Entries from SaxonMath

They're the same age--both going into 6th grade--and my sense is that math is probably this boy's strong suit.

I just gave him the Saxon placement test, and he placed into Saxon 7/6, which is the 6th grade book.

That would be great, but here's the hitch: he has been taught almost nothing about fractions at all. (He had a good math teacher--he and Christopher were in the same Phase 3 class for the first half of this year--who left to have a baby. So it seems that the subject of fractions & decimals fell through the cracks.)

So.....if anyone has thoughts, I'd like to hear them. I'll probably go ahead with 7/6, but that means I'm starting a 6th grade book with a child who's been taught virtually nothing about fractions and decimals.

update

whose job is it, anyway?

Why should I be the person figuring out that this boy hasn't been taught fractions & decimals?

Why shouldn't the school be figuring this out? (Yes, the school might say he was taught fractions and decimals, but didn't learn them. However, it's clear to me that there are certain topics he simply hasn't even heard of, because with some topics he'll say, 'I kind of remember that.' In other words, he can tell me which topics he failed to learn, or didn't learn well enough to retain, or whatever it is. With topics like adding fractions, he simply doesn't know anything about them, and has no memory of having been taught.)

So, yes, the school might say, 'He was taught, but he didn't learn.'

But so what?

If he was 'taught' and 'didn't learn,' then he wasn't taught as far as I'm concerned. It's the school's job to perform formative assessment to know what students have and have not learned.

Then it is the school's job to re-teach if a child has not learned.

Then, if the child still isn't learning, it's the school's job to figure out what else he needs.

common sense from The Education Wonks

That's one of my major concerns with NCLB. When students don't do their homework or study for exams, or even attempt to do classwork, it's still considered to be the teacher's fault if the students don't achieve their federally-mandated level of proficiency in reading, math, and science.

And yet NCLB doesn't give me, as a teacher, the authority to require student's who aren't even attempting the work to stay after school and complete their assignments. Unless the kid has committed some breach of the school's disciplinary policy, I can't keep them any later than the school regular dismissal time.

The No Child Left Behind Act holds me solely accountable for my students' academic progress but doesn't give me the authority to help make that happen, especially for children that are considered to be "at risk" of failing to meet minimal standards of academic progress.

Sadly, under the law as it is now written, a large number of children are going to be left behind.

But that's not the case with Christopher's friend. This boy has done all of his work; he's a serious student; his parents are serious parents.

It's the school's responsibility to know whether this boy has or has not learned how to add, subtract, multiply, and divide fractions, and to teach or re-teach the subject if he hasn't.

I've been wanting to get this book for a while, but funny -- it's never at our (beloved) local used bookstore. That is usually the sign of a good book -- people are hanging on to their copies. Of course, it's too bad for me. I may have to break down and buy it new. Everything I'm seeing about this book is telling me it's a great read.

I am a fan of the whole concept of Core Knowledge, being (I suppose) an educational traditionalist. Ben went to a Core Knowledge elementary school, and he learned quite a bit in spite of himself.

Thomas/Vorblik writes:

The philosophy behind these movements is often described in the literature as "constructivism''. Thus, Jack Price, President of the NCTM, says in [6]: "the standards are based on research and on a constructivist theory of learning . . . Critics may not agree with the theory, but they cannot say that the standards are not research based.'' But, as The Schools We Need shows, they can and do.

Is it possible that the ideas recommended by the NCTM are the very ideas that already pervade the schools they are supposed to reform?
Such a hypothesis is reinforced by the teaching methods that the NCTM and other reform groups advocate for achieving higher-order thinking skills. These "new'' methods include attention to individual needs and learning styles, discovery learning, and thematic learning. But these teaching techniques are essentially the project-oriented, child-centered methods that have long dominated educational thought and have prevailed for decades in our schools.
--Hirsch p. 132

If Hirsch is right about the entrenchment of constructivism, then we are the radical reformers; and it feels kind of nice for a change.

This supports my feeling that constructivism, in some form, is (like the poor) always with us, at least since Rousseau. And I suspect before Rousseau as well ... perhaps Ed knows what predated Rousseau's ideas, or perhaps Hirsch has even written about it; the reviews say that he treats the historical genesis of constructivism at length.

Fie on Rousseau, anyway. One reason I tend toward educational traditionalism is that, deep down, I believe mankind has not evolved or changed that much since we were all tribesmen. I really do feel that civilization is precariously thin, and not to be taken for granted. I guess Lord of the Flies made a big impression on me (either that, or I'm just paranoid).

Hirsch recommends a math curriculum which he says is "reasonably close to what research is telling us about how students learn". Surprise: it's Saxon math.

The new Saxon Math is here.

In place of the Winner Homeschooling Family from Oklahoma or wherever it was, standing in their yard beaming because they got picked for Homeschooling Family of the Year, we have Out-of-the-box Beaming Learners from Hemera Technologies or some such. The site takes hours and hours to load, and when it finally does load, you can't see any books.

I'm trying to remain calm.




I take it back.

I am not calm.

This is a horror.

We've also worked through the entire PRIMARY MATHEMATICS 3A & B lessons on fractions, and we've done all the workbook problems. Now we've started on the fraction lesson in PRIMARY MATHEMATICS 4A. I've been planning to post something about this, because we're doing the Singapore lessons with a friend of Christopher's.

We also did one bar model story problem from Primary Mathematics 3A almost every day, and we'll carry on doing one a day for good. As simple as the early 3rd grade problems are, there were plenty Christopher couldn't do.

When I say 'couldn't do' I mean that he specifically couldn't conceive of or draw the bar model.

He could do the problem alone, without the bar model. He could set it up and solve it. But he couldn't represent it visually or spatially at all.

Interestingly, he also can't even come close to doing perimeter problems of figures like this:



Asking him to figure an L-shaped perimeter is like asking him to do calculus. Completely beyond his capacities.

This is a kid whose one adult aspiration is to be an architect.

So we're gonna keep working on geometry. Nets are next!

does geometry predict math ability?

I think I'm noticing the same pattern in the 4 children whose math work I know best.

Two of these kids are Christopher & Lew. Lew lives next door and is one of Christopher's good friends. Christopher and Lew both seem to be strongest in verbal skills, and both are befuddled by perimeter problems once you get past finding the perimeter of a square or a rectangle.

Meanwhile, the two other boys I know & have worked with--I'll use their initials, 'G' and 'P'--are the exact opposite.

These two boys may have real math talent. I don't know how to spot Real Math Talent, but that's what I think I see in them.

Both G & P, when I showed them a more-complicated perimeter problem, instantly got it. They SAW it; it 'popped' for them, the way a hidden figure in a hidden figures test 'pops' for Temple (Grandin).

I think that's pretty interesting.

Anyways...this is one of the reasons I love the bar models. I'm pretty sure bar models develop visual-spatial 'seeing' or 'understanding' in verbal kids (& in verbal adults like me).

I've mentioned before that the first girl ever to win an international Olympiad was a Singapore student brought up on Singapore bar models. Singapore, as far as I can tell, is the only country to use bar models systematically throughout the first 6 years of elementary school math.

using bar models to teach fractions in Saxon 8/7

Christopher spent the whole summer complaining about his daily bar model. Then, yesterday, when we used a bar model to find If 2/5 of the the 270 fans wore green to the game, how many fans wore green? he said, 'These bar models do help!'

They really do.

Here's an example of a Saxon bar model used to solve a diferent fraction-of-a-group problem:






From now on I'm going to have him draw the model and link it to the computation. I'm going to have him explain to me why & how 2/3 x 270 is the same thing as dividing 270 by 3 and then multiplying 90 by 2, as the bar model has you do.

Then we're going to keep doing that until it makes sense.

I am happy to say that I was wrong when I wrote that there was no evidence supporting Saxon. There was a large study done in California, in which some pilot schools adopted Saxon math and saw their scores take off relative to those from other schools, which stuck by their existing curricular choice (Everyday Math). Apparently this study was not covered in the What Works Clearinghouse Middle School Math study, probably because it addressed not middle school but elementary school math.

The results are summarized here. -- Carolyn, October 3, 2005.

Original post

Apropos of the analysis of the research on Connected Math I did yesterday...

in the interest of fairness in reporting, it doesn't appear the evidence is overwhelmingly in favor of Saxon Math either. Saxon Math is a favorite 'traditional' curriculum of mine and Catherine's, sort of an 'anti-Connected Math' -- and it's served us very well in teaching our own boys -- but here is a roundup of what the (scarce) research says about it.

First, there is one study of Saxon Math that fully meets WWC's (the What Works Clearinghouse's evidence standards for a quality research study. Here is what WWC has to say about the study.

Peters, K. G. (1992). Skill performance comparability of two algebra programs on an eighth-grade population. Dissertation Abstracts International, 54 (1), 77A. (UMI No. 9314428).

WWC: Peters (1992) reports that students in the intervention and control groups showed gains on the Orleans-Hanna test during the course of the school year (that is, from pretest to posttest). However, the test score gains of the two groups did not differ significantly. There was no evidence that the Saxon Algebra curriculum (intervention) was more or less effective than the University of Chicago Mathematics Project curriculum (control).

The WWC warns that the number of kids involved in the study (only 36 kids total) was so small that it was very difficult to tell whether there had been a significant effect.

A second study of Saxon math met the WWC's evidence standards with reservations (meaning there was some not-quite-fatal flaw in the study's design).

Crawford, J., & Raia, F. (1986, February). Analyses of eighth grade math texts and achievement (evaluation report). Oklahoma City: Planning, Research, and Evaluation Department, Oklahoma City Public Schools.

WWC: Crawford and Raia (1986) found that students in the intervention group scored significantly higher than students in the comparison group on math computation, but not on total math or math concepts.

Here's what they did: they had one class working from Saxon Algebra, and one from Scott-Foresman Math (is this another non-fuzzy program? I wish someone would compare Saxon to a fuzzy curriculum!). They used a test (the California Achievement Test -- CAT) that breaks up the math scores into a math computation and a math concept subscore; they gave both a pretest and a posttest.

When they were ready to do their analysis, they matched students from each group one-for-one according to their total CAT score. That means that Fred may have been matched to Annie, because they both got X as a total score on their tests; but Fred may have far outscored Annie on the concepts subtest (with Annie stronger on computation).

They then looked at the mean posttest subscores of the matched children as a whole. They found that the mean scores were improved significantly in math computation, but not in math concepts.

The problem with this analysis is that, because the kids were matched by their total scores, one class may have had more conceptual skills to begin with as a whole. So interpret these results with care. However, here's a

Non-statistical side comment: I have read some parent reviews of Saxon that agree that it is great for developing computational skill, and not so great for developing problem-solving skills. These results are consistent with those comments.

To Summarize the Research: Of the two studies the WWC has reviewed on this topic, the second study has (to my mind) a flawed design, and the first shows an insignificant effect from the Saxon Math curriculum over the University of Chicago curriculum.

I'd love to be able to say that science proves Saxon is clearly superior; so far, it doesn't. However, Catherine and I have both said in the past that we would definitely recommend Saxon for kids who are struggling in math, and need to build their confidence by experiencing success; and I'll stand by that.

With Saxon, your kids won't miss out on any critical skill, either.

I received an email today from Joanne Cobasko of Save Our Children from Mediocre Math (SOCMM). She drew my attention to a couple of articles, describing the improvement in California test scores after the new California standards were adopted.

I looked at the attachment and skimmed the second article. It's not a research study (i.e., it would not meet the WWC's standards of evidence for a well-designed study); but it is definitely one situation where Saxon went head-to-head with fuzzy math, and won.

Here's the letter (thanks, Joanne!):

Hi Carolyn:

Both these studies show fantastic classroom results achieved in CA classrooms which are attributed to Saxon Math. I believe Bishop & Hook down play the Saxon Math connection in favor of the "CA Key standards" so as not to promote any particular curriculum over another, they choose to promote the math standards employed.

You will find references to the curriculum in their write ups though.

http://www.nychold.com/talk-hook-040404.pdf
http://www.nychold.com/report-wbwh-040619.pdf

There is also a great district comparison of standardized test results from Manhattan Beach, CA and Palos Verdes, both well to do communities (the comparison was provided to me by Martha Swartz from Mathematically Correct). [Note: Joanne points out that Manhattan Beach uses Saxon Math, and Palos Verdes uses Everyday Math. -- Carolyn]

Palos Verdes has the edge with a 26% Asian population, and one Kumon or other type tutoring facility for every 429 grade 2 - 6 elementary age student (the tutoring info was my informal review of the school population per the state testing info and a print out from the Kumon & other centers indicating their locations within a 5.22 mi radius).

Manhattan Beach, with a 7% Asian population and only 1 KUMON facility in town for 2,1113 grade 2-6 students, outscores Palos Verdes on the 2004 test scores.

Jo Anne Cobasko
Save Our Children from Mediocre Math (SOCMM).

We have used both curricula, and will continue to talk about both of them frequently in these pages.

However, different kids need different curricula, and we wanted to help parents make a choice that benefits their child best (and also matches best their own ability and willingness to support their kids' math learning; Saxon is probably a lighter-maintenance curriculum than Singapore).

Along those lines, I've been planning to post these links to Saxon and Singapore word problem comparisons at Paula's Archives for a while. Have a look. Although it's not really an accurate comparison to pull word problems out of each book and compare them at random, I think it's generally true that Singapore has exceptionally good word problems. Any kid reared on these problems is going to be mathematically in darn good shape.

Here, too, are Paula's Archives on Saxon Math and on Singapore Math individually.

When assessing your kids for their placement in Singapore Math, you'll almost certainly find that their placement level doesn't agree with their grade level. Somewhere on this site, though, someone posted that the Singapore math levels through 6B are actually completed when Singapore kids are in the 8th grade (or the age-equivalent grade in Singapore, whatever it is) -- something to tell your kids if they refuse to back up and do something they think is meant for younger kids (I sure wish I could find the post where that comment was).

Here's a post Catherine did on a price comparison between the Saxon and Singapore product lines.

And let me say, once again (repetition is key!), that if you have a kid who is getting lost or has gotten lost in math, and needs to make up lost ground and rebuild their confidence, I still don't believe anything beats Saxon math's approach.

I'm relieved, I have to say. I've been semi-sanguine about the possibility of having two math curricula in your child's life, a fuzzy one at school & a non-fuzzy one at home.....but the fact is, I haven't (really) had to face that situation.

Last year, in 5th grade, Christopher had SRA Math at school, and Saxon Math 6/5 at home. SRA Math is a very tough textbook to teach from (impossible for me, and experienced teachers have told me the same). But it's not hardcore fuzzy.

David Klein points out that most U.S. textbooks are fuzzy to some degree. That was certainly the case with SRA. Time and again I'd read a passage--this was when I was just setting out to reacquaint myself with math--and not have a clue what it meant. Invariably this was because the text would lay out a couple of observations and then pose a question to the student, who was supposed to draw the appropriate conclusion.

I remember one day I was trying to figure out how to find the equation for the slope of a line, and there was just no way. Finally my neighbor came over, read the passage, and said, 'You'd have to know how to do it to understand this explanation.' Then she showed me how.

Still and all, SRA Math wasn't a b*s book. Not at all. The math was real, and Christopher had two good teachers who'd had plenty of experience getting math into kids' heads in spite of the problems.

I'm pretty sure that in Christopher's case it was a net plus that he had two separate math curricula. He had far more time-on-task, and he had the benefit of seeing the same subjects from slightly different vantage points (which always helps me, and is probably good for everyone).

But I wasn't having that feeling about Ben at all. SRA & Saxon, OK. Connected Math & Saxon? Blech.

So, long story short, I was getting worried about Ben. I'm glad Connected Math is gone.

Saxon into the breach

I've now read quite a few negative assessments of Saxon, by people whose judgment I respect. These are folks on the web--a couple of obviously intelligent homeschoolers, as well as Robert, who writes the brightMystery blog. Robert told me he wants to like Saxon, but just does not--and that students who come to his college courses having been homeschooled in Saxon aren't ready. (That's a paraphrase, so take it with a grain of salt.)

I have misgivings myself. Sometimes I worry Saxon is TOO 'structured'; I worry about pattern training--that Christopher is going to be a Saxon Boy who can only do Saxon Problems typed in Saxon Font.

Thus far that has not been the case. As far as I can tell, all of Christopher's Saxon knowledge has transferred to SRA (and, now, to Prentice Hall).

