kitchen table math, the sequel: 3/23/08 - 3/30/08

Saturday, March 29, 2008

Engage Me!

So is this what schools mean when they insist on developing 21st century skills and go on and on about preparing our children for the jobs that don't exist yet?

I'm speechless.

If you're a glutton for punishment, and watching Engage Me! just wasn't enough, I dare you to read Engage Me or Enrage Me: What Today's Learners Demand.

If you still have words left at that point, please share them.


way off-topic:

I've just this moment read the last line of Robert Harris' Ghost. Incredible. I may have to study this book. The narrative structure strikes me as perfect. Maybe I'll pick up a copy of James Bell's Plot and Structure and read the two together. That's a fancy way of saying Don't open the book unless you're prepared to stay up nights reading.

bonus: His portrait of a writer having a breakdown is worth the price of admission alone.

For me it was.

Friday, March 28, 2008

"Scandinavian Excellence": Tinted Glasses

Liz posted here about Scandinavian excellence in schools. According to that article, "They found that educators in Finland, Sweden, and Denmark all cited autonomy, project-based learning, and nationwide broadband internet access as keys to their success.".

funny. The WSJ posted this article, "What Makes Finnish Kids So Smart?" a month ago. They quote the same consortium. But that article didn't cite the sophisticated technology, the project based learning. It also didn't say anything about age 4-7.

It did have a quote from someone from COSN:
"Officials from the Education Department, the National Education Association and the American Association of School Librarians saw Finnish teachers with chalkboards instead of whiteboards, and lessons shown on overhead projectors instead of PowerPoint. Keith Krueger was less impressed by the technology than by the good teaching he saw. "You kind of wonder how could our country get to that?" says Mr. Krueger, CEO of the Consortium for School Networking, an association of school technology officers that organized the trip."

Other choice excerpts:
"Visitors and teacher trainees can peek at how it's done from a viewing balcony perched over a classroom at the Norssi School in Jyväskylä, a city in central Finland. What they see is a relaxed, back-to-basics approach. The school, which is a model campus, has no sports teams, marching bands or prom."

For those wondering about the homogeneity of the student body,
"Despite the apparent simplicity of Finnish education, it would be tough to replicate in the U.S. With a largely homogeneous population, teachers have few students who don't speak Finnish. In the U.S., about 8% of students are learning English, according to the Education Department. There are fewer disparities in education and income levels among Finns. Finland separates students for the last three years of high school based on grades; 53% go to high school and the rest enter vocational school. (All 15-year-old students took the PISA test.) Finland has a high-school dropout rate of about 4% -- or 10% at vocational schools -- compared with roughly 25% in the U.S., according to their respective education departments"

Definitely looks like the article writers at least are writing what they want to echo. Who knows what the various groups took away.

State Tests

The St Paul Pioneer Press recently published a sample test for the Minnesota Comprehensive Assessment II. The MCAII Sample Test covers grades 3-11.

Reading: The reading selections appear to be all actual newspaper articles. Check out the 3rd grade selections on pages 3-4. Does your 3rd grader have the background knowledge and the reading ability understand an article entitled "Of 1200 Toys Tested For Lead, 35 Percent Had Lead: Coalition Releasing Consumer Guide Today"? "The American Academy of Pediatrics recommends a level of 40 ppm of lead as the maximum that should be allowed in children's products..." Not to mention how much sleep your child is going to lose if she does happen to understand it! And I always thought newspaper articles were aimed at the 8th grade level. Silly me.

Math: The typical potpourri of too many topics, heavy emphasis on probability and patterns, and advanced topics before the basics are mastered. Pick a page at random and see if you can tell what grade level it is. A closer look reveals some real zingers. For example, check out problem 9 in the 5th grade section on page 29. You see a visual of two pies. The apple pie is divided in to 5 pieces and 3 are shaded. The peach pie is divided into 4 pieces and 3 are shaded.

You are instructed to "[u]se the figures" to answer the question.

The question: Raphael's mom made 2 pies for the family. If they ate 3/5 of the apple pie and 3/4 of the peach pie, how much more peach pie was eaten than apple pie?

