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Carolyn's dead right about Saxon: the program moves students along at a brisk clip.<sup>*</sup> 

I was thinking about it just last night, while I was doing my own Saxon lesson.

I'd put money on it I'm learning <I>lots</I> more than Christopher, whose book is, technically speaking, more advanced.

And I'd put money on it he'd end the year knowing more than he's going to know with Prentice-Hall if he were using Saxon, too. 
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<font size=+1>slow and steady wins the race<font size=2>

The conventional wisdom about 'behavioral' programs like Saxon Math is that they're remedial; they're for slow learners.

Well, it's true. 

If I were teaching a class of slow learners, I'd choose Saxon Math in a heartbeat. 

But Saxon also moves fast learners through material at a fast clip. If you're a fast learner, you just work through the material more quickly. Back when Christopher and I were using Saxon 6/5, the 5th grade book, we were doing complete full lessons a day for a time. 

Only recently have I realized that <B>Teaching to Mastery means accelerating a student's rate of learning</B>.

High achievers move faster with Direct Instruction:

<blockquote>
Tarver and Jung reported that the Direct Instruction program was equally effective for lower and higher performing children who participated in the study. Other studies provide additional evidence that Direct Instruction programs accelerate the learning of high-performing students in language (Robinson & Hesse, 1981), reading (Schaefer, 1989; Sexton, 1989), and science (Vitale & Romance, 1992).%BR%<font size=-2>source:%BR%Watkins & Slocum, The Components of Direct Instruction,  JOURNAL OF DIRECT INSTRUCTION, summer 2003, p. 75-110<font size=2>
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<font size=+1>low achievers move faster, too<font size=2>

Direct Instruction is, <I>expliticly</I>, a teaching approach designed to produce 'maximum acceleration' for all students at all levels. <font size=-2>(see: Student-Program Alignment and Teaching to Mastery by Siegfried Engelmann)<font size=2>

Not only can low achievers be accelerated, when they <I>are</I> accelerated their learning curves look like those of fast learners:
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http://www.kitchentablemath.net/twiki/pub/Bloggers/AccelerationNotRemediationLogPage/DIlearnercurves.jpg%BR%
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I find this counterintuitive and almost bizarre.

<font size=+1>When taught to mastery, low IQ students learn at the same clip as high IQ students?<font size=2>

Hard to believe.

On the other hand, I wouldn't be surprised. So many of our decades-old beliefs about students and learning are just pure ideology.

So I hope Engelmann's right. 

Here's what he has to say:
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<blockquote>
Even students who would be predicted to have low levels of achievement benefit greatly from Direct Instruction. Gersten, Becker, Heiry, and White (1984) examined the yearly achievement test profiles of students in Direct Instruction classrooms to determine whether annual gains made by students with low IQ scores differed significantly from the gains made by students with average or superior IQ scores.

Figure 2.11 [above] shows the yearly gains made by students in reading as measured by the Wide Range Achievement Test. As shown in this figure, students with higher IQ test scores started at higher achievement levels and ended with higher levels than their peers with lower scores. However, the pattern of growth of students with low IQ scores is remarkably similar to that of other students. The group with the lowest scores (under 70) gained nearly as much each year in reading as students with much higher scores. By the end of third grade, those students with the lowest IQ scores were performing at the 70th percentile, or a grade equivalent of 4.3.

The results are even more pronounced in math as seen in Figure 2.12 [below]. This figure shows the students’ performance on the Metropolitan Achievement Test. The growth rate for all groups of students corresponds to one grade equivalent for each year in school.
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http://www.kitchentablemath.net/twiki/pub/Bloggers/AccelerationNotRemediationLogPage/DIslowlearnermath.jpg%BR%
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<blockquote>
These results provide evidence that Direct Instruction is appropriate for, and effective with, a wide variety of individuals including those with low IQ scores, those with IQ scores in the average range, and those with high IQ scores. In addition, because children in this study were taught in small homogeneous groups (having students with relatively the same skill levels), the gains of students with lower IQ scores were not made at the expense of other students nor the other way around. 

Several reviews of research focusing on the use of Direct Instruction with special education populations have all converged on the finding that Direct Instruction is measurably effective with these students. White (1988) reviewed 25 such studies and found that all comparisons favored the Direct Instruction group. Forness,
Kavale, Blum, and Lloyd (1997) conducted an analysis of various intervention programs for special education and determined Direct Instruction to be one of only seven interventions with strong evidence of effectiveness.

Perhaps because Direct Instruction programs have been so successful with students who have failed in other instructional programs, their use is commonly associated with children who are behind, who are failing, or who are at risk for failure. And some have questioned their appropriateness for general education. However, Figures 2.11 and 2.12 provide direct evidence of the effectiveness of Direct Instruction for students with IQ scores in the middle range and those in the upper range.