Other times I've felt the Saxon books are too scattered & fragmented. The fragmentation of topics is a deliberate strategy on Saxon's part, the intention being to use the principles of spaced repetition and distributed practice. That makes sense, but when I taught the Primary Mathematics Grade 3 chapter on fractions to Christopher and his friend Greg it was so much more satisfying and rich, or seemed so.

So.....I've been a heavy-duty Saxon user; I owe Christopher's move to Phase 4 math to Saxon 6/5. And I know the knowledge he's gained from Saxon is conceptual as well as procedural.

But in spite of all these good things, I have Nagging Doubts.

Usually I pay attention to Nagging Doubts. But in this case I think my doubts are either wrong or, more likely, misdirected. Because I keep coming back to Saxon every time I'm in trouble, and Saxon keeps bailing me out.

Saxon vs Dolciani

Christopher has another quiz today, on algebraic expressions.

I was reading along in Prentice Hall, which said that in an expression like x + 7 the x and the 7 are terms.

In an expression like 2x + 7, 2x and 7 are terms, and 2 is the coefficient.

Well, right away I was confused.

Does a term mean you're either adding or subtracting?

Does multiplication mean you don't have a term, you have a coefficient?

That seemed wrong.

So I got out my copy of Mary Dolciani's Pre-Algebra: An Accelerated Course.

I'd been thinking, OK, I'm done with Saxon. There are just too many negative opinions out there, Mary Dolciani's a genius, my neighbor's son liked Dolciani's book, it's shorter than Saxon & we're pressed for time......this year I'm going with Dolciani.

She was no help at all:

In the expression 9 + a, 9 and a are called the terms of the expression because they are the parts that are separated by the +. In an expression such as 3ab, the number 3 is called the numerical coefficient of ab.

Saxon on coefficients

Saxon 8/7 has an entire lesson on algebraic terms. Lesson 84, page 571. I haven't read it yet--I've skimmed--but it's obvious that when I do, my question will be answered.

Here's how he opens:

We have used the word term in arithmetic to refer to the numerator or denominator of a fraction.

Right off the bat, he's made the smart metacognitive move. We have used the word 'term' to refer to numerators and denominators, and it's a good thing to point this out to the student, because otherwise, at some point (probably not now, but later on, when it will really ball things up) the student is going to think, Wait! Doesn't TERM mean DIVISION? Does it mean FRACTION? Does it mean NUMERATOR & DENOMINATOR?

OR WHATTTTTTT?????????

I'm going to go out on a limb and say that Saxon Math is the most metacognitively aware textbook I've encountered to date. Constantly, the books remind you of what you have learned, and point out to you that you are now learning an extension of that concept or you are learning a new and possibly quite different meaning of the same word.

Back to Lesson 84.

Next the book has a table of monomial, binomial, and trinomial algebraic expressions. Wonderful.

THEN the text says:

Terms are separated from one another in an expression by plus or minus signs that are not within symbols of inclusion.

Thank you, John Saxon. I needed that.

More examples follow, and eventually we get to this:

Each term contains a signed number and may contain one or more variables (letters). Sometimes the signed-number part is understood and not written. For instance, the understood signed-number part of a^2 is +1 since a^2 = +1a^2. When a term is written without a number, it is understood that the number is 1. When a term is written without a sign, it is understood that the sign if positive.

Perfect.

At least, perfect for me. What do you think?

deer in the grass

So I looked out, too, and sure enough: the young deer grazing in our lawn is darker than the young deer who was living here a month ago.

But Martine thinks it's the same one. She thinks they get dark in the fall.

It's probably time to give him a name.


(a^2 means a squared - right?)

update

The story problems!

Barry reminded me: they're dreadful. They're just wildly too-easy.

I had meant to put that in the original post, and forgot.

However, the story problems aren't the reason for my 'nagging doubts'.....the story problems are an obvious problem you can remediate easily through supplementation.

It's the other stuff.....

The short answer is yes, but for the record, here is a fuller explanation. I think the best quick introduction to constructivism and its recent history in U.S. educational practice is Barry Garelick's An A-maze-ing Approach To Math, which appeared in Education Next this year. I'll excerpt a little piece of it to answer Andy's question, entirely without Barry's permission (but hopefully with his blessing).

Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students' not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering (and taking ownership of) the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.

Texts that are based on NCTM's standards focus on concepts and problem solving, but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches "mathematical thinking" rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo without first teaching him to swim.

The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics (PSSM 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.

So how does this fine-sounding idea play out in the classroom? Kids tend to spend too much deriving everything from first principles. What gets sacrificed is time spent learning advanced skills, as Barry shows:

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I've seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student's going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method (one that is held in disdain because it uses rote memorization and is not discovered by the student) the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.

Now, consider the constructivists' argument for allowing this lack of 'domain knowledge' to persist -- kids develop deeper understanding, 21st century skills, bla bla bla -- after having read KDeRosa's "Terminator essay" on math education.

That essay just puts this nonsense to death, don't you think?

p.s. from Catherine

Here are the 2 best passages.

Smart constructivism says:

A common misconception regarding 'constructivist' theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1940)--even listening to a lecture involves active attempts to construct new knowledge.**

Radical constructivism says:

It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.

Here's a question on from Saxon 8/7 that Ben and I discussed for twenty minutes tonight.

Regurgitate the answer to this one, if you will. This problem addresses quantization error, which is at the heart of a lot of engineering problems.

Jeffrey's ruler is marked in eighths of an inch. Assuming a measurement is done correctly, what is the maximum measurement error possible using Jeffrey's ruler?

Here's an example of a number bond flash card:





You can download these cards from DonnaYoung.org, a homeschooling resource that looks pretty good, and has a page of mostly terrific paper math manipulatives, including lots of circular fractions, terrific large-print math facts drill sheets, graph paper, play money, scale paper for household furniture arrangements, and some cool-looking empty worksheets with number lines on top.

It also has triangular addition and subtraction flash cards (pdf file).

from the directions:

To use the cards, hide one of the corner numbers with your thumb or finger and let the child tell you what the hidden number is.

Saxon's fact families

Saxon Math does not use triangular flash cards.

Saxon uses four-fact families combined with Extreme Practice. If there is One Thing Christopher & I have overlearned from Saxon 6/5, it is FOUR FACT FAMILIES:

1, 2, 3

1 + 2 = 3
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2

Same deal with multiplication and division.

Here's a typical four-fact family problem from Lesson 2 in Saxon 7/6:

23.
Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.
12 + 24 = 36

Christopher can do that in his sleep.

So can I.

I probably have done it in my sleep.

I've been doing so much grade school math I sometimes dream about it.

four weeks into Saxon 6/5

Quoting from a post I wrote on this subject awhile back:

About a month after Christopher and I began working with Saxon Math 6/5, he told me,

Multiplication and division are the big brothers, and addition and subtraction are the little brothers.

Then he said,

And multiplication and division are cousins.

This is a 9-year who, just 6 weeks earlier, had been flunking math.

You have to do a lot of four-fact fact families to come up with a thing like that.

I vote for fact families

Triangular flash cards and number bonds are everywhere these days, but I don't like them. Here's why:

• First of all, the potential for confusion is huge. An addition & subtraction number card looks extremely similar to a multiplication & division number card, and separating factors from addends in a child's mind is a challenge under any circumstances.

• Second, triangular number bond cards aren't all that easy to 'read.' Kids don't naturally undestand visual displays of data; far from it. There's too much info on these cards, IMO.

• Third, number bonds are incredibly static, and I don't think math is static. Math is something you do, not something you look at. Four-fact families are action-packed; you get so good at them you can whip one of those babies out in a couple seconds flat. They're fun, and they absolutely (I'd bet money on it) prepare kids for the time when they're going to start solving problems like 2 + a = 5. When Christopher segued to 2 + a = 5 in Saxon 7/6 he didn't have a second's difficulty. He'd been inverse-operationing 2 + 3 for a year at that point, so 2 + a was just obvious.

• Last & certainly not least, I haven't had any luck with flash cards, period.

Not nearly as beautiful as Doug's number lines, but a good idea.

oops

I've just noticed that Donna Young prefers sites not link to her printable forms, and in fact these links won't access the forms. Just go to her homepage, click on math, and then find what you're interested in. The math page is clear & easy to use.


Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes

---

INeedAPlan 18 Sep 2006 - 17:21 CarolynJohnston




(blank Saxon answer sheets are attached in the Comments thread.)



Ben is not liking his new math plan. He's been doing Saxon math with an aide for around 3 weeks now, and is doing okay, but he complains a lot about wanting to go back to Ms. Math Teacher's class. She's a nice teacher, that's one thing, but also the other kids are there and he doesn't like feeling singled out. Especially not when the singling-out means he's working a lot harder than he used to in the regular class.

The other problem we are having is one we could be having with any math curriculum; Ben is trying to whip through his math homework, as fast as he can, in order to finish as quickly as possible. As a result, he makes a lot of careless mistakes. I really want to get him working more meticulously, because any work habit Ben acquires will be very hard to get rid of. Here is the plan I've implemented in order to get him checking his work more carefully. It's behavioral, a reward system; I've used behavior plans with Ben since he was a young spacey toddler because if I design them well, they work.

With Saxon 8/7, most of the homework on any given evening is review; it's mostly stuff that he could do correctly the first time if he were careful. So I've taken the risk of telling him that I expect him to get it 85% correct the first time through. If he does that, he gets rewarded (the current favorite reward is getting to watch The Simpsons on DVD when he's done with homework). If not, he does without. But I'm not crazy about this plan, and I'm looking for a replacement.

One thing I don't like is that the reward is tied directly to performance. If he ever gets a homework that is on a topic he finds more difficult than usual, the 85% plan isn't going to work very well.

The other thing I don't like about it is that it isn't working very well. He still makes careless errors the first time through his work. When he is done with that first pass through, and I say "you'd better double check your work now before you hand it in," he usually passes up the opportunity to check his work again. Checking Your Work Again is the brussels sprouts of schoolwork; I remember loathing it myself. Apparently it is SO loathsome that he'd rather risk losing his Simpson's reward than Check His Work Again..

I call this an almost-failed intervention. It's not a total failure; his error rate has dropped considerably since we started, but his first-time-through success rate is topping off at just under 85%. He still has a baseline careless error rate of around 10%, I'd say; he usually has one or two problems he's skipped over completely. You really want a behavioral plan, too, that the kid accomplishes successfully most of the time, not one where he's always just barely missing the target.

Can anyone think of a better metric for success?

And, clearly, Checking One's Work Again from the beginning, at the end of an assignment, is too disgusting to contemplate. How do you get a kid to Check His Work Again on a problem-by-problem basis?

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FractionManipulativeLessonOnReciprocals 14 Sep 2006 - 14:26 CarolynJohnston


The other night, Ben was working on his math, and I was doing something else. He paused in his work and asked me: "Mom, what's a reciprocal?"

I guess it's no surprise that he doesn't know what a reciprocal is, since it wasn't taught in his Everyday Math classes in elementary school, and since (apparently) it's not introduced in the early part of Saxon 6/5, the curriculum I was supplementing from last year. But in Saxon 8/7, which he's using this year (he tested into it, I swear), knowledge about reciprocals, and the role they play in division of fractions, is assumed.

So I'm doing reactive teaching again, but at least this time I'm reacting to the curriculum of my own choosing.

Saxon has had problems in the mixed practice the last few nights that go straight to the heart of why the reciprocal gets involved in fraction division. The questions are like this: how many 3/8s are there in 1? How many 4/5ths are there in 1?

Here's a demonstration I devised for him on the 3/8ths problem, using the tile fraction manipulatives that Catherine and I have recommended here (warning: pies won't work for this very well).

This sort of question seems to throw him off, so I start by asking other questions that sound more familiar, like: How many 2s in 8? and How many 3s in 9? Then I point out that he is getting the answer by dividing, so by analogy, we'd want to divide 1 by 3/8.

Everyone knows the rule for fraction division: Ours is not to reason why, just invert and multiply. But of course, we are modern traditionalists here, and procedural knowledge is only the beginning of our demands. We want our kids to have an understanding, too, of why they are inverting and multiplying. I taught Ben the invert-and-multiply rule, but then I wanted to convince him that the answer that the invert-and-multiply rule gives you, 2 and 2/3rds, is the right answer.

We attacked the question of how many 3/8ths go into 1 directly, using the manipulatives. The manipulatives were all placed on a sheet of paper, so I could write curly braces and labels next to the tiles. I drew a diagram below of what we do with the tiles (note that the 3/8th tiles are not really single blocks, they are 3 1/8 blocks in a row; I have to tell him to think of them as a single unit.. The labels and curly braces help with this).

It's easy for him to see that two 3/8ths will fit into the 1; I stick them below the 1 tile, and label them as "2 3/8ths". A third 3/8th will overhang the end, though. So I take the extra 3/8ths unit and break it apart into thirds (pointing out that that's what I'm doing). Two of those thirds will fit into the rest of the space in the 1. So this gives us a total of 2 and 2/3rds 3/8-units that will fit into the 1.

I don't expect that this is the end of this; we'll do this a bunch more times and hopefully it will sink in. The trick is to get the kid thinking of the divisor (in this case 3/8ths), however weird a fraction it is, as being a unit. I hope Saxon keeps this sort of problem coming for a while.

Doug Sundseth's downloadable fraction manipulatives & number lines

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SuchALittleDetail 31 Oct 2005 - 18:37 CarolynJohnston


I've been working with Ben on these problems that keep recurring in Saxon Homeschool 8/7. They are always something like this:

You have a protractor that is marked in degrees. What is the largest possible error you can make in measuring an angle, assuming that you are using the protractor correctly?

You have a ruler that is marked in half-inches. What is the largest possible error you can make in measurement, assuming that you are using the ruler correctly?

Every time we encounter one of these, I end up explaining it again. I have to explain that we're assuming, for example, that you can't eyeball a half a degree or halfway between the tick marks, even though you probably actually can. Then I have to draw an object that comes midway between two tick marks, and shade in the difference between the measured and actual length of the object. Then he gets it. A couple of days later, though, when we encounter the same problem again in slightly different form, it has to be explained again.

So tonight I asked him the same question, put slightly differently.

Assume you have a ruler. What's the greatest error you can make if you're measuring an item to the nearest half-inch?

This time, the question was no problem. So here's my question: what's so confusing about the other way to phrase the same problem?

And does this mean Saxon should rephrase the question?

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IsSaxonPlusSingaporeTooMuch 07 Nov 2005 - 23:47 CarolynJohnston


We had a request today for some information about supplementing Saxon Math with Singapore Math...

I found this site several weeks ago and I LOVE IT! I started homeschooling my two sons last year after taking them out of public school. I have been using Saxon math. Last year they were in second and third grade and I had them in Saxon 2 and 3. This year, I have them both in Saxon 5/4. I like the Saxon program because it seems to be very thorough and they have plenty of practice. Neither I nor they are very strong in mental math and I have wondered about supplementing Saxon with Singapore Math. I'd like some advice on this. Would it be overkill? To let you know about where they are now: It takes them about an hour a day to do their math lessons. They are at lesson 28 in Saxon. (It's all pretty much review—nothing they haven't had before.) They have had four tests and have done well on all of them. (They both scored 100 percent on the first three.) My older son knows his multiplication facts through 12s pretty well. My younger son is shakier on these and hasn't learned sixes, eights and twelves. I tried giving them the Singapore 3a placement test and they just couldn't do it. I started giving them the Singapore 2a placement test and they are handling that fine (though with a lot of complaints because they have to THINK about what to do in the word problems.) They both like to have me walk them through problems instead of making a stab at it on their own. Thanks in advance for any help anyone can give me. Diane

First responders on the scene (with math tourniquets) were Susan and Dan...

Susan's response:

A homeschooler friend of mine once told me that many homeschoolers use both Singapore and Saxon at the same time. I'm presently using Saxon as the core supplement curriculum for my public school child, but I add Singapore problems to whatever chapter I'm on.

Singapore's word problems are better than any of the other books I've seen because they start with one and two steps and move up to 4+ steps by their level 5.

I don't know if you've seen The Well Trained Mind book, but it has an easy to follow schedule for homeschooling all subjects throughout the years of your child. You might get some ideas of how much to do from there. Since I'm an "after-schooler," as they call me, I haven't ever looked closely at the way they set up the teaching schedule, but it looks fairly thorough.

Dan's response:

I haven't homeschooled, so I feel a little uncomfortable commenting...but only a little.

I just wanted to ask if you were testing the multiplication (and, for that matter, addition) facts with timed tests. I'm pretty sure that timed fact tests are part of the Saxon school curriculum. It seems to be a consensus opinion here at KTM that these facts must be mastered to the point of automaticity. I certainly agree, and have found any lack of automaticity to be a major hindrance as students try to move forward.