Great, as long the visuals are omitted! Can someone tell me how the pictures can be used to solve this problem? Seriously, I'd love to know. They seem to me to actually obstruct any thought process that might lead to the correct answer.

Susan Wise Bauer on writing

Susan Wise Bauer has posted additional preview chapters to The Complete Writer: Strong Fundamentals over at Peacehill Press. In Chapter 4, The Three Stages, she has this to say:

What You’re Not Doing

But what about journaling, book reports, and imaginative writing?

In Years One through Four, it’s not necessary for the student to do original writing. In fact, original writing (which requires not only a mastery of both steps of the writing process, but the ability to find something original to say) is beyond the developmental capability of many students.

There is plenty of time for original writing as the student’s mind matures. During the first four years, it is essential that students be allowed instead to concentrate on mastering the process: getting ideas into words, and getting those words down on paper.

Some children may be both anxious and willing to do original writing. This should never be discouraged. However, it should not be required either. Students who are required to write, write, write during elementary school are likely to produce abysmal compositions. Take the time to lay a foundation first; during the middle- and high-school years, the student can then build on it with confidence.

In the Well Trained Mind forums, in response to the common lament of children shutting down when it comes to creative writing, she had this to say,

"[I]t's absolutely NOT necessary for an educated person to write creatively. You either enjoy it, or you don't, and forcing a child who doesn't have a bent for creative writing to do creative assignments can result in a child who loathes ALL kinds of writing."

Here's another favorite nugget from The Three Stages:

It’s important to resist the my-child’s-writing-more-than-your-child pressure.

Your neighbor’s seventh grader may be doing a big research paper, while your seventh grader is still outlining and rewriting. Don’t fret. Those research papers have been thrown at that seventh grader without a great deal of preparation. He’s probably struggling to figure out exactly what he’s doing, making false start after false start, and ending up with a paper which is largely rehashed encyclopedia information. I’ve taught scores of students who went through classroom programs which had them doing book reports, research papers, and other long assignments as early as third grade. This doesn’t improve writing skill; it just produces students who can churn out a certain number of pages, when required.

As someone who’s had to read those pages, I can testify that this approach is not, across the board, working.

Finally, someone really gets it.

New from Susan Wise Bauer:
Writing with Ease: Strong Fundamentals
Writing with Ease Workbook 1
Writing with Ease Workbook 2
First Language Lessons 4

More fun with number lines and fractions

Attempting to understand fractions as numbers, and that arithmetical operations on fractions follow naturally from arithmetical ops on whole numbers, we started to familiarize ourselves with arithmetic on the number line. Picking up from where I left off in this previous post, we remind ourselves that the critical key is not to be counting the Fence Posts, but the jumps. We could explain 7 - 4 by jumps, rather than counting the hash marks on the number line. Once that is crystal clear in our minds, we can then abstract away from the jumps, and think instead of motion on the number line as continuous motions, where we moved along the number line progressively:

This idea is a kind of intermediate step. For now, let's think of this motion to facilitate us thinking about the number of Unit Segments we've moved. That is, instead of thinking of addition in terms of jumps, we can think of addition as a continuous motion whose number of Unit Segments corresponds to the number. What's a Unit Segment? It's the length of the segment between 0 and 1.

We add 4 and 3 by starting at Zero and moving 4 units to the right. Then we move 3 units to the right. We get 7. Graphically, we see that on the number line, we reached 7 by moving right for 7 segments. So 7 is made up of 7 unit segments, just as it was reached by 7 jumps.

Subtraction, then, means moving the correct number of unit segments to the left. 7 - 3 means moving 7 units to the right, and then moving 3 units to the left. The number we reach, 4, is made up of 4 unit segments, just as it is 4 jumps from zero.

Another way to look at this is that the number 7 can be represented EITHER by the position of the point on the number line, OR by the length of the segment from 0 to 7. They are equivalent representations (isomorphisms, actually.) (It's not clear to me that this is the most child-friendly way to think about it. Someone with 4th graders needs to find out if it's useful to present this to children--I think it's easier to think of the jumps being of the unit length for now, and we'll adapt the lengths of the jumps.) Wu puts this as "we identify this standard representation of the number with its right endpoint" of the segment that starts at Zero. But if this is too hard, the jumps still work: the number is represented as the location on the number line that we end up at after we've completed all jumps.

