming genetic pattern of transmission. I was taught with a bit of phonics, then with whole word methods using sight words. My parents sounded out words for me when I struggled with words at home. A parent who did not know phonics would not be able to provide this kind of help for their children. My dyslexia page has more information about dyslexia, including links to articles and presentations about the brain changes that occur when dyslexic students are taught phontic reading and spelling.

Note 3: Spelling Books in the 1700's and early 1800's were used for both phonics and spelling purposes, and were used to teach children to read. Noah Webster himself explains this in his 1828 American Dictionary of the English Language. The entry for spelling-book reads, "n. A book for teaching children to spell and read."

### Equity in Mathematics Education (January 2008)

It's amazing what you find on the NCTM site.

"Equity in Mathematics Education (January 2008)"

NCTM Position

"Excellence in mathematics education rests on equity—high expectations, respect, understanding, and strong support for all students. Policies, practices, attitudes, and beliefs related to mathematics teaching and learning must be assessed continually to ensure that all students have equal access to the resources with the greatest potential to promote learning. A culture of equity maximizes the learning potential of all students."
[This sounds vaguely nice enough, but they don't leave it at that.]

"A culture of equity depends on the joint efforts of all participants in the community of students, educators, families, and policymakers:"

"All members of the community respect one another and value each member’s contribution.
[Except for the contributions of parents and mathematicians and anyone else they disagree with. We're not allowed to contribute. All we get are open houses and the opportunity to be "informed". Even the national math panel only gets to define "a first step". NCTM gets the rest.]

"The school community acknowledges and embraces all experiences, beliefs, and ways of knowing mathematics."
["Ways of knowing mathematics"? Did the national math panel define this? Did they define various ways to know algebra? They did the opposite. They defined what algebra is.]

"All necessary resources for optimal learning and personal growth of students and teachers are allocated."
[This is the more money escape clause.]

"High expectations, culturally relevant practices, attitudes that are free of bias, and unprejudiced beliefs expand and maximize the potential for learning."
[As long as they are in charge of defining what all of this means.]

All students have access to and engage in challenging, rigorous, and meaningful mathematical experiences."
["Meaningful mathematical experiences"? How about having access to rigorous curricula, quality teaching, and no excuses? How about making sure that kids actually learn math, not experience it?]

"Such practices empower all students to build a relationship with mathematics that is positive and grounded in their own cultural roots and history."
[OK, I reject the zero because it wasn't grounded in my own cultural roots and history. I want Roman Numeral Math.]

"Different solutions, interpretations, and approaches that are mathematically sound must be celebrated and integrated into class deliberations about problems."
["Must be?" As long as they are not the traditional algorithms.]

### spelling test

I was feeling smug until I got to number 8.

Number 15 threw me, too, though I'm pretty sure I spell that word correctly when I'm writing as opposed to taking a multiple choice test.

sigh

I'm thinking I could use Megawords myself.

DWT Spelling Test

### What Algebra? When?

I came across this message from Mr. Fennell when I went to the NCTM site to see if I could find any information about the math panel. I only found their press release about the panel report, but there was a big emphasis about data on the home page: "Focus on Data/Probability". Keep that in mind when youy read this message.

"President's Messages: Francis (Skip) Fennell"

"What Algebra? When?"

NCTM News Bulletin, January/February 2008"

"Currently, about 40 percent of eighth-grade students in this country are enrolled in first-year algebra or an even higher-level math course (for example, geometry or second-year algebra)."
[Really? What kind of algebra? Surely not the math panel type.]

"As Chambers (1994) notes, algebra for all is the right goal—we just need to make sure that we’re all targeting the right algebra in our teaching. This algebra would focus on topics like expressions, linear and quadratic equations, functions, polynomials, and other major topics of algebra. (Note that these ideas will be discussed in the National Math Advisory Panel’s report on algebra topics.)"
[OK, what is the percentage now, and how many of these kids get help outside of school. How difficult is that kind of research?]

"At a time when maintaining our nation’s competitive edge means encouraging more students to consider math- or science-related majors and careers, should we address the challenge by moving more students into higher levels of mathematics earlier? Well, I am not so sure."
[If students aren't ready, then it can't be the fault of the school or curriculum? It's also not a matter of when you get to algebra, but how well you are prepared for algebra. Many kids can't even handle algebra when they get to college! So, the problem is that many kids can't get to algebra at ANY time, but he sees the problem as a "when" problem. The rule is (apparently) that if you want to deflect criticism, redefine the problem.]