And Diane replied..

DanK, Yes, I am using timed tests for addition and subtraction, and I use multiplication fact worksheets for drill, though I don't usually time them. We are just now moving into timed multiplication tests with Saxon.

SusanS, I have read "The Well Trained Mind" and I just revisited her suggestions for scheduling. An hour a day for math seems pretty typical for what most other homeschoolers I know are doing.

I am leaning towards getting Singapore and supplementing with it. Some of my friends who use Saxon with their kids just have the child work every other problem. I've been having my sons do every problem, and, as I commented earlier, it takes them about an hour. I don't want them to get overwhelmed by having an hour and a half of math every day, so I guess I would have to cut out some of the practice problems in Saxon.

So I'll weigh in now with a few thoughts...

I think an amalgam of Saxon and Singapore is a good choice for homeschooling. With Saxon, especially in the early grades, you can be sure that you're not missing out on any essential skills. I think Singapore has a good emphasis on word problems, and I like the way they get kids thinking algebraically very early.

I home-supplemented my son a lot the last two years (we had a constructivist curriculum in 4th and 5th grade—Everyday Math), and even though I'm knowledgeable about math, there were days when I felt up to the task of 'constructing his curriculum' (so to speak) and days when I just didn't. Saxon is a great support for homeschoolers who don't want to be carefully preparing their kids' lessons every day. Singapore takes a greater background knowledge of math, and is much harder for the kids to do independently than Saxon, so to do Singapore, you'll be making a commitment to get really involved with your kids' math. Not every homeschooler wants to do this.

I'd be reluctant to cut out every other Saxon problem on a regular basis, because I think those mixed practice problem sets are the genius of Saxon. They'll revisit a skill intermittently, and if your kids are only doing even problems, they'll miss getting the practice they need if the skill only appears in odd problems (it would be genius indeed if they had enough forethought to put a given skill alternately in even and odd problems!).

You could start by trying to add Singapore word problems to each math session, and see whether that worked; you might find the kids tolerate it pretty easily. If not, you might try switching off days. You wouldn't get through either curriculum as fast, but Saxon has a lot of repetition from one year to the next, so even if you didn't get all the way through a Saxon book you'd have little cause for worry.

Another thing you might consider doing is making Saxon your main text, and supplementing from one of the Singapore books that specializes in word problems, since that's where I think Singapore really has the most to offer. Singapore has a workbook series called Challenging Word Problems Books 1 - 6 ($7.80 plus shipping; 129 pages), in a U.S. (as opposed to British English) edition. You can start at the workbook that's at the level your kids placed into; the problems are marked at a mixture of difficulty levels. This is definitely what I would do if I were constructing a homeschool program.

One more thought—my son, who has Asperger's Syndrome, got balky in second grade about doing math timed tests. He would basically refuse to deal with them, in class; although he knew the facts, he wouldn't do the timed tests because he was reluctant to deal with the time pressure. We ended up doing some heavy bribing to get him to move on those tests (once he did, he was fine). I think adding the time pressure factor is important to nudge the kids toward automaticity. Rewards in the form of treats or outings or privileges are good, I think. Competition can also be good, if it's friendly competition and not cutthroat (and if they're siblings that close in age, it could get ugly).

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CreativeSaxonWordProblem 07 Nov 2005 - 23:32 CarolynJohnston


Problem Number 3 on page 147 in Saxon 8/7 homeschool:

One hundred twenty six thousand scurried through the colony before the edentate attacked. Afterward only seventy-nine thousand remained. How many were lost when the edentate attacked?

From Merriam-Webster online:

edentate: primitive terrestrial mammal with few if any teeth; of tropical Central America and South America

So apparently ants were lost, not moon-colonists; thank goodness.

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JohnSaxonAndFrankWang 02 Dec 2005 - 21:04 CatherineJohnson



Incredible story of Frank Wang and John Saxon:

For Saxon president Frank Wang, getting good at mathematics was the answer to a personal crisis. In 1970, a doctor and school officials came to the conclusion that he had "neurological impairment" and could not be educated. This diagnosis was a great blow to his parents, recent Chinese immigrants to the US. Wang had his own solution: He noticed that what counted for intelligent in his school was an ability to do mathematics. This was the key to convincing school officials that he had a mind worth educating, he reasoned.

"I didn't want to live out this prophecy," he says. "I really wanted to prove to the doctors that I had intellectual capacity. And getting good in mathematics looked like the way to do it."

He began by studying past New York State Regents exams in mathematics - quietly, on his own time, one question at a time. It was tough at first, but he just continued working problems until he understood the principle, then moved on to another topic.

Finally, he told his eighth-grade algebra teacher that he already knew all the material in the course. The teacher sent him to the principal, who sat him down with an old Regent's exam (he'd already studied) to test the boast. Wang scored a 96.

"He asked me how I had learned all of this. I shrugged my shoulders and said, 'I don't know. It just came to me.' I outright lied, but it was such a delicious feeling. All of a sudden people's thoughts of me changed from a disabled child to someone with potential," he says.

The fact that experienced educators believed this child when he told them an entire year of eighth-grade algebra 'just came to him' is the most alarming part of this story.


Saxon

Wang met Saxon founder John Saxon after his family moved to Norman, Okla., where his father took up a position as professor of mathematics at the university. Saxon needed a research assistant, and 16-year-old Wang volunteered.

"He just struck me as a very eccentric fellow, but someone with a very strong and powerful sense of mission. He had very grandiose plans at that time. He thought that he had a better way of teaching mathematics, and the world should know about it," says Wang.

Saxon, once dubbed "the angry man of mathematics," was a retired Air Force pilot who flew 55 missions in Korea and later taught electrical engineering at the US Air Force Academy. Brash, outspoken, and never one to dodge a fight, he started his own publishing company to challenge the math orthodoxy of the day.

Smaller is better

Saxon's concern wasn't that math books were too full of pictures, chatter, and not enough problem-solving. (That came later.) In the early 1980s, Saxon argued that children should not be expected to learn math in big thematic chapters. He argued that math needed to be taught in smaller increments, with lots of practice and reviewing.

It turns out, that's exactly how Wang had taught himself mathematics. In the end, the youngster hired to punch papers and do errands contributed so much to the book that Saxon acknowledged him in the preface - and later invited him to take over his company.

"The Saxon pedagogy was incremental development: Teach in small pieces, continual review of those increments, and frequent cumulative testing. There would be no asking: Is this going to be on the test? Every Saxon test was cumulative, and every test gave kids a chance to redeem themselves," Wang says.

Saxon in Oklahoma

In 1992, Saxon offered to donate his program free to seven Oklahoma City elementary schools. A district follow-up found Saxon students outscored a control group of non-Saxon students in every math category on the Iowa Test of Basic Skills. Asked to cite weaknesses of the plan, some teachers said that lessons were too time-consuming.

Much of the evidence in support of the Saxon method is anecdotal, but compelling enough to have forged a strong following among some school administrators and parent groups.

Test scores at Falconer Elementary School in Chicago, for instance, went up so dramatically that the central office suspected its students were cheating. Students retook the test and scored at the same level. (76.9 percent of its third-, fourth-, and fifth-graders scored at or above national norms on the Iowa Test of Basic Skills. Prior to the use of Saxon only about a third scored at that level.) Another example: Saxon students at Riviera Elementary School in Kelseyville, Calif., one of the state's poorest districts, now outscore students in affluent Laguna Beach schools.

Someone needs to write a book about Saxon Math.


our hero

John Saxon was one of the first to oppose the recommendation of the National Council of Teachers of Mathematics to integrate calculators into math classes. The 1989 NCTM standards that urged students to "construct their own understanding" gave Saxon textbooks a new target.

"John Saxon used to say that understanding more often than not follows doing rather than precedes it," Wang says. "If I'm going to teach you how to drive, I don't lecture you on the theory of the internal-combustion engine. I get you behind the wheel of the car and drive around the block."

He adds: "We're not saying we're against critical thinking. But we feel that creativity comes from a well-prepared mind. What we want to give every child in America is the ability to work to develop a well-prepared mind."

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TwoWaysOfTeachingMath 19 May 2006 - 21:12 CatherineJohnson

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PanBalanceInSaxonMath 20 Feb 2006 - 17:44 CatherineJohnson



I LOL'd when I read one of Carolyn's patented dry observations on the follies of 21st century math instruction:

Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations.

I distinctly recall being charmed the first time I saw a pan balance in Algebra to Go.

And of course I loved Carolyn's pan balance drawings:








I also had a lot of fun playing with the pan balance problems in the National Library of Virtual Manipulatives.








But I continued to experience a disconnect between pan balances and 'isolate the variable,' or 'do the same thing to both sides,' until I finally did Investigation 7 in Saxon 8/7: "Balanced Equations."








back from fun-filled Con Ed hiatus

We had an ice storm Saturday night, then a wind storm Wednesday morning, and there are so many trees down all over Westchester it's like Hurricane Katrina without the water.

Also without the trilliions of dollars in property damage, the loss of life, the breakdown of civil order, the helicopters, Wolf Blitzer, and the international expressions of shock and opprobrium.

Apart from that, it's exactly like Hurricane Katrina.

Anyway, the electricity went off at noon; the garage door is electric; the car was in the garage; the road to town was blocked; the side roads were blocked; when the electricity went back on the internet connection didn't; and so on.

All in all, about what you'd expect.



where was I?

Something about a pan balance.

Right.

I have no idea what I was planning to say about pan balances....apart from the fact that -- it's coming back to me now -- John Saxon can write a Pan Balance lesson like nobody's business.

The reason John Saxon can write a pan balance lesson like nobody's business is that he doesn't just slap down a drawing of a pan balance and expect the student to see the light.

Instead, he carefully develops his pan balance analogy, presenting the student with a sequence of 3 or 4 drawings of pan balances, one after the other, each one representing a step in the solution of an equation.

And he explains the whole thing in words. Words, pictures, numbers, and variables. Kit and caboodle.

Here he is:

Equations are sometimes called balanced equations [ed.: wonderful!] because the two sides of the equation "balance" each other. A balance scale can be used as a model of an equation. We replace the equal sign with a balanced scale. The left and right sides of the equation are placed on the left and right trays of the balance. For example, x +12 = 33 becomes

• (insert drawing of pan balance with x + 12 on the left side and 33 on the right)
  

Using a balance-scale model we think of how to get the unknown number, in this case the x, alone on one side of the cale. Using our example, we could remove 12 (subtract 12) from the left side of the scale. However, if we did that, the scale would no longer be balanced. So we make this rule for ourselves.

Whatever operation we perform on one side of an equation, we also perform on the other side of the equation to maintain a balanced equations.

We see that there are two steps to the process.

Step 1: Select the operation that will isolate the variable.

Step 2: Perform the selected operation on both sides of the equation.

Click.

This is perfect.

Instead of plopping a pan balance down in the middle of the page and expecting the student to discover its meaning, Saxon explains what the image means, and why it works.

Then he takes you through the steps which can only be implicit in a static drawing of a lone pan balance.

Then he has you draw your own pan balances.

I'm sick Christopher isn't using Saxon.

I'm so sick he isn't using Saxon, that I may try to squeeze Saxon back into our 'schedule.'



Saxon - Prentice-Hall smackdown Part 2

I've mentioned Christopher seems to be not only not gaining new knowledge, but to be losing the knowledge he already had.

Here's why.





The Saxon pan balance 'Investigation' opens with addition & subtraction equations.

Then the same Saxon Investigation proceeds to multiplication and division equations, reminding students in passing that multiplication and addition are related.

Prentice-Hall splits all of this up into separate lessons, and never the twain shall meet.

Addition and subtraction go together.

Multiplication and division go together.

Integers go together.

Decimals go together.

Fractions go together.

They're all in their separate lesson-boxes.

If the student doesn't make the connection, the connection doesn't get made.

I see why David Klein says all American textbooks are constructivist.

Technically, Prentice-Hall is a traditional book.

But nothing is explained, beyond the bare minimum. It's like a website with a lot of info to sort through (David has made that observation before), or a reference book with problem sets.

I don't know why they don't just buy these kids a Dictionary of Mathematics and let it go at that. There's a bunch of them out there.

-- CatherineJohnson - 20 Jan 2006

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BarModelsInKumon 04 Feb 2006 - 21:02 CatherineJohnson



I just looked ahead in this week's packet of KUMON worksheets, and found KUMON bar models!




Saxon uses bar models, too

I keep meaning to mention the fact that Saxon Math 8/7 uses bar models to teach fraction word problems. A Saxon student sees bar models in a number of lessons, then practices drawing them to mastery.

Saxon, Singapore, & KUMON.

We have a consensus.



Sybilla Beckmann's terrific article on bar models

Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4-6 Texts Used in Singapore (pdf file) by Sybilla Beckmann. (pdf file) by Sybilla Beckmann>





Sybilla Beckmann Kazaz


also by Sybilla Beckmann:

• Mathematics for Elementary Teachers, Volume 1
• Sample test problems for elementary teachers (pdf file)
• Mathematics for Elementary Teachers: Making Sense by Explaining 'Why' (pdf file)

-- CatherineJohnson - 04 Feb 2006

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CarolynOnMasteryLearning 07 Feb 2006 - 19:54 CatherineJohnson



I was just doing some Librarian work on ktm (linking like posts with like, dropping 'back doors' into existing posts, posting links in the book-style index) — and I discovered that Carolyn wrote a post on mastery learning back in May!

How good is mastery learning? Two of the review studies looked at mastery learning by itself and with combinations of other curricula, and found that mastery learning by itself produces better results than what was termed 'conventional instruction'. However, mastery learning got its best results when used with other teaching techniques. One study got decent results for "mastery learning with corrective feedback" (meaning -- electric shock? The review didn't say), but got its best results from mastery learning with 'enhanced cues' -- extremely detailed instructions to the students on how to do problems.

Another study found that mastery learning and cooperative learning strongly enhanced each other (note: cooperative learning is structured working-together among students, as opposed to simply being stuck in groups to do your homework together: see part two of this series).

It's interesting, reading this post now, not least because I recognize one of the author's names: Doug Carnine.


Report to the California State Board of Education

-- CatherineJohnson - 06 Feb 2006

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RoteKnowledgeInEverydayMath 23 Feb 2006 - 12:06 CatherineJohnson



Great comments on the advice from a top high school student thread. (This was the student whose father countered his son's rote learning of math by having him derive every formula he used.)


from Steve:

Perhaps you don't have to derive everything, but you do have to be able to understand and explain why you can do something using basic definitions and rules. And don't forget mastery. There is linkage.

The biggest fallacy of the latest fad math is that it teaches conceptual understanding. It does no such thing. My son is taking 4th grade Everyday Math, which is supposed to be one of the better fad math curricula. One of its rapid spiral "Home Link" assignments lately was a "Fraction of" sheet with problems like:

4/5 of 25 is _

This is before they know anything about multiplying fractions. How does the teacher explain how to do this problem? You take the whole number (25), divide it by the number on the bottom of the fraction (5) (notice that it is evenly divisible), and then multiply it by the number on top of the fraction. All rote understanding. Perhaps the students have to try and get a Zen-like understanding of four-fifths of 25. But then what do they do with a problem like:

4/5 of 7/8 ?

or

4/5 of 2.3 ?

Another Home Link spiral sheet talked about something like "Part of One" ?!? which is supposed to be the opposite of the example above:

20 is 4/5 of _

Once again, my son was taught a rote procedure for solving this problem.

I am getting really sick of this modern math conceptual understanding rubbish. Can anyone give an example of teaching real mathematical understanding in MathLand, TERC, Everyday Math, CMP, or their cronies? The so-called problem with traditional math was that it was all about drill and kill and no understanding. Well, nowadays, modern fad math does neither.

One of our school committee members told me that her younger daughter (in 6th grade, I think) doesn't have the math skills that her older brothers had at her age, but she has better conceptual understanding because of MathLand and CMP. I honestly don't have a clue what that means.



from Susan:

Exactly. Problems like these show up in Saxon in the form of word problems with the aim being to "see" the numerator and denominator in the form of a bar model, or to practice and clarify unerstanding the role of the numerator (3/8's of the girls had blonde hair, what fraction of the girls did not have blonde hair.) There are several problems like that, but I there is no rote procedure to solve them except in drawing out the vertical bar model to see the segments.

The Saxon version seems to be shooting for conceptual understanding without anyone locking in the procedure of dividing by the denominator, then multiplying by the numerator as the most efficient way of doing it.