We are now ready for fractions. No, really, we are! We begin with a specific fraction, and we work up to a generalization.

The fraction 1/3 has a numerator and a denominator. We will show how it is a fraction of our unit segment, the segment from 0 to 1. We now partition our unit segment into thirds. 1/3 is then symbolized by starting at 0 and moving to the right one partition of one segment:

The position of the right endpoint of that segment is the location of 1/3 on the number line.

But just as we could partition the unit segment into thirds, we can partition all of the segments between consecutive integers into thirds. The hash marks of these partitions are the fractions whose denominators equal 3. Again, this is saying "we identify this standard representation of the number with its right endpoint." These thirds function as our new "units", and we could make jumps on them just as before. The place we land on our jump is the number.

So, now that we know how to imagine the number line in thirds, we can generalize:

The fraction m/n is the point on the number line, when we partition each unit into equal nths, and then make m jumps to the right of Zero on those new nth-hash marks, or equivalently, partition each unit into n equal segments, and then move m nth-segments to the right.

There you have it--a definition of a fraction! Now, an important point glossed over in these examples is that a point on the number line is unique. Its representation, however, can be varied. This is worth stressing: a number, be it whole or fraction, corresponds to the unique location on the number line (or, as Wu would put it, to the unique line segment whose right endpoint ends at that location.)

With whole numbers, most people don't think that's very interesting. But with fractions, there are many representations for the same number. This can be confusing. The best way to clarify it is to say: they are still the same number-it's just a different way of reading the same number. This leads to a clear understanding of equivalent fractions.

Consider the fraction 4/3. we will show that (5 x 4)/(5 x 3) = 4/3 as follows: First, locate 4/3 by its unique spot on the number line. We do this by breaking each unit into thirds, and then jumping 4 such 3rd-units to the right:

Now we partition each 3rd-unit into 5ths. Doing so immediately gives us 15-ths for each unit. Now we count how many 15ths our point 4/3 is: the answer is 20. We could do this for any partition--7ths, 162nds, etc. It doesn't matter. No matter how you partition into equal bits, you haven't moved off of your point on the number line. Nothing about repartitioning changed your location.

Similarly, improper fractions are a breeze: we can see that 10/3 is 10 jumps to the right on the 3rd-hash-marks. It's also 3 jumps on the Unit Hash marks and 1 jump on the thirds: 3 and 1/3. Since the same point on the number line can be read in either fashion, they must be the same number, because a number is defined by its unique point on the number line.

We'll pick up with common denominators next.

Scandinavian Excellence in Elementary Education: Is It Project-Based Learning and Teacher Autonomy, or Other Factors?

How do the Scandinavian educational systems differ from the US? A group of educators went on a fact-finding tour.

A delegation led by the Consortium for School Networking (CoSN) recently toured Scandinavia in search of answers for how students in that region of the world were able to score so high on a recent international test of math and science skills. They found that educators in Finland, Sweden, and Denmark all cited autonomy, project-based learning, and nationwide broadband internet access as keys to their success.

What the CoSN delegation didn’t find in those nations were competitive grading, standardized testing, and top-down accountability—all staples of the American education system.


Kati Tuurala,[snip] credits Finland’s success to its major reforms of the 1970s, which included an emphasis on primary education for everyone in the country. “That’s the reason for our present-day success,” Tuurala said.

In all three countries, students start formal schooling at age seven after participating in extensive early-childhood and preschool programs focused on self-reflection and social behavior, rather than academic content. By focusing on self-reflection, students learn to become responsible for their own education, delegates said.

Barbara Stein, manager of external partnerships and advocacy for the National Education Association, said Scandinavian countries “encourage philosophical thought at a very young age. … Grading doesn’t happen until the high-school level, because they believe grading takes the fun out of learning. They want to inspire continuous learning.”

“My teacher” and “the teacher” are terms of respect, not only when used by the students, but also by the school leader or headmaster. The teacher is most often viewed as a mentor, someone who has both knowledge and wisdom to impart and plays a key role in preparing students for adulthood.