Engelmann and Carnine (1989) found that typical second graders who had received 2 years of Direct Instruction scored an average 4.6 grade equivalent in reading on a standardized achievement test. The children’s average scores in science and math were 4.0 and 3.4, respectively. Other researchers have arrived at similar findings. Tarver and Jung (1995) investigated the effects of a Direct Instruction math program (Connecting Math Concepts) and a discovery learning math program on the math achievement and attitudes of general education students in the primary grades. They found that, at the end of second grade, the children in the Direct Instruction program scored higher on measures of math computation and math concepts than children in the comparison group. In addition, children in the Direct Instruction program had significantly higher scores on a survey of attitudes about math. Finally, Tarver and Jung reported that the Direct Instruction program was equally effective for lower and higher performing children who participated in the study. Other studies provide additional evidence that Direct Instruction programs accelerate the learning of high-performing students in language (Robinson & Hesse, 1981), reading (Schaefer, 1989; Sexton, 1989), and science (Vitale & Romance, 1992).
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<font size=+1>acceleration for all students through Direct Instruction in a nutshell<font size=2>

   * The group with the lowest [IQ] scores (under 70) gained nearly as much each year in reading as students with much higher scores. 

   * The results are even more pronounced in math as seen in....performance on the Metropolitan Achievement Test. The growth rate for all groups of students corresponds to one grade equivalent for each year in school.  

   * because children in this study were taught in small homogeneous groups (having students with relatively the same skill levels), the gains of students with lower IQ scores were not made at the expense of other students nor the other way around  

   * Perhaps because Direct Instruction programs have been so successful with students who have failed in other instructional programs, their use is commonly associated with children who are behind, who are failing, or who are at risk for failure. 

   * Engelmann and Carnine (1989) found that typical second graders who had received 2 years of Direct Instruction scored an average 4.6 grade equivalent in reading on a standardized achievement test. The children’s average scores in science and math were 4.0 and 3.4, respectively. Other researchers have arrived at similar findings. 

   * <B>discovery versus Direct Instruction</B>: Tarver and Jung (1995) investigated the effects of a Direct Instruction math program (Connecting Math Concepts) and a discovery learning math program on the math achievement and attitudes of general education students in the primary grades....at the end of second grade, the children in the Direct Instruction program scored higher on measures of math computation and math concepts than children in the comparison group. In addition, children in the Direct Instruction program had significantly higher scores on a survey of attitudes about math. Finally, Tarver and Jung reported that the Direct Instruction program was equally effective for lower and higher performing children 

   * <B>more on high achievers</B>: Other studies provide additional evidence that Direct Instruction programs accelerate the learning of high-performing students in language (Robinson & Hesse, 1981), reading (Schaefer, 1989; Sexton, 1989), and science (Vitale & Romance, 1992).
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<font size=+1>[[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/KumonAverageStudentsBeyondGradeLevel][KUMON is an acceleration program, too]]<font size=2>

Interestingly, KUMON adds the element of teaching children to become 'self-learners,' i.e. self-teachers:

<blockquote>
Our aim should be to educate our students so well through the Kumon Method that they don't have to depend solely on classroom activities to be able to deeply understand the course content. Students who develop this capacity will have a good chance to enter leading universities. To make this possible, we must help students <B>acquire the ability of self-study from an early age</B> and accelerate their level of study beyond their school grade. (Emphasis added)
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Here is the irony.

When Ed and I told our 'Team' that we want the school to be responsible for Christopher's learning, as opposed to Christopher being responsible for Christopher's learning, the principal objected. Christopher has to learn to be responsible, he said. He'll need it in high school. 

It was another helicopter parent moment, though neither hostile nor critical. 

The essential meme in middle schools everywhere seems to be that helicopter parents don't 'allow' their children to grow up and become responsible for themselves and their studies.

But KUMON says that a Teach-to-Mastery approach <I>builds responsibility in children</I>.

I don't understand quite how that happens.

But I believe that it does.

I think this is one of those [[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/BlookisForBayes][Bayesian issues]] where parents have the right idea, without knowing <I>why</I> they have the right idea. 

A parent sees his child floundering and failing, and knows this is a <I>bad thing</I>.

The parent <I>knows</I> the child will be far better off if the school continues to 'coddle' and 'protect' him while he learns the material his teachers are teaching. 

But <I>how</I> do we know this?

What are we basing it on?

It's the same problem parents have 'knowing' fuzzy math is bad.

The minute I heard about fuzzy math, I knew it was bad.

But could I say <I>why</I> it was bad?

No. 

Same thing with 'responsibility.' 

Obviously, I want Christopher to grow up to be a responsible person.

And yet, somehow, I'm in the position of arguing 'against' Christopher being responsible. 

I know &mdash; in the Bayes way of knowing &mdash; I'm right.

But I don't know why.

%MAROON%<B>UPDATE 10-20-2006: Now that my child has spent one year in a math class in which full responsibility was placed upon parents for reteaching and students for learning, this issue is no longer a mystery.%BR%</B>%BLACK%
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<sup>*</sup> ed. update 4-21-2006: <B>Dan</B> has some reservations on this score. It's certainly true that the Saxon books have a tremendous amount of repetition from one book to the next.

[[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/MikeFeinbergKIPP][Mike Feinberg of KIPP on spiral curricula]]%BR%
[[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/SteveAndSusanJOnSpiralCurricula][Steve and Susan J on spiral curricula]]%BR%
[[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/AccelerationNotRemediation][acceleration versus remediation]]%BR%
[[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/SpirallingStories][parents' stories about spiralling curricula]]
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-- Main.CatherineJohnson - 26 Jan 2006

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