Multiplying fractions as a procedure comes quite a bit later. While using the word "of" in previous chapters as another word for "times," this chapter is where they first bring it to learn and practice in a straightforward, rote way. The timing, I think, makes a big difference and is probably less confusing. My son has learned all of this with no bumps in understanding. One piece just fits into the next.

Cancelling is not mentioned at this point. Reducing, at this point, only happens in the answer. I'm dying to just teach this to him, but I'm sure I'd be piling on too much, too soon, and I've learned my lessons the hard way about doing that. A couple of chapters later is the GCF chapter, so I know that's why that next step is postponed a bit.

It's just hard to be an adult and go backwards. I want to show him the easy way when he needs to soak in the new stuff a little at a time.



aside from Catherine:

I am ONE with Susan on this point.

I've mentioned that I worked every single problem in Primary Mathematics Challenging Word Problems Book 3, and that I'm doing every problem in Saxon 8/7 as well.

That's a lot of bar models.

As a direct result, my brain has changed. When I read a fraction problem, bar models pop into my mind's eye unbidden.

I imagine some of you will feel skeptical about this, but for me that's a good thing.

Also, Steve's question — But then what do they do with a problem like: 4/5 of 7/8 ? or 4/5 of 2.3? — has an answer. In Singapore & Saxon the sequence of fraction problems is such that you 'see' that you need a common denominator — that is, you need a bar model divided into 40 segments.

I can't remember whether either Saxon or Singapore asks students to draw bar models of a problem like 4/5 x 7/8 — judging by the fraction problems I just did in KUMON Level F, for god's sake, Singapore may.

Saxon would, I think, use bar models to have kids do a problem like 2/3 x 3/5.

You see from what you've drawn that 2/3 'of' 3/5 is 6/15 which equals 2/5.

The funny thing is that, because I'd learned the multiplication operation with zero conceptual understanding attached (zero conceptual understanding of how the algorithm worked, I mean) I spent quite a long time being befuddled by the computation itself.

I just couldn't 'get' why you multiply the numerators and the denominators. You guys all tried to teach me & I still didn't get it! (I should rustle up those posts. Rudbeckia sent me a wonderful explanation; Dan created a graphic that everyone loved & I was the only person who didn't understand - - - )

Finally, the idea that 'clicked' for me came to me in the car.

This will sound incredibly unschooled & dumb....but tant pis. (French for t**** s***.)

I'd always sort of 'gotten' the idea that you multiply the numerators for the same reason you always multiply; you're finding '2 of 3' or '2 sets of 3' which is six.

But I couldn't put that together with why you multiplied the denominators.

Finally I realized that the denominators are divisors. 2/3 of 3/5 means you are successively dividing 3/5 by 3; you're doing two divisions in a row.

Two divisions in a row means you're dividing by 2 x the factor. (If you divide 12 by 2 and then divide the quotient by 2 again, you're dividing 12 by 2 x 2.)

I don't think this works as a verbal explanation for somebody still trying to figure this out, but it probably makes sense to all of you.

I have NO idea why I was so stumped by this.

I'm guessing I spent too many years overlearning the algorithm without the faintest idea why the algorithm worked. But I don't know.



conceptual understanding without skills

I think I do know what conceptual understanding without skills may be.

I think it would be quite possible to gain conceptual understanding of fraction multiplication — including conceptual understanding of problems like 4/5 x 7/8 — without acquiring procedural fluency in the multiplication algorithm.

It might even be possible to gain conceptual understanding of fraction multiplication with very limited understanding of factors.

I think it's the same observation Ken made a little while ago, after giving his son a Rubik's cube for Christmas.

It's possible to understand the directions for how to solve a Rubik's cube.

Actually solving the Rubik's cube is a different story.



from Kathy:

The biggest fallacy of the latest fad math is that it teaches conceptual understanding. It does no such thing.

Good-I was looking for an excuse to post my latest rant. My daughter is also in 4th grade Everyday Math. Tomorrow is her Unit 6 test, which covers long division, coordinate grids, something called "turns", map coordinates, angle measuring and drawing with a protractor, and word problems where you have to interpret a remainder. And all these topics relate to each other in what way??? With Meg's issues, all this jumping around is very confusing. She has figured out long division, thanks to much practice and tons of supplementation, but all this other stuff is causing much confusion.

Just for "fun" I was able to find the 4th and 5th grade Math texts I used back in the mid-70's and started to do a quick comparison to Everyday Math.

The thing that jumps out immediately is the sheer number of practice problems from my old books. For example, there are over 300 long division problems in the 4th grade 1970's text, just in the division chapter. They return to previously taught concepts in "Keeping Fit" sections which appear at least twice in every chapter.

How many practice division problems in EM's division unit? 20.

Measuring angles with a protractor wasn't introduced until 5th grade, and there's just 1 lesson on it, logically in the Geometry chapter. And decimals didn't appear until the very end of the 5th grade book; in EM they appear in 3rd grade, often through the introduction of problems which the kids are never taught to do.



from Carolyn:

Oh, Kathy, you're bringing it all back to me. 4th grade Everyday Math was the absolute worst, perhaps mostly because it was new and horrible to me, but also because 4th grade is a year when you have to learn and master so many critical things -- fractions, long division, multidigit multiplication. And there is all the jumping around in topics, and never never enough practice, and topics introduced in advance of their being taught.

I have to go lie down now.



a fraction problem from Intensive Practice 3B

Ms. Martinez ordered a pizza. The boys ate 2/5 of the pizza while the girls ate 1/2 of it. One of the boys, Mike, said that all of them ate 3/7 of the whole pizza. Was Mike correct?

3B is second semester, third grade; this problem comes from the 'Take the Challenge' section, so it's considered difficult.

Unfortunately, I sold my copy of 3B, so I can't look to see how kids are taught to solve such problems.

I'd put money on it they draw bar models along with using the addition algorithm, however.


advice from a top high school student
rote knowledge in Everyday Math



-- CatherineJohnson - 20 Feb 2006

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CoolDimensionalAnalysisProblemFromSaxon87 01 Mar 2006 - 17:15 CatherineJohnson




The Adams' car has a 16-gallon gas tank. How many tanks of gas will the car use on a 2000-mile trip if the car averages 25 miles per gallon?
(source: Saxon 8/7 Lesson 96 page 660 #3)







printable version


keywords: dimensional analysis unit multipliers

cool dimensional analysis problem Saxon 8/7 L96 #3
printable dimensional analysis problem Saxon 8/7 Lesson 96 #3
printable solution to Saxon 8/7 Lesson 96 dimensional analysis #3



-- CatherineJohnson - 28 Feb 2006

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MiniProblems 15 Jul 2006 - 16:33 CatherineJohnson



I've been complaining for months about the lack of word problems in Christopher's math class.

The kids memorize one procedure/rule/formula a day, do a few calculations, and march on. As a direct result, IMO, their knowledge really is rote as opposed to procedural. At least, Christopher's is. And I've had enough math talks with other kids in the class to know some of them are in the same boat.

Today I had a eureka moment reading a Comment left by Kathy Iggy:

The old math books I found (the same ones I used in grade school) have lots of what they call "mini problems" used to illustrate how a recently taught concept would be presented in a word problem. Megan likes these because of their brevity and she doesn't have to struggle with comprehension that much.

For example:

20 yards of ribbon. 1/4 used for dress. How much ribbon used?

That's IT!

mini problems


That's the concept, and the phrase, I've been looking for.




mini problems:word problems :: basic skills:higher order skills .

That's from Ken, and he's exactly right.

[update 4/23/2006: no! he's not right! Actually, he's write about using mini problems to teach word problems; I'm talking about mini problems to teach math - to teach the fundamental concept in a lesson. Awhile back I realized that word problems are the 'real manipulatives.' Now I know what I mean by that.

All concepts should be taught — illustrated — with mini problems. All concepts, every last one.

PRIMARY MATHEMATICS does this; SAXON MATH does it; KUMON does it. I'll post examples.

I've come to feel that the first word problems illustrating a new concept should be so simple children can do them in their heads.

For example, the very first ratio word problem a child does should be something like this:


Christopher bought one pencil for one dollar.
How many pencils can he buy for two dollars?


The question should be written this way, too: on two separate lines, so the child sees instantly that the first sentence is the set-up, and the second sentence is the question. Richard Brown's revision of Mary Dolciani's BASIC ALGEBRA, a book I like very much, does this for many of its word problems. I'll post some of those, too, as I get to it.

mini problems are applications

The problem with word problems is that, in the U.S., they're always hard.

Word problems are so hard people have apparently come to think that if a word problem isn't hard it isn't really a word problem.

I'm wondering if we ought to ditch the phrase 'word problem' (ditto for 'story problem') and adopt the word 'application.'

A better idea: we should think about the point of word problems.

Some word problems are written and assigned to give students practice.

Many word problems are written and assigned to assess whether students have developed flexible knowledge.

I'm talking about a third purpose, which is instruction. I'm talking about word problems designed to teach.


instructional word problems


A word problem is an application. A super-simple, starter word problem explains and demonstrates a mathematical concept by showing students how the concept is applied.

As a matter of fact, an instructional word problem shouldn't even be a 'problem.' It should just be a question, and the answer should be obvious.

A simple, instructional mini-problem should not test the child, should not challenge the child, and certainly should not trick the child.

It should teach.


examples to come


be sure to see Google Master's comment



how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 07 Mar 2006

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OnStudyingForTheStateTest 11 Mar 2006 - 23:10 CatherineJohnson



This is thrilling.

I noticed today that on February 19 Christopher couldn't do this problem from the Glencoe test prep book:


On Friday and Saturday, there were a total of 200 cars in the parking lot of a movie theater. On Friday, 120 cars were in the parking lot.

Part A
What percent of the total number of cars were in the parking lot on Friday?
Show your work.

Part B
What percent of the total number of cars were in the parking lot on Saturday?
Show your work.


Tonight he did it no sweat.

Thanks to Saxon Math.


-- CatherineJohnson - 09 Mar 2006

---

BarModelsInSaxonMath 13 Mar 2006 - 23:48 CatherineJohnson



(this is a repeat from last August that I want to get logged into its own post so I can send the link to a friend...)


using bar models to teach fractions in Saxon 8/7

Lesson 22, Problems About a Fraction of a Group, uses bar models to teach fractions of a group.

[update: Saxon 8/7 uses bar models throughout to teach fraction-of-a-group. 3-13-2006]

Christopher spent the whole summer complaining about his daily bar model. Then, yesterday, when we used a bar model to find If 2/5 of the the 270 fans wore green to the game, how many fans wore green? he said, 'These bar models do help!'

They really do.

Here's an example of a Saxon bar model used to solve a diferent fraction-of-a-group problem:






From now on I'm going to have him draw the model and link it to the computation. I'm going to have him explain to me why & how 2/3 x 270 is the same thing as dividing 270 by 3 and then multiplying 90 by 2, as the bar model has you do.

Then we're going to keep doing that until it makes sense.


-- CatherineJohnson - 13 Mar 2006

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TomFriedman 29 Mar 2006 - 23:32 CatherineJohnson



Tom Friedman makes me slightly crazy....

It's all the "Message to America" moments.

Like this one.

Message to America: They are not racing us to the bottom. They are racing us to the top.

source:
Still Eating Our Lunch
by TOM FRIEDMAN
NYTIMES $
September 16, 2005

Please.

I can't read another Message to America.

Can't and won't!

But this is a good one:

Friedman at a recent hearing of the House education panel, said that American parents used to tell their children to clean their plates because children were starving in China. Now, he said, parents should be telling their children to study their math and science because children in China want their jobs.

source:
A New Sputnik Moment?
by BRIAN FRIEL
NATIONAL JOURNAL ($)

The only thing wrong here is that I don't think anyone ever told their kids to clean their plates because there were starving children in China.

As a matter of fact, I was trying to remember this just the other day — which country was it?

Was it Africa or India?

I think it was Africa.



grump

Naturally Tom Friedman's child is one of five kids in the entire United States who gets to use Singapore Math books in her math class.

Here's a thought. How about Tom Friedman stops haranguing parents and kids about their lousy math skills, and starts haranguing schools and school boards about their lousy books?

That might help.

Math Literacy Scores, 15-year-olds, 2003
 
Country     Math literacy score
Mexico             385
Turkey             423
Greece             445
Portugal           466
Italy              466
United States      483
Spain              485
Poland             490
Hungary            490
Luxembourg         493
Norway             495
Slovak Republic    498
AVERAGE            500
Germany            503
Ireland            503
Austria            506
Sweden             509
France             511
Denmark            514
Iceland            515
Czech Republic     516
New Zealand        523
Australia          524
Switzerland        527
Belgium            529
Canada             532
Japan              534
Netherlands        538
South Korea        542
Finland            544

Source: Organization for Economic Cooperation and
Development

Average Math SAT Score

1995               506
1996               508
1997               511
1998               512
1999               511
2000               514
2001               514
2002               516
2003               519
2004               518
2005               520

SOURCE: College Board, 2005

source:
The Science Scare
by BRIAN FRIEL
NATIONAL JOURNAL $
1-14-2006



Tom Friedman
Tom Friedman piles on
Tom Friedman, Tom Friedman

-- CatherineJohnson - 26 Mar 2006

---

AverageProblemsInSaxon87 05 Apr 2006 - 21:50 CarolynJohnston


Ben had the flu a couple of weeks ago, and it really laid him low. He was out of school for an entire week.

When he got back, he had a week of review in math for a test at the end of the week -- during which I saw no homework coming home. On Friday, he had his test, and he blew it. The week following was spring break week -- and we've been catching up on math all week.

I didn't get very far before I realized he'd had a real regression in math, losing just about everything he'd covered in the ten days before getting the flu. There were topics that he'd not quite mastered, that were just gone once he started working on his math again after his illness. (If anyone has any information at all about academic regressions after illnesses -- I'd love to hear it!). Not only that, it seemed as though certain topics that he had a marginal grip on before his illness were actually harder to acquire the second time, rather than easier (which is what I'd normally expect from someone learning them twice).

One of the types of problems that were harder to acquire the second time is the topic of doing word problems involving averages. Here's an example:

After taking four tests, Jane's average score is 70. Jane takes a fifth test and gets a 90. What is her new average score?

And here is an example of a variant of this problem that was also covered in the same chapter:

After taking four tests, Jane's average score is 75. After Jane takes her fifth test, her new average is 70. What was her score on the fifth test?

Some of the problems he was having were due to the fact that these two very similar problems, requiring different approaches, were being introduced in the same section. But it seemed to me that before his illness, Ben could muddle through these problems procedurally. But afterward, the same approach I'd been taking didn't seem to work at all.

So I tried a visual approach. Even apart from the Singapore Math bar-modeling approach, of which I approve, I've found through years of doing math and teaching that any time I can draw a picture to support my or a student's algebraic thinking, I'm better off. I've tried to pass on that notion to Ben, with limited success. He just refuses to draw a picture -- or to do any extra work at all -- if he thinks he can get by without it (Catherine has found the same thing with Chris; he won't use the bar models if he thinks he can see how to do it without them). But this time, I insisted that he draw a picture every time; and he wasn't having a lot of success with the approaches he was using, so he buckled under.

I don't want to be religious about bar models, but they really have helped Ben to get these problems right.

The trick with these average problems is to recognize that if a kid has an average of 75 on 4 tests, it's the same as though he'd gotten a 75 on each of those tests; it means that the sum of the scores on the 4 tests is 4 times 75, or 300. It's not a tough trick; it's the setting-up-the-word-problem part that's tough.

Here's the bar models, with the problems they correspond to:

After taking four tests, Jane's average score is 70. Jane takes a fifth test and gets a 90. What is her new average score?

(Note that Jane's average score is actually the unknown in this bar model, labeled with ?, divided by 5).

After taking four tests, Jane's average score is 75. After Jane takes her fifth test, her new average is 70. What was her score on the fifth test?

If he can identify and label the unknown in the picture, he has no trouble following through by doing the correct computations (note that in each case the unknown is marked with a ?).

Remember that quote about how a teacher's job is to know three ways to present a new idea, and the student's job is to understand one of them? I've found that these pictures, at the very least, give you a consistent way to come up with at least a second way to explain how to do a word problem (you're on your own for the third!).

An afterthought: these examples illustrate why I think the combination of the Saxon math and the Singapore math approaches is unbeatable. Singapore supplies what Saxon is missing -- a powerful supplementary approach to solving word problems.