In Finland, for instance, teaching is one of the most highly venerated professions in the country—and only one in eight applicants to teacher-education programs are accepted. All teachers there have a master’s degree.

Go read the whole thing and come back and discuss.

I'm thinking that the teacher preparation, and the social learning (social cognition) curriculum for ages 4-7 has a lot to do with students' achievement.

The meaning of good work

I just received my gently used copy of Spelling Mastery, Level E by Robert Dixon and Siegfried Engelmann. Perhaps my favorite passage in the Teacher's Book is The Importance of Corrections.

Correct all mistakes immediately.

The very term "correction" has a negative connotation, but corrections are a positive, critical part of effective teaching. Corrections are the equivalent of MORE INSTRUCTION, and that instruction is prompted directly by needs that students demonstrate.

Bringing every student to mastery on every component of Spelling Mastery exercises is vital to the effectiveness of the program. Spelling Mastery departs from traditional spelling programs in that once a word (or generalization) is introduced in the program, IT DOESN'T GO AWAY. Everything in the program is reviewed cumulatively, as insurance of long-term retention and transfer to writing.


Using quick pacing and an upbeat, positive attitude toward corrections are major keys to effectively correcting an entire group. Don't forget to communicate to students how pleased you are when you are not having to correct them, particularly on difficult items.

Remember this about corrections. Students don't always know what constitutes a good job. When you hold them to a very high criterion of performance (especially early in the program) they learn exactly what your expectations are. They also see that by having high performance expectations, you are making it easier for them to learn and use what is being presented. Once they see the rules of the "game," their performance improves dramatically. They learn fast and generalize well -- things they would not do if they never learn what "good work" really means.

I think that Karen Pryor would approve.

Thursday, March 27, 2008

spelling & reading

A passage from Elizabeth's post on picture overshadowing and learning to read via sight words as opposed to phonics:

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.
Richard G. Parker

I asked palisadesk a couple of weeks ago whether in her experience good spellers were also good readers and vice versa. Here's what she had to say:

> Are your 7th graders who tested Level A less skilled readers than the kids
> who tested higher??

Yes, definitely, but the correspondence isn't exact. That is, some lousy spellers are excellent readers but none of the SUPER-lousy spellers were excellent readers. All the sixth and seventh graders who scored at level A (meaning they couldn't spell even regular words like plant correctly) were middling to poor readers, though some had B's in reading and most were not considered remedial readers. I found a consistent correlation between word decoding and word recognition and basic spelling.

In fact, some kids with poor orthographic memory learn to read via spelling. They don't really master distinctions like "supper" vs. "super" or "stared," "starred" and "started" until they master the spelling skills involved. Then it seems to click in.

I'm familiar with Ehri's work [post on Ehri's new study t/k] and the fluency research is supportive, too, of the need for visual representations of the word (Shaywitz's "word form area" in the, I think occipital, lobe, is key to fluent word recognition, and also connects to the right hemisphere visual and meaning areas).

Many normal kids, not "LD," get stalled at the point of multisyllable words such as the ones they encounter in more advanced (especially nonfiction) text -- words like photosynthesis, heterogenous, homeostasis, bicameral, etc. They have not been trained to "see" word parts (morphemes) and they have a strategy of just taking a wild guess based on the first letter or letters. Guessing from context or pictures no longer works well in middle school and beyond.

I call these kids "bumper car readers" -- they crash into the front end of the word and hope for the best, but can't segment it into its parts. Once they can easily do this, long words are no problem and are easy to read AND spell, with a few glitches here and there (is it occasional or ocassional?). Bob Dixon, who wrote the Spelling Mastery series and Morphs as well, calls this the morphographic principle, and says kids need to g through abnout half of Spelling Through Morphographs (about 70 lessons) to really internalize it. I bet a year of SM level D,E, or F would do the same. For reading, rather than spelling, a good quick fix is a program called REWARDS from Sopris West ( -- it teaches multisyllable word decoding, morphemes, word parts, some vocab and spelling, and fluency (rate). There's an intermediate level for kids in grades 4-6 and another for grades 7 and up -- the strategies are identical but the vocabulary is more advanced in the secondary version.

