Comments

The answer is 3 times h with exponent 5 all over k with exponent 5.

Exponents change sign when you move them from the numerator to the denominator or vice versa.

And you add the exponents which have the same base.

So when you move the h (h with exponent 1) to the numerator you get h with exponent 6 times h with exponent -1 or h with exponent (6-1).

When you move the k with exponent -3 to the denominator, you get k with exponent 2 times k with exponent 3 or k with exponent (2+3).

-- SusanJ - 06 Jan 2006

3 h^5 k^-5, or, if you prefer, 3 h^5 / k^5. Probably the latter is what they're looking for.

-- GoogleMaster - 06 Jan 2006

Is this it?

-- CatherineJohnson - 06 Jan 2006

yes!

that's it!

-- CatherineJohnson - 06 Jan 2006

3h5
---
 k5

-- GoogleMaster - 06 Jan 2006

What are you using to make your nifty JPGs?

-- GoogleMaster - 06 Jan 2006

OK, I goofed.

This is the real problem:

Here's the answer the book gives:

-- CatherineJohnson - 06 Jan 2006

Google Master

I figured out a fast work-around.

• I'm using Word (on a Mac), which comes with Equation Editor. (That's a free, short version of.....I forget. Sorry.)

• I write the equation in a Word document, then use the Mac Screen Grab command to 'grab' & save it.

• Screen Grab saves everything in TIFF.

• Then I use Mac's 'Preview' program to open up the TIFF file.

• I use the 'Save As' command to save the file as a jpeg.

voila

-- CatherineJohnson - 06 Jan 2006

I thought it probably involved Preview somehow. :)

The book has a typo.

-- GoogleMaster - 06 Jan 2006

Does Word on Windows have Preview??

I never used any of these things before I switched.

-- CatherineJohnson - 06 Jan 2006

The book has a lot of typos.

Some of which are obviously calculation errors.

Though you're right; in this case it's probably a typo.

-- CatherineJohnson - 06 Jan 2006

Catherine, the answer the book gave to that last problem is wrong. The exponent of k should be -1.

-- CarolynJohnston - 06 Jan 2006

That's a 6th grade math problem? I think I would have come unglued!!!

-- KtmGuest - 06 Jan 2006

3h^5/k

-- VerghisKoshi - 06 Jan 2006

The most difficult thing about this problem--assuming that the student has learned multiplication and division of powers, including negative exponents--is that the numerical coefficients are simply divided, whereas the exponents must be subtracted. Many algebra 1 students, after they have gotten the hang of the whole dividing powers thing, will give the answer 10h^5/k or 10h^5k^(-1), subtracting the 15 and 5 instead of dividing.

There are so many concepts and skills that are prerequisite to understanding and solving this problem! Have they worked with division of powers enough to know WHY you subtract the exponents? Do they understand what the problem would look like if it were written without exponents? I.e., do they know what the problem means?

-- GretaFrohbieter - 06 Jan 2006

"...do they know what the problem means?"

Probably not, but there's no need to make it harder than it is. One way to do it is this:

15(h^6)(k^-3)/5h(k^-2)

= 3x(hxhxhxhxhxh xkxk)/kxkxk xh

= 3(h^5)/k.

All it takes is knowing that a negative power implies the reciprocal. Not sure why this problem is useful, though.

But it's possible that I'm missing something.

-- VerghisKoshi - 06 Jan 2006

Er, isn't this algebra and not "pre-algebra"?

I suppose pre-algebra now is pick an algebra lesson (and I use that term loosely) at random, teach it poorly or not at all, and ask the student to memorize the answer solve the problem.

-- KDeRosa - 09 Jan 2006

Er, isn't this algebra and not "pre-algebra"?

I HAVE A WHOLE BIG LONG POST ON PRE-ALGEBRA I'M JUST BURSTING TO WRITE

(got sidetracked by Bayes)

-- CatherineJohnson - 09 Jan 2006

Actually, your one-sentence Comment IS THE POST I WAS PLANNING TO WRITE.

Ed figured it out Saturday night.

He'd been wrestling with this stuff all day, and suddenly he said: "This is spiraling."

They're going to teach this same stuff for the next 3 years.

-- CatherineJohnson - 09 Jan 2006

He'd been wrestling with this stuff all day, and suddenly he said: "This is spiraling."

On the nosey.

This must be how algebra gets taught in the spiral curriculum. Teach it over 2-3 years and keep on revisiting the topics and hope the student gets it by the end of the 2nd or third year.

We're all looking for the staircase of knowledge and successive building of math skills. But, there is no staircase. There is no spoon.

What a horrible way to teach algebra. What a horrible way to teach anything for that matter, but it really hits home when you see a problem like this given out of the blue to neophyte students and recall how much underlying algebraic knowledge it took to get to this level using the staircase approach.

Now I know why I always instinctively hated the spiral second-hand (since I never experienced it myself). I have to go back and re-read Engelmann's book where he mentions the spiral and how nasty it really is.

The good news is that you'll be able to reteach all this from the beginning once school is out and not picking it up now won't be fatal to higher learning.

Brutal.

-- KDeRosa - 09 Jan 2006

A disincentive to actual learning!

I think that's one of the biggest problems with the spiral approach. When kids don't get it, and get bad grades as a result, the risk is that they either tune out and decide they don't care any more, or completely lose confidence in their ability to learn.

How about this for the effects of the spiral approach? The downward spiral of kids who "don't get math" continues. . . .