-- CarolynJohnston - 04 Apr 2006

---

KippGoesToKindergarten 04 Oct 2006 - 16:11 CatherineJohnson



Trying to track down a Jay Matthews column on St. Anne's school in Brooklyn, I came across this column saying KIPP has started an elementary school in Houston.

That's good news.

And check this out.

They're combining Saxon Math with Everyday Math:

At SHINE, Brenner says, he is blending the more modern Everyday Math with the more traditional Saxon Math for first-graders. The proponents of those two teaching programs have been at war for 20 years; can combining them really work? I'd predict that joining such radically different elements would cause an explosion, like when I used to toss manganese shavings into the surf to illuminate beach parties.

Brenner seemed unfazed by my doubts. "Our kids are off the charts in math," he says. I haven't surrendered my skepticism, but I will visit his school, and then watch what happens when Laura Bowen brings all this here, where Washington can get a really good look at it.

I'm not surprised.

My friend with the kids in the fantastic private school told me her school combines Everyday Math with traditional math. They seem to do nothing but EM for the first couple of years; then they shift.

I was shocked when she told me this, and assumed that her kids were getting shortchanged.

Then she faxed me her son's math homework.

WAY past anything kids are doing in public schools. This boy was doing long division with a gazillion digits; no forgiving division anywhere in sight. The word problems were serious and challenging - challenging at his level. My friend was shocked that we have to reteach math at night. She and her husband never reteach any subjects at all. The kids in her school are way up at the top of U.S. kids, and they're learning everything they know at school.

Barry has mentioned before that James Milgrim thinks Everyday Math would be a good supplemental program when used with a traditional math curriculum.

Looks like he's right.

-- CatherineJohnson - 12 Apr 2006

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MySaxonActivity 19 Apr 2006 - 17:48 CatherineJohnson








Now I know why I've been putting off doing Saxon 8-7 Investigation 11.

It's going to take an entire day of my life.

• print out Adobe Acrobat files of 1-centimeter grid activity sheet

• restart printer repeatedly when printer hangs up trying to print out Adobe Acrobat files

• close and re-opent Adobe Acrobat files onscreen when printer hangs up trying to print out Adobe Acrobat files

• repeat restart process with printer when previous effort fails

• etc.

• set up materials on picnic table

• read directions

• find scissors

• sit down with materials at picnic table & read directions again

• find Scotch tape

• construct 1cm, 2cm, 3cm, & 4cm paper cubes with grid showing

• take picture of cubes for ktm

• edit picture, post to ktm

• retrieve now-damp materials from picnic table because it has begun to rain

I may be too old for projects.


-- CatherineJohnson - 14 Apr 2006

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FutureEducationPolicyWonk 05 May 2006 - 20:46 CatherineJohnson



She doesn't like Saxon Math.

You might want to leave comments on your own experience with Saxon. I said it's neither rote nor boring, but added that the books are highly repetitive.

-- CatherineJohnson - 05 May 2006

---

MasteryLearningAndIq 26 Jun 2006 - 16:59 CatherineJohnson




Through sheer serendipity, I've stumbled across the book with the answers:

Standards and Mastery Learning: Aligning Teaching and Assessment So All Children Can Learn

by J. Ronald Gentile, James P. Lalley.

Learning, in other words, occurs in phases or episodes, and this original learning phase includes (a) the readiness component (described above), (b) learning to initial mastery, and (c) forgetting....it is clear that forgetting is the inevitable result of initial learning, even when a high mastery standard of, say, 80% to 100% correct is required. When the degree of original learning is less than mastery, say, 60% to 80%, then forgetting is likely to occur more rapidly or be more complete. If it is less than 60%, it is questionable to speak of forgetting at all, because learning was inadequate in the first place.

why do we have to learn all this stuff?

Finally, an answer:

Students show that they understand this principle implicitly when they ask, “Why do we have to learn this stuff anyway? We’ll only forget it.” Our typical answers, “Because it will be on the test” or “Because I said so,” are not satisfactory. In fact, we have been able to find only one satisfactory answer to the question, and it was supplied in one of the first empirical studies of learning/forgetting (Ebbinghaus, 1885/1964). The answer is that relearning is faster—that is, there is a considerable savings of time in relearning compared with original learning. Furthermore, there is a positive relationship between amount of time saved in relearning and the degree of original learning, with essentially no savings when original learning is below some acceptable threshold (which we earlier argued was 60% or less).

fast learners, slower learners, memory, IQ

Suppose, however, that we ask how IQ relates to all of this. We already know, for example, that IQ is moderately but significantly correlated with memory. But suppose we randomly assign half the students to have to achieve a preset standard, while the other half (within the same IQ range) are exposed to the same material but do not have to achieve the preset standard. What happens to the correlation between IQ and surprise delayedretention test scores?

A dissertation study on this very premise was completed recently, under the senior author’s direction, by Marianne Baker (1999).... for original learning, a short story was read aloud to fourth and fifth graders individually, immediately followed by a free-recall test on specific items of information as well as comprehension of ideas in the story. For the mastery group, this process was repeated until each student scored between 75% and 90% correct. The nonmastery group heard the story once and did the free-recall test. A week later, both groups were surprised with a written test of memory for the same items. Then students relearned under their respective conditions and finally were tested for retention again after 14 days and 28 days.

Table 1.2 shows the remarkable results regarding intellectual traits and memory.5 Under nonmastery conditions—that is, a single exposure for original learning, recall after 7 days, a single relearning opportunity, and then recall after 14 and 28 days—the correlations between intellectual traits and recall are all positive and significant. That is, higher-ability students tend to remember more, as society has come to expect.

In stark contrast, imposing a mastery standard of 75% to 90% correct on original learning and then again at relearning renders those standardized intellectual measures nonpredictors of how much is recalled: The correlations hover around zero and are all nonsignificant.

What mastery to a high standard can do, in summary, is virtually bypass the effects of IQ for specified educational objectives. What is recalled about educational lessons is more dependent on how well the material is mastered than on such traits as rate of learning or general intellectual abilities.

I believe it.

I'll have more later. The preface and first chapter (pdf file)) are available online.

I'm ordering the book.



in a nutshell

• learning occurs in phases or episodes

• all initial learning results in forgetting

• learning to high mastery means immediate recall of 80% to 100%

• learning to "low" mastery means immediate recall of 60 to 80%

• below 60% you haven't learned; when you encounter the material again you'll be starting over again [ed.: hoo boy. we're looking at a big, honking Phase 4 math reteach-fest this summer.] [UPDATE 12-6-06: Actually, we're not. C. is now much faster at learning math; he's managing to absorb and, I think, hang onto some of the content in Phase 4 Grade 7.]

• IQ is moderately correlated with memory

• high-IQ allows one to recall more after one exposure: "higher-ability students tend to remember more, as society has come to expect"

• high mastery to a standard of 75% to 90% on original learning plus one relearning erases the correlation between IQ and memory
  

"What mastery to a high standard can do, in summary, is virtually bypass the effects of IQ for specified educational objectives."

MORE COMING ANON



-- CatherineJohnson - 24 Jun 2006

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TellingTimeInSingaporeMath 09 Jul 2006 - 22:17 CatherineJohnson




Rudbeckia asked whether schools ought to be expected to teach children to tell time.

I was under the impression that that has always been the school's job, but once she asked the question I wasn't sure so I checked Singapore & Saxon.

Singapore Math teaches the topic of telling time in the 2nd half of 2nd grade:





The series begins time calculations in 3B:





Saxon Math begins teaching time (and date of month) in Kindergarton, then continues in 1st grade and 2nd.

Christopher learned to tell time at school.



At the Phase 4 Parent Uprising Meeting a year and a half ago one of the parents brought up the fact that kids all over New York state were doing poorly on the "Measurement" scale on the state tests. (Christopher's low score on the Measurement scale almost cost him his 4.)

Lisa Urban, the legendary middle school math teacher, grinned and said with obvious relish, "Your children can't tell time!"

Then she elaborated.

"By that I don't mean they can't tell me what time it is on the clock. I mean they can't calculate time. They can't solve problems concerning time. I want your children to go out in the world and solve problems.

etc.

It's probably just as well she's retiring.

I'll kick myself for saying this next fall.


-- CatherineJohnson - 30 Jun 2006

---

BackToTheFuture 08 Oct 2006 - 22:45 CatherineJohnson




I've been spinning my wheels, trying to figure out what, exactly, I'm doing about Christopher's math this summer.

I'd been thinking I'd have him carry on with KUMON & do ALEKS lessons as well. That arrangement would give him the independence we think is working well for him.

Amazing how it turns out to be the helicopter parents who teach responsibility and independence, isn't it?

Amazing, too, how you don't teach responsibility and independence by taking points off for failing to center the title of a graph.

sigh

Anyway, I've been thinking ALEKS.

Then I looked at the ALEKS assessment, and looked at the Ms. Kahl assessment, and I called my neighbor, the clinical psychologist and statistician.

She said, "Back when you were remediating and accelerating Christopher successfully, you were using Saxon."

She's right.

We're going back to Saxon.

Things have certainly changed for the better around here. When I first dived into all this, exactly two years ago, Ed was skeptical. He didn't instantly see why our entire household should be consumed by reading about math, doing math, teaching math, and, inevitably, writing about math on the internet. (Men!)

When Christopher learned everything he'd failed to learn in 4th grade math and then jumped to Phase 4, Ed was impressed.

But he still wasn't on board for much in the way of afterschooling, and we had triangulation issues that resulted in one whopping big parental unit blow-up last fall, after which he did get on board, but mostly because he realized that allowing one's sixth grader to play one parent against the other is a bad idea.

So here we are at the end of the year, in receipt of yet another report card adorned with canned comments from the Comment Bank, one of which, next to Math, says "finds subject matter difficult."

Yes, Ms. Kahl opted to punch in "finds subject matter difficult" on Christopher's report card. She gets tenure, we get the Rosenthal effect.

We'll deal with Ms. Kahl, but at this point she's a sideshow. The truth of the matter is that Irvington Middle School isn't going to teach Christopher math at all. I've been debriefing parents, trying to calibrate my perceptions to something resembling reality, and, once again, I'm discovering that it's always worse than you think. Parents I hadn't talked to since last school year, it appears, have spent the past 10 months in various states of frustration and disbelief over Ms. Kahl's class. I've heard that one of the brainiest math kids I know, whose favorite subject had been math, has now lost interest. Another parent, a man who taught math for many years, is appalled. "What is she teaching? What is she doing?" etc.





science proves it's always worse than you think

I read a new study in SCIENCE NEWS this week. As people grow older, they stop processing the bad stuff consciously! The article showed scans of young people's brains reacting to bad things compared to middle-aged people's brains reacting to bad things. In young people’s brains a conscious center was burning furiously. In middle-aged people’s brains consciousness was kaput. IIRC, middle aged people were processing bad things, but they were using the unconscious to do it. (I approve.) ^*

It can't be a coincidence that Ed and I cooked up the saying “It’s always worse than you think” when we hit middle age. I would never have said “It’s always worse than you think” when I was young. When I was young, I thought everything was worse. Back then I used to remind myself that things weren’t as bad as I thought.

All of which brings me to the fact that, although I’ve spent the past 4 months bemoaning the Horror that is phase 4 math in Irvington Middle School, I didn’t quite believe myself.

Teens and Tweens has a post on this very subject: the irrational need to believe in your school. That's me. An irrational need to believe things can't possibly be as bad as they look.

It’s time to get real.

Christopher is not going to learn math in Irvington Middle School. He told us today that every week last winter & spring, when he went in for extra help, the place was packed with kids from Ms. Kahl's 7th grade Phase 3 class. Yikes. Talk about worse than you think.

So it's back to the future.





is Saxon Math brilliant?

I’m starting to think so.

I never trust my perceptions on this, because I don’t understand the structure of math. I don’t understand how it builds from arithmetic to algebra to calculus and beyond. I have no idea why calculus should come after algebra, as a matter of fact, though I think I'm starting to see why algebra comes after arithmetic.

And since I’m teaching myself, I don’t know whether I’m learning real math or Saxon-math. I’ve read that kids homeschooled in Saxon Math are dependent on Saxon, can’t generalize beyond it, can’t solve problems in math, etc.

I started to think these accounts must be seriously wrong a couple of months ago when I took the sample entrance exam for Thomas Jefferson High School for Science and Technology.

My old friend Donna had told me about Thomas Jefferson. It is apparently the top science & math high school in the country; Thomas Jefferson graduates are being recruited by good colleges as far away as Santa Cruz and given full scholarships. Thomas Jefferson is a public school, so parents move to Fairfax County to live in the district, then spend hundreds of dollars sending their kids to Kaplan and KUMON in order to prepare them for the entrance exam. People call the area “Juku City.”

I found the test and took it. At the time I’d worked through quite a bit of Saxon 8/7, the 8th grade pre-algebra book, though by no means all.

The test was a breeze. No question was hard, and I got every answer right. I kept thinking, “Is this it?”

Then I took the logic section of the test, for which I hadn’t studied a Saxon logic text, and couldn’t do one single problem.

That was a moment.

I had another moment this week when I took the algebra 1 assessment in ALEKS, which I did find difficult. But even though I’m only halfway through Saxon Algebra 1 & am teaching myself material I’ve never seen before, I was able to do assessment problems I hadn't gotten to in Saxon.

I’m becoming a believer.





back to the future

My neighbor is right. It’s time to go back to Saxon. Enough with the teaching to crammery and the extra-sensory guess-fests on what items Ms. K will put on the test that the kids have never seen and don't know how to do.

So I struck a deal with Christopher. If he works his way through all of Saxon Algebra ½, he can drop KUMON.

When I told Ed it took him 1 second to sign on for the plan.

I told Christopher he could read and study the lesson and do the 3 to 5 "practice problems" one day, then do the 30 mixed review problems the next day. 3 lessons a week, week in, week out, until he's done.

Ed pointed out to Christopher that if he wants to get done faster, he can do the whole lesson including the mixed review in just one day, then take a long weekend off.

Ed has been mugged by reality.

Christopher opened up his brand-new, still wrapped in cellophane Saxon Algebra ½ books this afternoon and read Lesson 1: Whole Number Place Value • Expanded Notation • Reading and Writing Whole Numbers • Addition. Saxon, like Engelmann & like KUMON, starts kids out with the stuff they already know and can do.

Sitting at the picnic table reading Lesson 1, Christopher was feeling cocky. He whizzed through the text, whizzed through the practice problems, checked all his answers himself—using red ink—and corrected the small error he had made in one.

He looked happy to be home again.

I know I am.








NEURAL FEEL. As people age, from
12 to 79 years old, they respond to
fear with greater and greater boosts
in medial prefrontal activity (left)
and to happiness with smaller and
smaller boosts (right).
Williams

^* I don't think this description is right, but I'm going to have to spend some time with the article to figure it out. Here's the jist: "Recognition of negative emotion (fear) showed a significant decline as a function of increasing age, whereas recognition for positive emotion (happiness) increased..." p. 6427

The Mellow Years?: Neural Basis of Improving Emotional Stability over Age Leanne M. Williams,1,2 Kerri J. Brown,1 Donna Palmer,1,4 Belinda J. Liddell,1,4 Andrew H. Kemp,1,2 Gloria Olivieri,1,3 Anthony Peduto,1,3 and Evian Gordon1,2,5 6422 • The Journal of Neuroscience, June 14, 2006 • 26(24):6422– 6430 (pdf file)
Emotional Memory and Aging
BioInfo Bank

teachtocrammery


-- CatherineJohnson - 01 Jul 2006

---

SingaporePlacementTest 10 Jul 2006 - 15:35 CatherineJohnson




So this morning I took half of what is supposed to be a 30-minute placement test given to a student who has finished New Elementary Mathematics Textbook D.

I think it took me 2 hours.

To do half.

I was thinking that was OK, because New Elementary Mathematics Textbook D is a 9th grade text.

It's not.

It's a 7th grade text.

In Singapore, I'm still in 7th grade.

I'm halfway through Saxon Algebra 1, I scored 80% correct on the ALEKS algebra 1 assessment, and I thought the placement test (pdf file) for Thomas Jefferson High School of Science and Technology, reportedly the best math and science high school in the country, was a breeze.

That translates to 7th grade in Singapore.




juku city

Thomas Jefferson, you may recall, is the public school that's so competitive and respected that colleges all over the country are recruiting its graduates and giving them 4-year scholarships. Parents move to Fairfax County so they're in the district, then send their kids to Kaplan & KUMON to prepare them for the test. People call the area "Juku City."

I passed that test.

Easily.

Not the logic part. I passed the mathematics part. Questions 1-120, p 36-56. I may not have missed a single answer, and all the items were easy.