BTW Bob Dixon wrote a great book for grownups that you can sometimes find cheaply on alibris -- it's called "The Surefire Way to Better Spelling." It's got a lot of the handy stuff from Morphs in it. But the actual programs are better for teaching kids, because they are carefully structured to provide the needed practice, transfer etc.

For years now I've had a feeling spelling was much more important than anyone knew (or than anyone knows today, I should say).

la même chose

Some fifteen years ago [circa 1990] I was working on problems and issues involved in the design of simulators for training in complex skills. Projects included applications in military and in paramedical and nurse training. In both contexts, the projects arose as a result of evidence that the investment in simulator design and development – many millions of dollars annually in the case of the military training applications – was not leading to the expected improvements in learning. Analysis showed that the design teams involved habitually followed certain principles, one of which was “fidelity” – the simulated experience should replicate as closely as possible the “real life” experience of doing the job.

However, a search of the literature on fidelity in simulator design revealed a chain of studies, initiating from some of the earliest work of Robert Gagné in the1950’s, clearly demonstrating that absolute fidelity to the real-life job situation in the simulated exercise was ineffective, especially in the early stages of learning. Partial simulation, that focused attention on the key skill elements that had to be learned, that enhanced relevant guidance and feedback information, and eliminated irrelevant “noise” that could distract the learner, was shown to be the most effective design for the early stages of learning. It was, indeed, quite possible to construct a research-based design theory for the incorporation of fidelity – indeed different aspects of fidelity (physical, perceptual, etc.) – in specific ways at specific stages of the learning process. However, the bulk of those involved in the design of simulators were completely ignoring this research base.


Maybe education is doomed to progress in ever repeating circles of reform that lead nowhere. Maybe this is a phenomenon of society in general, summarized so neatly in the French expression “plus ça change, plus c’est la même chose”- the more things change the more they stay the same.

"Constructivism Revisited: Progress in ever decreasing circles..."
by Alexander Romiszowski, Ph.D. Contributing Editor

I really don't like reading about bad teaching in places like the military.

Or in nursing.

One more reason to die in your sleep.

On our way to fractions: the number line

Prof. Wu's document Critical Concepts for Understanding Fractions tries to clarify operations on fractions by use of the number line. Wu stresses that fractions should be defined succinctly as numbers (rather than as pieces of wholes, or operations, etc.) To show that what students already know about whole number guides them to understand what is true of fractions, he shows the relationship between arithemetic on the number line for whole numbers and arithmetic on the number line for fractions.

That means we need to already have mastery of the number line for whole numbers in order to make fractions clear. That mastery means fluency showing addition, subtraction, multiplication and division with remainder on the number line, as well as flency with properties of commutativity, associativity and distribution. The number line should reinforce all of what we already know about whole number arithmetic. It should not be new--we should use our mastery of whole number arithmetic to arrive at mastery of the number line, and then we'll use mastery of the number line for whole numbers to achieve understanding of fractions.

So let's begin:
The standard number line for non-negative numbers is a ray, beginning at Zero and extending to infinity. We show a piece of it here:

By convention, this ray extends to the right. The non negative integers, or whole numbers, are shown at the has marks. The space between 0 and 1 is a fixed length. This length is the same length as between 1 and 2, and any two consecutive numbers. We don't care what the length is, but we do like to recognize that it represents a UNIT. Later, this will matter more, but for now, just keep clear that the actual distance between the 0 and 1 is arbitrary; the issue is that that spacing is consistent for each consecutive number.

The biggest confusion with the number line is that it starts at Zero. This is probably odd for kids because we don't hold up our first finger and say "Zero!" But here, the First Tick Mark is the Zero. So we have to learn to go from counting on our fingers, where we sort of assume "zero means nothing", including no place/position, to an abstraction where Zero exists.

On the number line, Zero is our starting point. This is actually how we counted on our fingers, but we didn't think about it very often.

Now, we use Zero, the place, on our number line as our starting point, and count by keeping track of MOTIONS, or JUMPS from the starting point to the next tick mark. We don't count the fence posts; we count the motions from fence post to fence post.

Practicing with the number line until it's ALWAYS clear that there's 1 more fence post than motions between fence posts is critical.