"Yes, we have more students taking higher-level courses in mathematics, and yes, the path to a good job often begins with algebra. But is mandating algebra for all seventh- or eighth-grade students a good idea? Teachers of algebra frequently tell me that far too many of their students are not ready for algebra, regardless of how it is defined (first- or second-year algebra, integrated mathematics curriculum, etc.)."
[Well, if they are not ready for algebra, then figure out why. Will they magically be prepared by ninth or tenth grade? No, our high school has to have lots of remediation to fix K-8 school problems. This is not a "when" problem. "When" allows them to avoid fixing problems.]

".. most teachers indicate that their students don’t know as much about fractions as they would like. By fractions, I mean fractions, decimals, percents, and a variety of experiences with ratio and proportion."
[Duh? And this is NOT a curriculum and/or a teaching problem?]

"Of course, we must not overlook the importance of integrating the essential building blocks of algebra in pre-K–8 curricula, especially during the middle grades. Work with patterns is probably overemphasized in some quarters as the defining component of algebra with younger learners, but early experiences with equations, inequalities, the number line, and properties of arithmetic (such as the distributive property) are foundations for algebra. Silver (1997) notes that integrating algebraic ideas into the curriculum in a manner that helps students make the transition from arithmetic to algebra also prepares them for what occurs later in algebra."
[So why does NCTM support curricula that don't meet these goals?]

"So is early access to algebra a good idea? Sure—for some—probably for many. More importantly, however, all students who are working to secure this valuable "passport" should begin their study of algebra with all the prerequisites for success, regardless of when the opportunity comes their way."
[They don't see that they have anything to do with this lack of preparation? They should have a big section called Focus on Algebra, not Focus on Data/Probability. Low expectations and blame shifting permeate this message from the president of NCTM. They lack any ability for critical self-analysis.]

## Monday, March 17, 2008

### students who struggle

I've just noticed this wording of a Core Finding listed on the Math Panel's 2-page fact sheet:

Explicit instruction for students who struggle with math is effective in increasing student learning. Teachers should understand how to provide clear models for solving a problem type using an array of examples, offer opportunities for extensive practice, encourage students to “think aloud,” and give specific feedback.

Explicit instruction for students who struggle with math ----- that population is approximately half of all students K-12, right?

If students who struggle with math need explicit instruction, it's hard to see where school districts can justify adopting constructivist curricula and then supplementing or remediating with Pinpoint Math and the like.

### Zig and Don Crawford on motivation, Rule 1

Rule 1: Always assume that there is a basis in evidence for the conception the students have about schoolwork and specific actions. If students behave as if the word-attack part of the lesson is aversive, there is a basis in fact for their belief. The word-attack portion of the lesson has become aversive to them. The solution is to find out why and correct it.

Fixing Motivation Problems
by Zig Engelmann & Don Crawford
Direct Instruction News Fall 2007

### black dogs

Hah!

Back when Temple and I were working on Animals in Translation, I came up with the idea that black dogs are better.

I came up with that idea because Temple told me skin color in animals matters; an animal needs some black skin in order to be neurologically typical.*

Temple was adamant that the color that mattered was the color of an animal's skin, not his fur, but I didn't believe her. I'd always loved black animals of any kind, clear evidence that black(-furred) animals are better, right? (Bayes strikes again.)

I started asking people about their dogs. I've been doing that for a few years now, and the survey always comes out the same. Black dogs are better. I'm sorry. They are. I've met maybe one person who'd owned both light and dark dogs who thought the yellow or white dogs they'd lived with were calmer & smarter than the black ones.

The best dogs of all: black mutts. This is my conclusion.

So today I came across a study of Labrador retrievers in suburban backyards that includes this passage:

The colour of the dog was the factor most highly correlated with the amount of time the dog spent on problem type behaviours, with gold dogs spending the most time on these behaviours especially if they were untrained. It has been noted that genetic influences have a strong effect on many behavioural responses (Wright and Nesselrote, 1987; Serpell, 1987). Dogs that are gold in colour may be more genetically predisposed to being strongly attached to their owner [very true of Abby!] and thus more anxious and prone to chew things when isolated than dogs that are black or chocolate coloured. In a study by Houpt and Willis (2001), it was found that chocolate Labradors were less likely to be presented for problem behaviour than the other colours. Yellow (gold) dogs, however, were over represented for aggression compared to the other colours. Other examples of dog colour associated with behaviour include red/golden English Cocker Spaniels being more likely to show aggression than black ones (Podberscek and Serpell, 1996).