Then it takes me two hours to do 11 questions on an 8th grade placement test in Singapore. And those were just the algebra problems; I didn't even bother with the 9 items on geometry. (I didn't understand some of the terms, which may be different in the U.S., and I haven't practiced geometry enough to remember various other terms I knew a couple of months ago.)

arrgh




I need a personal organizer

So I'm looking at these 11 items, asking myself why exactly they should consume 2 hours of my life.

They're not hard.

I conclude that I'm having a major problem organizing my work.

I write things down, then lose track of what they refer to, then go back to the beginning and try to figure out what part of the problem I was doing, then I run out of paper so I'm flipping back and forth trying to find the figures I wrote on the preceding page.....it's pathetic. If I had to take Ms. K's math class, the school would have to institute corporal punishment to deal with my level of math-paper chaos. Twenty points off wouldn't even begin to cover it.

The other problem is that I don't have enough insight into these problems yet to take shortcuts & trust that my shortcuts will work, so I'm writing out every step and then some, which makes everything worse.

I don't know what happened to me on item 9:

A man bought 450 books for $1,350. He sold half of them at a profit of 20%, 150 of them at a profit of 10%, and the rest at a loss of 4%. What was his gain percent, to the nearest percent?

I just could not do this problem.

I came up with one wrong answer after another, and the answer I finally settled on was wrong.

It's not a hard problem.

When I checked the answer key, and found out my answer was wrong, it took me about 5 minutes to do it right.

Looking at the problem now, I think my difficulties may have had almost nothing to do with actual math.

I think the obstacle was working memory, organization, and eyesight. Can't remember if I've mentioned this: I can't wear my glasses to do close work any more. My glasses are bifocals, so in theory I'm wearing reading glasses, but....I can't wear them. They strain my eyes. (My optometrist, Mel Kaplan, says it's not the glasses, it's me. He also says that progressive lenses are horribly stressful.)

I don't need glasses to read, but, otoh, I do need strong light to see by, and dark ink on the page. High contrast. Otherwise everything starts to look kind of gray. So naturally I opted to take the test in dim light using a pencil with soft, thick lead. That's because I have no common sense-y.




I've decided not to panic

I don't think this is quite as bad as I thought.

Saxon Algebra 1, which is algebra integrated with geometry, is supposed to be a 9th grade book, but I'm going to have Christopher using the book at the beginning of 8th. Singapore puts me midway through 7th grade; Saxon puts me midway through 8th.

I'm further behind when you take geometry into account, but not by much.

I do have a question about problem solving and word problems in Saxon. In 70 lessons, I've done almost no word problems. Algebra 1/2 focuses on word problems, and I skipped that book. If the next 50 lessons don't contain a lot of word problems, I'll do the problems in my other books: Dolciani (Dolciani Teacher Edition), Jacobs (Jacobs Teacher's Edition), Johnson.




maybe I'll panic just a little

So....what does this tell us about our best versus their best?

The kids who compete to get into Thomas Jefferson are our top math students.

Our top kids have to go to cram school & Kaplan to pass the entrance exam for Thomas Jefferson in 8th grade.

I'm thinking that in Singapore an average graduate of 6th grade wouldn't have any trouble with it.

That gives me an idea.

Why don't I take the 6th grade test?

Why don't I take the 6th grade test with the reading lamp TURNED ON this time?

Bonne idee.


[pause]


Well, I can't say the 6th grade test was a lot easier, though I did pass (89% correct), and I did all the geometry problems but one.

There's no way the kids in Ms. K's class could pass this test. The 2 to 4 mathematically gifted kids in her two classes might pass, though I wouldn't bet on it. Nobody else.

If I were placing myself in a Singapore Math text, I think I'd start at the beginning of New Elementary Mathematics Textbook 1. Grade 7. [update: maybe not. Following KUMON's & Engelmann's injunction to start before your level, I should place myself in 6A. sigh. Sometimes it seems like I'll never make it to calculus.]

Which I just so happen to have sitting here on my bookshelf. My neighbor bought it a year ago, then never used it. I'm going to take a look.

I'm thinking I should probably also skim through Primary Mathematics 6A & 6B and see if there are Units I ought to do.




Table of Contents

Singapore Math placement tests
ALEKS assessment



-- CatherineJohnson - 06 Jul 2006

---

JohnSaxonPrefaceAlgebra2 20 Jul 2006 - 17:25 CatherineJohnson




Preface to Saxon Algebra 2

This is the second edition of the second book in an integrated three-book series designed to prepare students for calculus. In this book we continue the study of topics from algebra and geometry and begin our study of trigonometry. Mathematics is an abstract study of the behavior and interrelationships of numbers. In Algebra 1, we found that algebra is not difficult—it is just different. Concepts that were confusing when first encountered became familiar concepts after they had been practiced for a period of weeks or months—until finally they were understood. Then further study of the same concepts caused additional understanding as totally unexpected ramifications appeared. And, as we mastered these new abstractions, our understanding of seemingly unrelated concepts became clearer.

Thus mathematics does not consist of unconnected topics that can be filed in separate compartments, studied once, mastered, and then neglected. Mathematics is like a big ball made of pieces of string that have been tied together. Many pieces touch directly, but the other pieces are all an integral part of the ball, and all must be rolled along together if understanding is to be achieved.

A total assimilation of the fundamentals of mathematics is the key that will unlock the doors of higher mathematics and the doors to chemistry, physics, engineering, and other mathematically based disciplines. In addition, it will also unlock the doors to the understanding of psychology, sociology, and other nonmathematical disciplines in which research depends heavily on mathematical statistics. Thus, we see that mathematical ability is necessary in almost any field of endeavor.

Thus, in this book we go back to the beginning –to signed numbers—and then quickly review all of the topics of Algebra 1 and practice these topics as we weave in more advanced concepts. We will also practice the skills that are necessary to apply the concepts. The applicability of some of these skills, such as completing the square, deriving the quadratic formula, simplification of radicals, and complex numbers, might not be apparent at this time, but the benefits of having mastered these skills will become evident as our education continues.

We will continue our study of geometry in this book. Lessons on geometry appear at regular intervals, and one or two geometry problems appear in every homework problem set. We begin our study of trigonometry in Lesson 43 when we introduce the fundamental trigonometric ratios—the sine, cosine, and tangent. We will practice the use of these ratios in every problem set for the rest of the book. The long-term practice of the fundamental concepts of algebra, geometry, and trigonometry will make these concepts familiar concepts and will enable an in-depth understanding of their use in the next book in the series, a pre-calculus book entitled Advanced Mathematics.

Problems have been selected in various skill areas, and these problems will be practiced again and again in the problem sets. It is wise to strive for speed and accuracy when working these review problems. If you feel that you have mastered a type of problem, don’t skip it when it appears again. If you have really mastered the concept, the problem should not be troublesome; you should be able to do the problem quickly and accurately. If you have not mastered the concept, you need the practice that working the problem will provide. You must work every problem in every problem set to get the full benefit of the structure of this book. Master musicians practice fundamental musical skills every day. All experts practice fundamentals as often as possible. To attain and maintain proficiency in mathematics, it is necessary to practice fundamental mathematical skills constantly as new concepts are being investigated. And, as in the last book, you are encouraged to be diligent and to work at developing defense mechanisms whose use will protect you against every humans’ seemingly uncanny ability to invent ways to make mistakes.

One last word. There is no requirement that you like mathematics. I am not especially fond of mathematics—and I wrote the book—but I do love the ability to pass through doors that knowledge of mathematics has unlocked for me. I did not know what was behind the doors when I began. Some things I found there were not appealing while others were fascinating. For example, I enjoyed being an Air Force test pilot. A degree in engineering was a requirement to be admitted to test pilot school. My knowledge of mathematics enabled me to obtain this degree. At the time I began my study of mathematics, I had no idea that I would want to be a test pilot or would ever need to use mathematics in any way.

I thank Tom Brodsky for his help in selecting geometry problems for the problem sets. I thank Joan Coleman and David Pond for supervising the preparation of the manuscript. I thank Margaret Heisserer, Scott Kirby, John Chitwood, Julie Webster, Smith Richardson, Tony Carl, Gary Skidmore, Tim Maltz, Jonathan Maltz, and Kevin McKeown for creating the artwork, typesetting, and proofreading.

I again thank Frank Wang for his valuable help in getting the first edition of this book finalized and publisher Bob Wroth for his help in getting the first edition published.

John Saxon
Norman, Oklahoma



Beautiful.

The third editions of the Saxon books seem to have done away with John Saxon's prefaces; at least, that's the case with the 3rd edition of Algebra 1/2.

Thanks to our ktm Book Fairy, I have a copy of the 2nd edition of Algebra 1/2, so I'll post that preface, too.

The books themselves don't seem to have been changed in other bad direction. If you're interested in buying the 2nd edition, though, Rainbow Resource seems still to have them. So does Seton Books. I'm sure other homeschooling stores do as well.



Wilfried Schmid on procedures and understanding

''I'm a professional mathematician, and I myself very often use mathematical methods that I understand only imprecisely,'' he said. ''It is while I use them that I begin to understand. After a while, the use and the understanding are mutually supporting.''

source:
The New, Flexible Math Meets Parental Rebellion (scroll down)
By ANEMONA HARTOCOLLIS (NYT) 2403 words
Published: April 27, 2000

Carolyn on procedures and understanding

Carolyn has said more than once that she believes in teaching procedures first. Conceptual understanding follows. (I can't find any of her posts on this, so if I've misremembered I'll delete this.)

I was always a little skeptical of this, although my working assumption is that where Carolyn and I disagree, Carolyn is right.

I've now spent enough time working my way through Saxon to see what Saxon, Schmid, and Carolyn are talking about. When you practice a procedure you don't understand over and over and over again, at some point it "naturalizes." It seems right and inevitable. And it makes sense.

John Saxon stresses this idea in book after book. Math isn't hard; it's different. It's unfamiliar.

When you've done so much math that it no longer seems strange, it starts to seem easy — or at least not harder than other subjects.

Of course, the irony is that this naturalizing process leaves me unable to explain procedures to someone for whom math is still strange. It does, however, make me understand why "math brains" tend to say things like, "It just is" when I ask for an explanation!

I'll add that Saxon (and probably Carolyn & Schmid, too) rarely teaches a concept stripped of all meaning or explanation — though he does do so far more often in Algebra 1 than in the earlier books. A student using Algebra 1 must take a lot on faith.

If nothing else, meaning helps memory; it's easier to remember a procedure you understand. (I have references for this observation, but don't want to spend the time to dig them up just now.) I'd be willing to bet that meaning increases student motivation, too. I recall Steve H saying that students always want an explanation if they can get one. (Steve - am I remembering that correctly?) Every one of Saxon's explanations in 6-5 through 7-6 has been pure pleasure to read, and has made me want to learn more math. In contrast, my motivation sometimes flags as I work with Saxon's highly abstract Algebra 1, my motivation sometimes flags.

In short, I think it's probably always good to try to teach some conceptual understanding along with procedure. I also think, after living through Ms. K's Phase 4 math class, that it's essential to include mini word problems — although Saxon does not do so in Algebra 1. But John Saxon can get away with it, because he's a genius math textbook writer. If you're not a genius math text writer, or a genius math teacher, you can't.

Nevertheless, these caveats aside, math is first and foremost something people do. Barry says that constructivist math ends up teaching math appreciation, not math, and I agree.

Teach procedures supported by meaning where possible, and, where not possible, teach the procedure and practice it to mastery. Understanding will follow as "totally unexpected ramifications appear."







John Saxon & John von Neumann on math
preface to Saxon Algebra 2



-- CatherineJohnson - 15 Jul 2006

---

SaxonPlacementGuide 19 Jul 2006 - 19:16 CatherineJohnson






source:
Saxon Publishers placement test for Algebra 1 (pdf file)




I came across a ktm post from September 30 2005 reporting that I was starting the final chapter in Russian Math and had flunked the Algebra 1 placement test for Saxon Math.

Here's today's progress report, 9-30-05 — 7-16-2006:

• final chapter of Russian Math

• all of Saxon Math 8/7

• 81 lessons of Saxon Algebra 1 (120 lessons in all - geometry is integrated into the algebra texts)

• UPDATE 10-14-2006: Review lessons A, B, C & 1st 22 lessons Saxon Algebra 2

Now that Christopher has replaced his daily KUMON worksheets with daily Saxon Algebra 1/2, I'm reading through 1/2 as well, partly to make sure I didn't skip anything I need to practice, and partly for the pleasure of encountering John Saxon's take on topics I already know. The familiar made strange. UPDATE 10-14-2006: Christopher is making his way very slowly through Algebra 1/2. Maybe 20 lessons thus far.

Algebra 1 integrates algebra and geometry, though without proofs. I'll start Saxon Algebra 2 3rd edition in September.

Seeing as how I acidentally paid for a month of ALEKS when we did our assessments, I think I'll check out their statistics course.

-- CatherineJohnson - 16 Jul 2006

---

HowToTeachYourselfArithmetic 21 Sep 2006 - 21:51 CatherineJohnson




This may seem like a strange question coming from me.....can you teach yourself arithmetic?

UPDATE 10-19-2006: The answer is yes. You can. Christian is doing it now. Starting in Saxon Math 5/4.

I ask because Christian just got his placement test results — he passed reading!

I don't think we can credit the Yonkers school system for that, but the Mamaroneck schools may have had a hand in it. I say "may" because Christian's mom is college educated and has always subscribed to the New York Times, which meant that as a child Christian, like the rest of us it seems, was getting most of his vocabulary and exposure to print at home. He went to Mamaroneck schools through middle school, then moved to Yonkers where his 12th grade English teacher used the same book Mamaroneck used in 7th.

So I'm not giving Yonkers a lot of credit.

The bad news is math. We're looking at a pre-algebra placement.

(Can we sue the schools for not teaching yet?)

I'm in no mood to pay for two zero-credit remedial courses at Westchester Community College, and I don't know whether financial aid exists for pre-algebra. Even if it does, Christian needs two semesters' worth of remedial math (pre-algebra and high school algebra) before he can take a math or science course for credit. If that's what he has to do, then that's what he has to do, but he also has to support himself and stay motivated. The college completion stats don't show a lot of people who have to take two semester's worth of remedial math making it through.

I'd like to find another way if possible.

Naturally I'm thinking Saxon. Christopher's been moseying through Saxon Algebra 1/2 this summer. He's up to Lesson 15 and he's been getting all the answers right. Christian could probably teach himself fractions, decimals, and percents using Algegbra 1/2.

On the other hand, the Saxon books are huge. "Huge" meaning long and time-consuming. Long and time-consuming may be the only way to go here, seeing as how there's no royal road to geometry. But if anyone has thoughts, I'd like to hear.

update: I've just realized I'm going to have to get Christian to take the Saxon placement test.

If Saxon puts him into 8/7 or 7/6....I'm going to have to find another way.




computers & test anxiety

Christian says his mother was shocked that he passed the reading test.

I didn't get that at all until he told me he's always had a hard time taking tests. It sounds like he has some test anxiety; plus he's got some kind of fine motor "issue" (Carolyn's favorite word!) that tripped him up for years. He was classified special needs, along with all the other black kids, and his mom was constantly trying to get the school to provide him with a keyboard. Plus he's lefthanded.

So basically, he's never been able to take tests.

Apparently the reason he did well on the WCC test was that it's done on a computer terminal. He took the Accuplacer test, which I gather is being used in colleges all over the country. I had no idea the College Board is also in the remedial placement testing business. Apparently there's a whole Accuplacer test prep world out there, too. (It's aways worse than you think.)

Doing the test on the computer made Christian feel as if he wasn't doing a test. He was the second person finished; he just whipped through it.




ALEKS?

This is making me wonder whether ALEKS might be a good idea for Christian.

I'm certain Christian has math baggage (scroll down for Rudbeckia, Steve H, Carolyn, & Susan) and it seems pretty clear that looking at math on a computer will help him "break set."

On the other hand, I've been using ALEKS for a few weeks and while I find it highly motivating - addictive, almost - I don't find it highly illuminating. It's pretty much the ultimate in fragmented content, and the program offers no "metacognitive pointers" as Saxon does. You're on your own.

By "metacognitive pointer" I mean the kind of pointers people give when they're telling a delivery person how to get to their house. ALEKS doesn't give pointers. ALEKS just gives you the procedure, along with a lot of hyperlinks to other pages filled with other procedures & definitions, and that's the end of it. It's like learning algebra from Hal.