Addition, then, looks like this: jumps to the right. (Again, this is convention.)
We add 4 and 3 by starting at Zero and making 4 jumps to the right. The first jump takes us to 1, then fourth to 4.

Then we make 3 more jumps to the right. The first of these takes us to 5. The last takes us to 7. We end up at 7.

Commutativity of addition should be obvious now: whether you made 4 jumps to the right and then 3 to the right, or 3 to the right and 4 to the right, you always ended up at 7. Associativity should be just as clear: 3 + (4 + 5) is the same as (3 + 4) + 5 for the same reason.

Subtraction is then defined as well. Subtracting a number from another number, you jump to the left (by convention.) 7 - 3: We start at zero, and take 7 jumps to the right to arrive at 7. Then to subtract 3 FROM 7, we make 3 jumps to the left. We end at 4.

Multiplication then is just a set of grouping of jumps. 4 X 10 is 4 Tens. That means, you learn to make a set of ten jumps, and then you call that grouping something like "a ten jump". And now you make a total, from 0, of 4 of these grouped jumps.

(Exercise: How do you do division with remainder? Answer: For X divided by Y, you start at zero and make grouped jumps of length Y. When you reach X, you stop. You count the number of grouped jumps you were able to complete, and the remainder is the number of singleton jumps left to reach X.)

The critical key no matter what the operations is to remember not to be counting the Fence Posts, or the hash marks between the numbers (inclusive or exclusively), just the jumps.

If the jumps are clear, then the next thing to realize was that you could have visualized those motions between numbers not as jumps but as continuous motions, where we moved along the number line progressively:

This is the simplest way to connect it back to fractions, because when we look at fractions, we now will consider the space between the hash marks.

Wednesday, March 26, 2008

collect and correct

The terrific Huffenglish blog, which I think Liz Ditz may have introduced me to, pointed me to research on homework.

Turns out the question of whether collecting and correcting homework produces more learning than simply assigning homework & not looking at it has been asked and answered.

Collect-and-correct wins hands down.

Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement by Robert J. Marzano, Debra J. Pickering, & Jane E. Pollock

The average effect size for assigned-but-not-graded-or-commented-upon homework is .28; average effect size for graded homework is .78; average effect size for homework with teacher's comments or feedback is .83.

.83 is a very large effect size:

Effect size = (mean of experimental group - mean of control group)/standard deviation Generally, the larger the effect size, the greater is the impact of an intervention. Jacob Cohen has written the most on this topic. In his well-known book he suggested, a little ambiguously, that a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small (Cohen, 1988). The usual interpretation of this statement is that anything greater than 0.5 is large, 0.5-0.3 is moderate, 0.3-0.1 is small, and anything smaller than 0.1 is trivial. There is a good site that describes all this that is worth a visit for those really interested.

And, from Robert Coe, of The CEM Center:

Interpreting Effect Sizes

Provided our data have the kind of distribution shown in Figure 1 (a ‘Normal’ distribution), we can readily interpret Effect Sizes in terms of the amount of overlap between the two groups.

For example, an effect size of 0.8 means that the score of the average person in the experimental group exceeds the scores of 79% of the control group. If the two groups had been classes of 25, the average person in the ‘afternoon’ group (ie the one who would have been ranked 13th in the group) would have scored about the same as the 6th highest person in the ‘morning’ group. Visualising these two individuals can give quite a graphic interpretation of the difference between the two effects.


Another way to interpret effect sizes is to compare them to the effect sizes of differences that are familiar. For example, an effect size of 0.2 corresponds to the difference between the heights of 15 year old and 16 year old girls in the US. A 0.5 effect size corresponds to the difference between the heights of 14 year old and 18 year old girls. An effect size of 0.8 equates to the difference between the heights of 13 year old and 18 year old girls.

What is an effect size? A brief introduction
Robert Coe

As far as I can tell, collecting and correcting (which would cost nothing to implement in a school district not collecting and correcting) is a far more powerful force for student achievement than small class size (which costs a bundle) or SMART Boards ($4000 a pop).

So --- bonne idée!

Class Size: Counting Students Can Help (pdf file)

Robert Marzano's book, which is published by the ASCD, is extremely well-known in the field of education.