The fact that having no training was only likely to increase problem behaviours in dogs if they were gold may say more about the temperament of gold dogs than about the effects of training. Podberscek and Serpell (1997) found that the type of training had no significant effect on whether dogs had high or low aggression. Voith et al. (1992) also found no affect of obedience training on the likelihood of dogs showing problem behaviours.

I happen to own a yellow Lab: a yellow Lab (Abby) and a black mutt (Surfer).

There's a reason why Marley & Me was written about a yellow Lab.

* This doesn't apply to "black" and "white" people, she said, because white people aren't white. White people are beige.

## Sunday, March 16, 2008

### is this ironic?

Here is a page with what seems to me a nice explanation of seductive details (aka page splatter) that is itself filled with seductive details.

hmmm....

Just looked at two of the other pages: one seductive detail per page.

I give up.

### my field trip to Barnes and Noble

Lots of goodies!

Physics for Entertainment by Yakov Perelman

Earth Science SparkChart!* (Picked up an SAT Math SparkChart, too, just because.)

Algebra 101 (terrific layout with no seductive details)

looks good, remains to be investigated further:

Theory of Almost Everything by Robert Oertner

* Speaking of earth science, I read an earth science article in the TIMES the other day, and I understood (practically) every word. If this keeps up I'll do great on my Regents Earth Science exam!

### Around the Edublogosphere: Homework #1, Homework #2, Classroom Management

Homework #1--How Mrs. Bluebird gets her students to do homework

Mrs Bluebird teaches middle school in a low SES school. A couple of years ago, she attended a national conference for middle school teachers, and came away with this gem.

One of the workshops that Mrs. Eagle and I attended was on increasing student motivation. As anyone who has ever taught middle school knows, these kids can be slugs. We had a lot of problems with kids turning in work, especially homework, and were looking at some innovative ways to motivate them. (I wish I could remember who the presenter was, but alas, I don't.) The presenter put forth a lot of good ideas, but the one that resonated with us was something we call the Homework Helper. He said that the number one reason kids don't do homework is because they don't understand it.

His solution is to give the kids the answers to the homework.

Okay, I know what you're thinking because you could have heard a pin drop in that room as we all looked at each other and went, "What????" Homework is, after all, practice. If a kid doesn't get it, and does the homework wrong (if he does it at all), then he's repeating the wrong thing. He's learning and remembering something that is wrong. However, if you give the kid a key to check the work, then they're doing it correctly, and learning it correctly.

Wouldn't it be great if all teachers were as insightful as Mrs. Bluebird?

Homework #2--Dana Huff wades into the studies on homework and achievement

It all started with a discussion over at The Teachers' Lounge on homework (Homework Myth? Kids Need a Break, Ban Bad Homework, Alfie Kohn Weighs In , Drudgery or the Pursuit of Knowledge, Spotlight on the Home, The Dichotomy of Homework, and a Final Word from Grant Wiggins).

Dana Huff (who teaches high-school English at a private school) carried on with digging into the research on the value of homework. She and I share two frustrations:
1. What counts as homework? There doesn’t seem to be much rigor in the definition.

2. Does the age of the student affect the results? In other words, is homework equally effective in first grade and tenth grade?

The Blogger Formerly Known as RdKd, now blogging as Catching Sparrows, sums up the classroom management discussion

I do, in fact, stand at the door between each period, but what is so illuminating about what Mr. K says is that I do often feel as if I’m establishing a contract with students in which certain behaviors will not be indulged in my classroom, and I know the other effective teachers out there do the same. Over at Joanne Jacobs, I commented about the gray areas between “show ‘em who’s boss” management styles and the fallacy that its only alternative is some sort of “let’s all gather and hold hands” soft approach.

It’s a social contract, with the heavier responsibilities and behavior restrictions actually lying (rightfully so) on the teacher’s shoulders. In return for the students’ agreement to behave civilly and responsibly, to communicate their needs and dissatisfaction in an appropriate manner, and to reject inappropriate behavior within the classroom walls, the teacher’s portion of my personal, internal contract (the one I relay to them through my actions throughout the year) reads something like this:

You will have to go read her post to get her list of behavior expectations--for herself.

I have been observing a 4th grade student for one of my gradschool classes. In second grade, this young man had a host of behavior issues. Miss C., his teacher this year, is brilliant in several dimensions, not the least of which is classroom management. I don't know if Miss C. has a contract with her students similar to Catching Sparrow's, but she behaves as if she does. Miss C's classroom is a calm, orderly, respectful place. My young student is thriving and learning. Miss C's approach lets him stay out of frustration and fear, which means many, many fewer behavioral problems. Fewer behavioral problems means the student is experiencing academic success, which again contributes to better behavior.