Years ago, when I interviewed nearly 100 couples for a book on marriage, I ended up dividing people into two categories:

• people who give good directions

• people who don't ^*
  

People who give good directions always tell you where you're going to be tempted to go wrong, how to tell if you have gone wrong, and what to do about it when you realize you did go wrong. A really good direction giver will say "You can't really see the driveway from the road, so if you get to the traffic light across from the church and the Sunoco station you've missed it."

That kind of thing. That's what the Saxon books do. Saxon lessons routinely tell students what mistakes they're likely to make and how not to make them. Often these pointers give you greater insight into the topics you've been studying.

Saxon Algebra isn't going to be addictive for most people.

But it is illuminating.



Any thoughts?

^* When I first met Temple, she made exactly the same observation & for the same reasons.

Christianlearnsmath



-- CatherineJohnson - 31 Aug 2006

---

MileStone 07 Sep 2006 - 00:02 CatherineJohnson




I'm finishing Saxon Algebra 1 tonight!







In the nick of time, I might add. I leafed through Christopher's 7th grade math textbook last night, taking a closer look.

It's algebra.

Also a chapter on logic & a chapter on statistics.

And it's not written by John Saxon.


[pause]


Carolyn & I could probably use this as a ktm personal logo (not about Saxon of course):


A parent says "Yuck", September 27, 2005




started Saxon Algera 1: April 14, 2006
finished: September 1, 2006

-- CatherineJohnson - 31 Aug 2006

---

UnitMultiplierProblemFromSaxonAlgebra1 02 Sep 2006 - 17:15 CatherineJohnson




This problem was in one of the lessons in Saxon Algebra 1:



Juanita exercised for one hour.

How many seconds did Juanita exercise?



I love this problem. Don't know why. (Though I do remember, as a child, being tickled by the fact that a short period of time, like one hour, could have a very large number of even shorter periods of time inside of it...)

Anyway, I pulled this one out for Christopher to do.

-- CatherineJohnson - 02 Sep 2006

---

NctmReformsAgain 14 Sep 2006 - 16:52 CatherineJohnson




In today's Wall Street Journal ($):

Arithmetic Problem
New Report Urges Return to Basics In Teaching Math
Critics of 'Fuzzy' Methods Cheer Educators' Findings;
Drills Without Calculators Taking Cues From Singapore
By JOHN HECHINGER
September 12, 2006; Page A1

The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.

In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.

The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.

Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country.

The new approach puzzled many parents. For example, to solve a basic division problem, 120 divided by 40, students might cross off groups of circles to "discover" that the answer was three.

Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.

Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor.

Francis Fennell, the council's president, says the latest guidelines move closer to the curriculum of Asian countries such as Singapore, whose students tend to perform better on international tests.

So maybe it wasn't such a great idea after all for IUFSD to ban my Singapore Math course.



new timeline

According to their report, "Curriculum Focal Points," which is subtitled "A Quest for Coherence," students, by second grade, should "develop quick recall of basic addition facts and related subtraction facts." By fourth grade, the report says, students should be fluent with "multiplication and division facts" and should start working with decimals and fractions. By fifth, they should know the "standard algorithm" for division -- in other words, long division -- and should start adding and subtracting decimals and fractions. By sixth grade, students should be moving on to multiplication and division of fractions and decimals. By seventh and eighth grades, they should use algebra to solve linear equations.

Here's the Singapore sequence.




Lutherans turning into Catholics

A recent study by the Thomas B. Fordham Foundation, a Washington nonprofit group, found that only two dozen states specified that students needed to know the multiplication tables. Many allowed calculators in early grades.

Chester E. Finn Jr., the foundation's president and a former top official at the U.S. Department of Education, blamed the earlier math-council guidelines for state standards that neglect the basics. He described the new advice as a "sea change," saying that "it's a little bit like Lutherans deciding to become Catholics after the Reformation."

Understanding math, rather than parroting answers to poorly understood equations, was the goal of the council's controversial 1989 standards. Those guidelines called on teachers to promote estimation, rather than precise answers. For example, an elementary-school student tackling the problem 4,783 divided by 13 should instead divide 4,800 by 12 to arrive at "about 400," the 1989 report said. The council said this approach would enable children using calculators to "decide whether the correct keys were pressed and whether the calculator result is reasonable."

"The calculator renders obsolete much of the complex pencil-and-paper proficiency traditionally emphasized in mathematics courses," the council said then. In 2000, in another report, the council backed away somewhat from that position.

Still, in response to the earlier recommendations, many school systems required children to describe in writing the reasoning behind their answers. Some parents complained that students ended up writing about math, rather than doing it.

As the debate heated up, concern grew about U.S. students' math competence. In 2003, Trends in International Mathematics and Science Study, a test that compares student achievement in many countries, ranked U.S. students just 15th in eighth-grade math skills, behind both Australia and the Slovak Republic. Singapore ranked No. 1, followed by South Korea and Hong Kong. Fueling concern about the quality of elementary and high-school instruction: one in five U.S. college freshmen now need a remedial math course, according to the National Science Board.

low-income students

This is very exciting. The AIR report (pdf file) led me to believe that Singapore Math had been a flop in low-income schools because the student mobility is so high (and see Hirsch on this subject, too):

If school systems adopt the math council's new approach, their classes might resemble those at Garfield Elementary School in Revere, Mass., just north of Boston. Three-quarters of Garfield's students receive free and reduced lunches, and many are the children of recent immigrants from such countries as Brazil, Cambodia and El Salvador.

Three years ago, Garfield started using Singapore Math, a curriculum modeled on that country's official program and now used in about 300 school systems in the U.S. Many school systems and parents regard Singapore Math as an antidote for "reform math" programs that arose from the math council's earlier recommendations.

According to preliminary results, the percentage of Garfield students failing the math portion of the fourth-grade state achievement test last year fell to 7% from 23% in 2005. Those rated advanced or proficient rose to 43% from 40%.

Last week, a fourth-grade class at Garfield opened its lesson with Singapore's "mental math," a 10-minute warm-up requiring students to recall facts and solve computation questions without pencil and paper. "In your heads, take the denominator of the fraction three-quarters, take the next odd number that follows that number. Add to that number, the number of ounces in a cup. What is nine less than that number?" asked teacher Janis Halloran. A sea of hands shot up. (The answer: four.)

Ms. Halloran then moved on to simple pencil-and-paper algebra problems. "The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?" (The answers: 21 and 42.)

In this class, the students didn't use the lettered variables that are so prevalent in standard algebraic equations. Instead, they arrived at answers using Cuisenaire rods, sticks of varying colors and lengths that they manipulate into patterns on the tops of their desks. The children use the rods to learn about the relationship between multiplication and geometry. The goal: a visceral and deep understanding of math concepts.

"It just makes everything easier for you," says fifth-grader Jailene Paz, 10 years old.

Cuisinaire rods for bar models!

That's so cool!




TERC time

The Singapore Math curriculum differs sharply from reform math programs, which often ask students to "discover" on their own the way to perform multiplication and division and other operations, and have come to be known as "constructivist" math.

One reform math program, "Investigations in Number, Data and Space," is used in 800 school systems and has become a lightning rod for critics. TERC, a Cambridge, Mass., nonprofit organization, developed that program, and Pearson Scott Foresman, a unit of Pearson PLC, London, distributes it to schools.

parents don't get it part 1

Ken Mayer, a spokesman for TERC, says many parents have a "misconception" that Investigations doesn't value computation. He says many school systems, such as Boston's, have seen gains in test scores using the program. "Fluency with number facts is critical," he says.

parents don't get it part 2

Polle Zellweger and her husband, Jock Mackinlay, both computer scientists, moved to Bellevue, Wash., from Palo Alto, Calif., two years ago so their two children could attend its highly regarded public schools. She and her husband grew suspicious of the school's Investigations program. This summer, they had both children take a California grade-level achievement test, and both answered only about 70% of the questions correctly. Ms. Zellweger and her husband started tutoring their children an hour a day to catch up.

"It was a really weird feeling," says their daughter, Molly Mackinlay, 15. "I do really well in school. I am getting A-pluses in math classes. Then, I take a math test from a different state, and I'm not able to finish half the questions."

Eric McDowell, who oversees Bellevue's math curriculum, says parents misunderstand Investigations.

If it weren't for the parents, teaching would be a great job.




math wars and war wars

In the Alpine School District in Utah, parent Oak Norton, an accountant, has gathered petitions from 1,000 families to protest the use of Investigations. His complaints began more than two years ago, when he discovered at a parent conference that his oldest child, then in third grade, wasn't being taught the multiplication tables.

Barry Graff, a top Alpine school administrator, says the system has added more traditional computation exercises. Over the next year, Alpine plans to give each school a choice between Investigations or a more conventional approach. Mr. Graff, who says Alpine test scores tend to be at or above state averages, expects critics to keep up the attacks and welcomes the national math council's efforts to provide grade-by-grade guidance on what children should learn.

"Other than the war in Iraq, I don't think there's anything more controversial to bring up than math," he says. "The debate will drive us eventually to be in the right place."

wow

I bet things are hopping over at math-teach & math-learn.

[pause]

hmm

No action thus far.

Once Wayne Bishop posts this baby, we'll be in a shooting war.

update: Bishop's got it!

let the fun begin



what Singapore students can do at the end of 7th grade

-- CatherineJohnson - 12 Sep 2006

---

FardellsNookesAndHides 16 Sep 2006 - 02:32 CatherineJohnson




from the Yahoo Group called Homework Help:

My son and I both came up with different answers to this problem...I am rusty at math but still think that I am right:)

How many fardells are in 8 hides?...
2 fardells = 1 nooke
4 nookes =1 yard
4 yards = 1 hide

This is a medieval Britain story problem. I got 256 fardells as my answer...sure seems like a lot but it's hard to visualize :)

Help!

Start teaching those unit multipliers now!



I'm going to see if Christopher can do this.

[pause]

Nope, he couldn't. He got 32. "I did it in my head."

After he got his wrong answer I started him out with unit multipliers by writing:

8 hides X

on the left side of the paper.

Then I asked him what needed to be in the denominator of the next ratio. (Ratio? Rate? What's the proper term?)

He knew it had to be hides, but he couldn't figure out what would be in the numerator.

sigh

About 5 seconds after that he got with the program, wrote out the series of ratios, and said, "Now what do I do, multiply?"

me: "Haven't you written out a string of multiplications?"

Christopher: "Yes."

me: "Then multiply."

He got 256.

That's one thing about distributed practice. It can't be too distributed. I was thinking that about this at the U.S. Open.

Every year for the past 4 or 5 years we've gone to the U.S. Open.

And every year Ed has to explain the entire scoring system in tennis all over again.

One year between exposures to content is too long.



Next year I plan to remember "All," as in "15 All," and "Deuce," as in .... as in I don't know when people say deuce. I've forgotten. I just know that once you say "deuce" the players have to win by two points. At least, I think that's what I remember.

I'm going to remember "all" because "all" means everyone and "all" is a tie. All have the same score.



OK, so basically what I'm going to remember is "All."

Also, I think I'm going to remember that men play 6 sets.

Or maybe 5.

See what I mean?

One year is too long.

-- CatherineJohnson - 14 Sep 2006

---

ProgressReportPart3 29 Oct 2006 - 01:38 CatherineJohnson




Christian came in the other day and said he'd gotten a 95 on his first paper.

Then he got a 98 on his first test, and the professor invited him to attend a screening of a movie on hip hop artists made by his son. (I think it was hip hop artists.) Shortly after that Christian ventured out of his quiet overachiever hiding place and challenged the brainy female student who'd been dominating the class — and the professor sided with him!

Ed said, "He's getting straight As and he's the teacher's pet."

That's good.

Christian needs to spend some time being teacher's pet.

My favorite Christian story — my favorite Christian's mom story, that is — was the time in high school when somehow his entire team of teachers decided to call Christian's mom on the carpet.

Something like that.

She went in for the meeting, sat down alone in whatever room they put her in, and one by one each teacher walked into the room expecting to tell her all the bad things they knew about Christian.

Not all of the teachers came.

As far as I can tell from this distance, they were all supposed to come.

But some of them refused. Christian's English teacher told him, "I have no problems with you. I'm not going."

That's another thing. There are a lot of hero teachers out there. My kids have had some hero teachers, of course; more often they've had terrific teachers in settings where heroism wasn't called for one way or the other, thank heavens.

But lately I'm hearing other people's stories of hero teachers. One of these stories makes me cry just thinking about it.

Anyway, Christian's mom sat down alone in the room and one by one each teacher came in with his list of complaints and sat down facing Christian's mom.

The teacher would start to talk and Christian's mom would cut him off. "What are you doing for my son?" she said.

The teacher would start to talk again and she'd cut him off again. "What are you doing for my son?"

She just kept doing it until the teacher gave up and left. At least, that's what she did in her son's retelling of the tale.

Then she did the same thing all over again with the next teacher.


I'm sure that went nowhere, but it's a great story.




Saxon math placement test

So I gave Christian his Saxon math placement test and the news was grim: if he were a kid he'd be starting in grade 3.

Since he's an adult I ordered Saxon 5/4, the fourth grade book.

Christian went to 4 grade schools in 5 years. That's called "student mobility," and it's death to achievement,^* particularly math achievement. (I think I'm channelling an earlier post.)










I don't know which of these categories Christian was in — either the 2nd or the 3rd.

That's another story. Christian's mom was fighting with the special ed people to get Christian something, a keyboard I think, a reasonable request given that his entire 504C classification was apparently based on bad handwriting, and the special ed person told her to have Medicaid pay for it.

Christian's mom said, "I work, bi***."

That's the difference between Christian's mom and me.

When my school told me to have Medicaid pay for assistive technology, I went for it.^** I spent months shlepping the kids to WIHD for Medicaid-funded assessments. Then two years later Ed spent months trying to clear up the billing problems when Medicaid didn't pay for it after saying they would.

Our district does now pay for assistive tech for Andrew.

As far as Christian can remember, none of these schools ever checked to see what he knew when he came in. They just plopped him into whatever classroom had space, gave him whatever "services" his 504C standing entitled him to, and went about their day.

Anyway, whatever his income category, Christian went to 4 schools in 5 years.

So today he has 3rd grade level math.




my trip to the edu-attorney

I've mentioned that Ed and I saw an education attorney a few weeks back.

We were there on special ed business, but I was eager to ask about the legal status of typical kids.

Do typical kids have any kind of legal entitlement to an education?

Obviously they do; typical kids have a legal right to a free public education.

But do typical kids have any kind of legal entitlement actually to learn the material the teachers are teaching?

That's what I wanted to know.

The answer is no.

I asked the attorney, "How close are we to being able to sue school districts for negligence or malpractice?"

"People can sue doctors," I said, "people can sue lawyers, people can sue accountants.^*** Why can't people sue schools?"

He stared at me blankly.

I stared back.

"What do you mean?" he said finally. "On what grounds would you sue a school district?"

"Well," I said, "say a student goes all the way through school and graduates without knowing how to read. Could a parent sue because the school has passed her child through 13 years of school without teaching him how to read?" I'd read somewhere that this would be the first successful lawsuit brought by a parent against a district. That's why I brought it up.

No.

A parent could not sue on these grounds.

"If he's gotten all the way through school without being able to read," the attorney said, "then he should have been referred to special ed at some point."

I wasn't quick enough on my feet to ask whether a parent could sue a district for failing to refer her child to special ed, but I gather the answer to that question, too, is no.

The reason I gather that the answer is 'no' is that I now know children to whom this has happened.

I asked whether NCLB altered the legal landscape, but didn't quite follow his answer, which was, in essence, that "NCLB is a special ed law."

I'd never heard it put that way, but I suspect he's right.

I'm coming to the conclusion that parents don't understand the first thing about the law.




case law and custom

For some reason, I had been assuming that the reason parents can't sue schools was that there are state laws protecting school districts from legal action. It made sense to me that a coalition of teachers' unions and school districts would have been able to lobby for such legislation and get it passed.

The reality is far worse.

The reality is that parents have been suing schools for many, many years in many, many states, and they have always lost.

The courts have always ruled in favor of the schools.

Never in favor of the children.

Many decades of case law and custom tell us that no school is accountable for an individual child learning anything at all. I'm happy — in fact I'm eager — to revise this characterization if it's wrong. So let me know.

As our attorney put it, the courts have ruled against parents, because the reason a particular child failed to learn "could be something about the child."

I wasn't quick enough on my feet to ask whether a class action lawsuit would get around the "something about the child" issue (could it be something about every child?), but I doubt the answer would have been any better.




parents step in

For a couple of years now I've been getting the message that it's up to me to make sure my child learns the material covered in school.