HyperStat: Measuring Effect Size
The Effective Use of Effect Size Indices in Institutional Research by Christi Carson (pdf file)
Food for Thought by Howard S. Bloom (pdf file)
Statistical Power Analysis for the Behavioral Science by Jacob Cohen
Evidence-Based Education
What is an effect size? by Robert Coe

The Homework on Homework, Part One

Cognitive Load Theory (CLT) For Beginners

Cognitive Load Theory (CLT) from Greg Kearsley's site, Theory into Practice:

Sweller's Cognitive Load Theory: Overview

This theory suggests that learning happens best under conditions that are aligned with human cognitive architecture. The structure of human cognitive architecture, while not known precisely, is discernible through the results of experimental research. Recognizing George Miller's research showing that short term memory is limited in the number of elements it can contain simultaneously, Sweller builds a theory that treats schemas, or combinations of elements, as the cognitive structures that make up an individual's knowledge base. (Sweller, 1988)

The contents of long term memory are "sophisticated structures that permit us to perceive, think, and solve problems," rather than a group of rote learned facts. These structures, known as schemas, are what permit us to treat multiple elements as a single element. They are the cognitive structures that make up the knowledge base (Sweller, 1988). Schemas are acquired over a lifetime of learning, and may have other schemas contained within themselves.

The difference between an expert and a novice is that a novice hasn't acquired the schemas of an expert. Learning requires a change in the schematic structures of long term memory and is demonstrated by performance that progresses from clumsy, error-prone, slow and difficult to smooth and effortless. The change in performance occurs because as the learner becomes increasingly familiar with the material, the cognitive characteristics associated with the material are altered so that it can be handled more efficiently by working memory.

From an instructional perspective, information contained in instructional material must first be processed by working memory. For schema acquisition to occur, instruction should be designed to reduce working memory load. Cognitive load theory is concerned with techniques for reducing working memory load in order to facilitate the changes in long term memory associated with schema acquisition.
From Kevin McGrew's blog, a guest post by Walter Howe, Cognitive Load Theory for School Psychologists:

  • Have you ever done something successfully, but not known exactly how you did it? It’s a common experience. It works, but we generally either cannot repeat this feat readily or transfer this performance to other, similar situations. We have performed a particular task successfully, but we haven’t really learnt a lot.
  • In CLT, this one-off success isn’t learning (in other theories it is regarded as learning, and termed implicit learning or procedural knowledge). Learning only occurs when we have abstracted a series of steps and rules that we can repeat in similar situations or even teach others so they, too, can be successful. These rules and procedures are called schemas or schemata and they are stored in long-term memory. Novices, by definition, either don’t have a schema for a particular learning task or it is very unsophisticated. Experts, on the other hand, have many, very sophisticated schemas, which they apply without thinking (i.e. the application of these schemas has become automatic).
  • CLT is concerned with how we learn or (in CLT terms), how we develop schemas and automate them and become experts. It applies to learning relatively complex material, as schema acquisition and development are generally unimportant for simple tasks, although how simple a task is depends both on the task itself and the individual who is learning how to do it successfully, as you will see.
Go read the whole thing, and while you are at it, poke around with Kevin McGrew's other posts on cognitive load theory and math, for starters.

CLT has obvious implications for the design of instruction in mathematics, among other things.

Kevin McGrew keeps a number of wonderful resources: IQ's Corner ("An attempt to share contemporary research findings, insights, musings, and discussions regarding theories and applied measures of human intelligence. In other words, a quantoid linear mind trying to make sense of the nonlinear world of human cognitive abilities.") Tick Tock Talk: The IQ brain clock ("An attempt to track the "pulse" of contemporary research and theory regarding the psychology/neuroscience of brain-based mental/interval time keeping. In addition, the relevance of neuroscience research to learning/education will also be covered.")

Reading Wars in Texas

This week, the Reading Wars continue in Texas. (They have actually been going on since 1826.)

Here are a few excerpts from the online article:

On the other side are supporters of a more traditional approach proposed by Donna Garner, a retired teacher and conservative education activist. Those board members say students need more spelling drills, phonics lessons and practice in grammar identification to learn to read and write properly. This side also favors developing suggested reading lists with a heavy emphasis on classical literature.