When I say "getting the message" I mean "getting the message" in a global, big-picture kind of way.

I don't get this message from individual teachers, with a couple of exceptions. No teacher at Dows Lane or Main Street School ever gave me the impression that she wasn't reponsible for her students learning. The middle school, last year, had an official Grade Contract message assigning full responsibility for learning to students, but when we had our "team meeting" it was obvious every teacher there felt personally responsible for students in her class actually learning the material she was teaching.

So...this isn't a "teacher message."

It's a "school message;" an "administration message;" a "district message."

Why has my district shown no interest in formative assessment or teaching to mastery?

Because decades of case law and custom say there's no reason for them to be interested in formative assessment and teaching to mastery.

They are in the inputs business. The inputs business and the compliance business. They must provide teachers, buildings, books, lessons; and they must comply with countless thousands of pages of edu-law. When I was the parent rep on hiring committees, one of the key questions we asked every candidate concerned his or her familiarity with education law.

And that's it. So when schools innovate or "implement" reforms, they provide more teachers, buildings, books, and lessons. Character ed, differentiated instruction, portfolio assessment — whatever it is.

More stuff.

Parents have to understand that it's up to us to make sure our children learn reading, writing, and arithmetic. It doesn't just seem that way. It is that way.

Individual teachers take on responsibilities beyond what they have to take on; individual schools may do the same thing.

But if you don't have one of those teachers or schools, assessing your child's learning and remediating gaps is your job.

Saxon into the breach

So last week Christian started Saxon Math 5/4 Homeschool Edition.

MORE COMING






^* Probably the best article on this is Hanna Skandera & Richard Sousa's "Student Mobility and the Achievement Gap" in the Hoover Digest, but the link isn't working at the moment.

^** Christian's mom didn't have Medicaid because she made too much money. We had Medicaid because we got a Medicaid waiver.

^*** Tough to sue a writer, I've noticed. Free speech is a beautiful thing.

christianlearnsmath

-- CatherineJohnson - 17 Oct 2006

---

HowToCureMathAnxiety 23 Oct 2006 - 00:00 CatherineJohnson





So I mentioned that the Yonkers school system graduated Christian from high school with a 3rd grade level of math knowledge — and that last week I'd started him on the 4th grade Saxon Math book, Saxon 5/4.

It's working.

He breezed through the first lesson.

Then he breezed through the second lesson.

Then he breezed through the third.

Now he comes in, deals with the kids, and, after dinner, pulls out his Saxon Math books and does the lesson in 15 or 20 minutes while keeping an eye on Andrew & watching whatever Christopher has on TV.

This is the way you cure math anxiety.

I think.

You cure math anxiety the same way you remediate gaps, by starting a student back before the point where math got hard. That's the Siegfried Engelmann way; that's the KUMON way.

I don't know that Christian had a full-blown case of "math anxiety."

I don't even know what "math anxiety" is, other than the predictable result of years of lousy math instruction.

Christian was against math when he first started working with the kids.

Now he's neither for math nor against it, as far as I can tell.

He just comes in and does his math.

Long live Saxon.



Speaking of living long, I'm a bit concerned.

The Harcourt website seems to have pulled its online placement tests.

yikes

We need those.

I have copies myself, and I've found two other websites that have them posted (links below). Harcourt has an online test, which is good, but it doesn't give you the flexibility you need and reading online is difficult.

I hope they post those tests again.

Engelmann on student placement in curriculum

Rule 1: Hold the same standard for high performers and low performers. This rule is based on the fact that students of all performance levels exhibit the same learning patterns if they have the same foundation in information and skills. The false belief that characterizes the conventional wisdom about teaching is that lower performers learn in generically different ways from higher performers and should be held to a lower or looser standard. Evidence of this belief is that teachers frequently have different “expectations” for higher and lower performers. They expect higher performers to learn the material; they excuse lower performers from achieving the same standard of performance. Many teachers believe that lower performers are something like crippled children. They can walk the same route that the higher performers walk, but they need more help in walking.

These teachers often drag students through the lesson and provide a lot of additional prompting. They have to drag students because the students are making a very high percentage of first-time errors. In fact, the students make so many mistakes that it is very clear that they are not placed appropriately in the sequence and could not achieve mastery on the material in a reasonable amount of time. The teachers may correct the mistakes, and may even repeat some parts that had errors; however, at the end of the exercise, the students are clearly not near 100% firm on anything. Furthermore, the teacher most probably does not provide delayed tests to assess the extent to which these students have retained what had been presented earlier.

The information these teachers receive about low performers is that they do not retain information, that they need lots and lots of practice, and that they don’t seem to have strategies for learning new material. Ironically, however, all these outcomes are predictable for students who receive the kind of instruction these students have received. High performers receiving instruction of the same relative difficulty or unfamiliarity would perform the same way. Let’s say the lower performers typically have a first-time-correct percentage of 40%. If higher performers were placed in material that resulted in a 40% first-time-correct performance, their behavior would be like that of lower performers. They would fail to retain the material, rely on the teacher for help, not exhibit selfconfidence, and continue to make the same sorts of mistakes again.

If students are placed according to their first-time-correct percentages, they tend to learn and behave the same way, whether they are “lower performers” or “higher performers.” In Project Follow Through, we mapped the progress of students of different IQ ranges. The results showed that regardless of students’ entering IQ, the rate of progress was quite similar across all children and across different subjects. Lower performers learned as fast as higher performers. They simply started at a different place, with material that higher performers had long since mastered. Note that this conclusion may be somewhat biased because we paid particular attention to the instruction for the lower performers. They tended to have better teachers and their instruction tended to be monitored very closely. In any case, they learned at a very healthy rate, one that paralleled that of students with IQs 40 points higher.

The typical practices of placing and teaching students are completely opposed to appropriate placement and teaching procedures. At the University of Oregon, we place teaching-practice students in special-ed classrooms that use direct-instruction programs. During the years that we first offered these practica, we typically worked with teachers who were teaching DI but had not generally received much training. Before we arranged for a placement with a new supervising teacher, therefore, we made sure that the classroom was “appropriate” for our students, which means that the children the practicum students were to work with were placed appropriately and that the teacher was using and modeling appropriate practices. As part of the review of the new classrooms that were candidates for receiving practicum students, we checked the program placement of the students and changed their placement if necessary.

Our estimate is that in the first 40 or more classrooms we used, the children were moved back in DI reading programs an average of 100 lessons—sometimes 120 lessons. The children, in other words, were placed about 3/4 of a school year or more beyond the optimum first-time-correct percentages. Nearly all teachers had children that were seriously misplaced. Furthermore, I don’t recall a single classroom in which children’s percentages required us to move children ahead in the programs. Children were always “over their heads.”

source:
Student Program Alignment and Teaching to Mastery (pdf file)
by Siegfried Engelmann

Children were always "over their heads."

When you design instruction so that children are always in over their heads, you have poor instruction and bad student outcomes.

Period.




in a nutshell

• students of all performance levels exhibit the same learning patterns if they have the same foundation in information and skills

• when placed correctly in the curriculum - at a level determined by what they already know and can do - "low performing" or "learning disabled" students learn "at a very healthy rate, one that paralleled that of students with IQs 40 points higher"

• teachers must provide delayed tests to assess the extent to which these students have retained what has been presented earlier

• children who are struggling in school are almost always placed far ahead of themselves in the curriculum. They are struggling the same way a straight-A student whose never studied physics would struggle if you dropped him into the middle of a college physics course.
  

remediating math at the college level

We've had a number of discussions about how to remediate the math knowledge of college kids. Rudbeckia Hirta has written about this often, along with Carolyn &, I think, Steve H.

When Ed and I first discussed what to do about Christian's math, I said I didn't particularly feel like paying for Westchester Community College to teach Christian the math his mom already paid Yonkers to teach him back in grades 6-12.

I said I was thinking about just giving him the Saxon placement test & buying him the books & seeing how it went.

Ed thought that was a bad idea.

(It's fun to be right! It's even more fun to be right on your own blooki.)

Actually, "I told you so" isn't (really) the point. At the time I thought Ed was probably right. I thought handing Christian a couple of Saxon books and telling him to go teach himself math — or trying to teach him myself — was probably nuts. He doesn't (didn't) like math; he couldn't do math; he had no interest whatsoever in the whole question of math....none of this sounds like a recipe for success in self-teaching.

The fact that I went ahead and gave him the Saxon placement test anyway was probably a Bayesian moment. I've been teaching myself math long enough now, and teaching Christopher math, and writing ktm, and reading the Comments that I probably "knew" at the level of the cognitive unconscious that Saxon was a better idea than a $500 "developmental" math course at a community college.

The reason Saxon is probably a better idea (it's early days, I realize) is that the community college doesn't offer courses in 4th grade arithmetic.

They're trying to remediate years of missing math instruction in one or two semesters. I just don't see how it can work no matter how good their teachers are (and I bet their teachers are good). In fact, the effort to close years of math gaps in one or two semesters seems almost inevitably destined to produce more math anxiety than a student had going in.

I'm thinking that just as there is no royal road to geometry, there's no royal road to Algebra 1. People whose math education went completely off the rails back when they were eight have to go back to when they were eight.

At least, that's what I'm thinking at the moment.

We'll see how it goes.






Saxon middle grades online placement test
downloadable Saxon placement tests at Learning Things
downloadable Saxon placement tests at Sonlight Curriculum
Rainbow Resource (least expensive source of Saxon Homeschool texts I've found)
teachingtomastery teachtomastery programplacement
Christianlearnsmath

-- CatherineJohnson - 20 Oct 2006

---

FindTheVariable 24 Oct 2006 - 23:48 CatherineJohnson




Christian just came in and said he has a friend, a young woman, in the same spot he's in: college level reading, grade school math.

She dropped out of school in 10th grade and passed the G.E.D. at some point. I didn't know you could pass the G.E.D. with grade school math, but Christian thinks she must have passed on the strength of her reading. Now she wants to go to college and she bombed the math portion of the placement test.

She was telling Christian about some problem on the placement test: "What's 4 plus d equals 8? I don't know what that is." She was flummoxed by the site of a letter inside an equation.

Christian told her, "d is 4. You have to find the variable." He said her mouth dropped when he told her that. She had no idea 4 + d could be anything remotely that simple.

Christian said he was pretty stunned himself.

"Variable," he said. "I never thought that word would come out of my mouth."

Obviously none of his 4 or 5 public schools thought so, either.



The cool thing is that he told her to do exactly what he's doing. Get the Saxon books and teach herself.

He also suggested KUMON!

Saxon is amazing. Think of the power these books give math-disenfranchised young people.

We're talking about brainy young adults with college level reading and 3rd grade level math.

I'm not sure I'll ever get over that. The only way this level of non-education could have happened was that Christian's schools took less-than-zero responsibility for finding out where new students were in the curriculum and catching them up to the rest of the kids.

They just threw them into the deep end of the pool and walked away.



So today Christian says he wants to pick up the pace. He's been doing one lesson a day 4 days a week, because I wanted to keep the books here long enough to see how things were going. I told him he should do 2 lessons at least. He's whipping through them, so he may as well.

I kept the books here for another reason, which was to be around while Christian established self-study skills. He's had no problem doing the work for the course he's taking or getting to class on time, and he's getting good grades. But studying math — even 4th grade math — completely on your own is another kettle of fish, and even though he's here a lot I didn't actually know whether an auto-didact math project was going to suit him or not.

Saxon worked its magic. The books hook you; they're rhythmic and soothing in their way.

My thinking now is that he should do all of Saxon 5/4, then go straight into 6/5 but skip the first 20 or 30 lessons, which are review. Given how easy this material is for him, and given the fact that he did sit in math classes for a good 10 years of his life, he's got some knowledge of arithmetic inside his head. We just don't know what it is, exactly, or what and where the gaps are. From what I can see, his knowledge of arithmetic is enough that he doesn't need to go directly from 5/4 to 6/5 and spend the first 30 lessons doing 5/4 again.

After that, I don't know. I'm thinking he should do 7/6, then skip to Algebra 1/2. On the other hand, 8/7 has all the required state topics, such as probability & stem and leaf charts and all the rest. Apparently Algebra 1/2 is more serious, but the focus is narrower.

Either book will almost certainly give him everything he needs to pass the placement test; 7/6 might give him enough, as a matter of fact.

On the other hand, since I'm semi-overseeing this process, and since he's probably going to become a certified teacher — and he's almost certainly going to be working as a classroom aide long before that day — I think he has a moral duty (seriously) to learn math and learn it well. I'm going to push him to work his way through Algebra 1, too.

I told him the other day, "If you're an aide, you're going to be teaching math to these kids. It's going to be you." I know this because that's true in our schools here. It's not that the special ed teacher doesn't teach math to the special ed kids; she does. But ideally the aide should be able to do one on one teaching of math, which aides here can do.

And as Liping Ma says, if you're going to do one on one teaching of arithmetic, you should know algebra. You should know what comes next.

When I said he'd be directly teaching math to ld kids Christian looked taken aback, but not horrified. Which is good.




Accuplacer

I just checked out some sample practice tests for Accuplacer, the College Board placement test all the colleges seem to use.

Elementary Algebra

Question 1:
If a number is divided by 4 and then 3 is subtracted, the result is 0. What is the number?
A. 12
B. 4
C. 3
D. 2

Question 2:
If A represents the number of apples purchased at $.15 each and B represents the number of bananas purchased at $.10 each, which of the following represents the total value of the purchases?
A. A+B
B. 25(A+B)
C. 10A + 15B
D. 15A + 10B

Question 3:
16x - 8 =
A. 8x
B. 8(2x - x)
C. 8(2x - 1)
D. 8(2x - 8)

Question 4:
If x2 - x - 6 = 0, then x is
A. -2 or 3
B. -1 or 6
C. 1 or -6
D. 2 or –3


ANSWERS (Elementary Algebra)
1. A
2. D
3. C
4. A


I'm going to go make Christopher & Christian take a look at these now.

I'm going to do that because I'm evil.

Christianlearnsmath

-- CatherineJohnson - 24 Oct 2006

---

NationalMathAdvisoryPanelLinks 21 Nov 2006 - 18:07 CatherineJohnson




meetings

• First Meeting (pdf file)

• Second Meeting (pdf file)

• Third Meeting (pdf file)
  

email updates

about the panel

homepage




where you can find links

I'm posting links to the Math Panel homepage, transcripts, & ktm posts here:

• recommended reading page
  

• book-style index
  

You can find both pages on the menu to the left.

If all else fails you can search posts using the keyword nationalmathematicsadvisorypanel with no spaces between words. (Works pretty well with spaces, too.)

I'm thinking this is about as findable and redundant as I can make the links now...unfortunately, you will have to remember some constellation of the words "national mathematics advisory panel" to find these links (that could be iffy for me these days....)

But I think I've just raised the odds of re-finding the transcript links considerably.


panel members w/links
Polite agreement or something we can use?
National Math Panel announcement
National Math Panel update
short story by Vern Williams

nationalmathematicsadvisorypanel


-- CatherineJohnson - 07 Nov 2006

---

ChristianFirstTest 18 Nov 2006 - 22:16 CatherineJohnson




Christian took his first test Friday - missed one out of 20!



   hah!




   source:
   Bitter Single Guy

He is going to learn math.

He's going to learn math if he keeps at it, which I'm expecting he will.

My bet is Christian will keep at it either by keeping at it, or by leaving his math books here long enough for me to spend the next couple of years telling him to go do his math.

Either way will work, and either way is fine by me.

It's becoming obvious he's got the "get up and go" to get through college. If he wants to borrow my frontal lobes to get through the math he should have been taught when he was nine years old, that may be the smart choice.

He's still got a little ways to go on what Rudbeckia would call the "baggage."

My copy of the Iowa Test of Basic Skills came in the mail yesterday.

aside: Giving the ITBS to Christopher is going to be a huge undertaking. The package is an inch thick - and there are many pages of directions. Six hours of testing.

Christian was dragging his heels about the Saxon test. First he said he didn't want to take it on Thursday; he wanted to take it on Friday, because Friday is his a lucky test-taking day.

I said, Fine, take your test on Friday.

So yesterday was Friday and he was still dragging his heels. Finally I told him he had to sit down and take the test.

He sat.

When I handed him the test booklet he said, "Is this it?"

"That's it."

"Ohhhhhh.....I thought it was that big test package there."

He was looking at the ITBS.

Christianlearnsmath



-- CatherineJohnson - 18 Nov 2006

---

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

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