"They teach reading by getting them to read every third or fourth word," Garner said of current methods. "But the problem is when they start getting the harder (words) at about Grade 3, and they hit a wall."

3rd grade slump? Maybe everything is bigger in Texas.

Dozens of passionate educators are expected to attend today's public hearing. Garner won't be among them; she says her presence is too polarizing. But that doesn't mean she isn't devoted to the cause.

"I've committed years to this cause, and I cannot believe that people can be so wrong about what kids need to learn," she said. "We've lost millions of Texas children in the last 10 years. We can't afford to lose millions more."

I'll be keeping Texas in my prayers.

Tuesday, March 25, 2008

palisadesk on school improvement and the Catholic school

A year ago I went to a large conference on school improvement (the focus was on using test results and assessment to determine instructional priorities, action plans, etc.). There were a number of presentations from schools that had made significant progress. My principal and I went and chose to attend presentations on low-income, high-diversity urban schools, since we figured that would be most relevant to our situation. We both noticed that nearly all of the presentations (teacher and principal teams presented what they had done and details of their implementations, etc.) were from Catholic schools.

What stood out was not so much the specifics of what they did, but the commitment of the faculty. They were clearly dedicated and willing to go more than the extra mile in a way that would be unusual in a public school, even one with very competent and caring staff. Whether it was their faith per se, or more of an ethos of service, it was quite strikingly different from what one would encounter in a public school, even a good one (and I have worked in a few).

Now of course these were "success stories," and it does not follow from this that every Catholic school is similarly graced with highly committed staff, but the factor of shared vision is probably an important one that should not be underestimated, and is difficult to replicate in a diverse public system, where many values are simply not shared.

help desk - quadratic word problem

Kirk and Montega have accepted a job mowing the soccer playing fields. They must mow an area 500 feet long and 400 feet wide. They agree that each will mow half the area. They decide that Kirk will mow around the edge in a path of equal width until half the area is left.

43. What is the area each person will mow?

44. Write a quadratic equation that could be used to find the width x that Kirk should mow.

45. What width should Kirk mow?

46. The mower can mow a path 5 feet wide. To the nearest whole number, how many times should Kirk go around the field?

Glencoe Algebra, p. 537
ISBN: 0-07-873316-2

I'm completely befuddled.
The equation I've come up with works when I plug in the book's solution (in Comments section), but I can't use it to arrive at the solution for x to begin with.

Too bad I don't have the answer key.

update: district has now said it will provide the answer key "ASAP. "

We will see.

can you FOIL the answer key?

Sunday, March 23, 2008

saving Catholic schools

Seton Hall’s Epics program — the acronym stands for Educational Partners in Catholic Schools — is one of about 20 similar efforts by Roman Catholic colleges and universities. Taken together, the initiatives have placed several thousand teachers in urban Catholic schools, and collectively they comprise a sort of religion-based version of Teach for America.

One major difference, besides the religious component, is in national visibility. Teach for America has become a virtual brand name on elite college campuses and a coveted item on graduate-school applications and corporate résumés. Programs like Epics have received far less public acclaim, and yet they are vital to the almost literally lifesaving role that Catholic schools play as an affordable alternative to chronically failing public schools in many low-income areas.


In 1970, more than half the teachers and administrators at Catholic elementary and secondary schools were unpaid clergy members. As of 2008, Ms. Helm said, clergy members account for about 5 percent of teachers. At the same time that nonwhite and often non-Catholic pupils are increasingly seeking out Catholic schools, the number of such schools in cities has been shrinking in the face of rising costs and insufficient revenue.

St. Patrick’s, where Mr. Encarnacion teaches, offers an archetypal example. The school, which runs from prekindergarten through eighth grade, is actually a merger of three schools. Roughly two-thirds of its 440 students are black Protestants. Of 26 full-time teachers and administrators, one is a member of the clergy.

Offering Teachers Incentives; and a Chance to Live Their Faith
by Samuel G. Freedman

We've had sad news here. The Catholic school down the hill from the high school is closing.

I hate to see these places go.

Catholic Schools and the Common Good