-- KtmGuest - 09 Jan 2006

Teaching algebra this way is the diametric opposite of mastery learning. It's as close to anti-practice as you can get. One of teh advantages of taching algebra traditionally isthat the each lessons builds on theh last and increasingly difficult problems get solved. So not only is the student learning the new stuff, but he's also practicing the old stuff by solving lots of problems. By the end of the year, students have probably mastered the first half of teh material due to all the problems they've solved even if there hasn't been enough practice for each topic as it's taught. A crude mastery if you will.

With this spiralling nonsense you don't even get that. And recall Engelmann on teh shelf life of knowledege being so short and see the classic sorting machine in practice, even worse, I bet, than under the traditional curriculum.

-- KDeRosa - 09 Jan 2006

"So not only is the student learning the new stuff, but he's also practicing the old stuff by solving lots of problems."

One could argue that this is a proper form of spiraling. However, the modern ed school approach to spiraling is tied directly to full-inclusion. Instead of spiraling to more complicated skills, understandings, and expectations, they use it as a pedagogical excuse to allow lower ability students to slide from grade to grade because they will see the material again. I have never heard any good explanation of what the more able students are doing when their teachers rehash the same material for those who didn't get it the first time. Let's call it rehashing instead of spiraling. I have mentioned in the past that our school still tries to get students to master their adds and subtracts to 20 in third grade. I can't imagine when they require mastery of the times table.

Each loop of the spiral covers the material superficially since they will present it again later on or next year. (Think Everyday Math) This superficial spiral approach precludes mastery. That's because they see no linkage between mastery and understanding.

-- SteveH - 09 Jan 2006

Ken

On the nosey.

This must be how algebra gets taught in the spiral curriculum. Teach it over 2-3 years and keep on revisiting the topics and hope the student gets it by the end of the 2nd or third year.

Well, that's what we're thinking.

The weird thing is that they throw everything in in the first go-round. (Is that right? I listed the topics here.

Spiral curricula in K-8 DON'T start out by dumping every single topic on kids in Kindergarten.

Ed says, though, that he thinks fuzzy math has taken major steps in that direction — teaching probability in Kindergarten, etc.

-- CatherineJohnson - 09 Jan 2006

What a horrible way to teach algebra. What a horrible way to teach anything for that matter, but it really hits home when you see a problem like this given out of the blue to neophyte students and recall how much underlying algebraic knowledge it took to get to this level using the staircase approach.

Now I know why I always instinctively hated the spiral second-hand (since I never experienced it myself). I have to go back and re-read Engelmann's book where he mentions the spiral and how nasty it really is.

The good news is that you'll be able to reteach all this from the beginning once school is out and not picking it up now won't be fatal to higher learning.

Brutal.

It's amazing, but you have to experience it firsthand to know how bad it is.

With spiralling in K-8 I always think about hideous boredom & repetition — though of course we're hearing stories about little kids bursting into tears when the textbooks come out.

Until I saw it myself, I just couldn't get how miserable this is.

In this case, both of us REALLY got it, because neither of us had ever been taught negative exponents that we recall.

Ed had to learn them on the spot; I had done a Saxon lesson on them just a couple of weeks ago.

Both of us could learn negative exponents pretty fast, but we had very shaky knowledge, and there we were trying to get this stuff inside Christopher's head.

We could feel some of his bewilderment and utter lack of interest in being flogged through this stuff for another day.

-- CatherineJohnson - 09 Jan 2006

It was funny, though.

Ed was doing big-time Discovery Learning.

He had no memory of ever laying eyes on a negative exponent, and at least four of the answers in the Teacher's Edition were wrong.

So he was doing 'multiple solution math'; he was coming up with extensions of principles he already knew -- like you subtract the denominator exponent from the numerator exponent to get the answer -- and applying them to negative exponents.

Contance Kamalii would have been proud.

-- CatherineJohnson - 09 Jan 2006

Today he said, 'I wonder if I could remember calculus if I look at one of the books.'

I said, 'You better start looking at those books.'

-- CatherineJohnson - 09 Jan 2006

This course shows me why so many people loathe math.

-- CatherineJohnson - 10 Jan 2006

re-read Engelmann's book where he mentions the spiral and how nasty it really is

Is that in the Academic Abuse book?

-- CatherineJohnson - 10 Jan 2006

ktm Guest

DOWNWARD SPIRAL IS IT!!!

I think that's one of the biggest problems with the spiral approach. When kids don't get it, and get bad grades as a result, the risk is that they either tune out and decide they don't care any more, or completely lose confidence in their ability to learn.

This is what I don't get.

We're going to deliberately not teach to mastery, BUT we're going to give tests & grades.

-- CatherineJohnson - 10 Jan 2006

Teaching algebra this way is the diametric opposite of mastery learning. It's as close to anti-practice as you can get.

That is a great term.

Anti-practice.

It is, absolutely, the diametric opposite of mastery learning.

-- CatherineJohnson - 10 Jan 2006

So not only is the student learning the new stuff, but he's also practicing the old stuff by solving lots of problems.

Right.

Exactly.

The RUSSIAN MATH book is fantastic that way. It is organized in chapters, unlike Saxon, but you get constant, ongoing practice in all skills taught. The book also has review of earlier material in every chapter.

-- CatherineJohnson - 10 Jan 2006

they use it as a pedagogical excuse to allow lower ability students to slide from grade to grade because they will see the material again

Is that why spiraling came into being??

And when DID it come into being?

Actually, I think the article Ken linked to gives the history (I just read the opening).

Oh heck.

I can't find it, and I don't remember it!

Ken do you remember where & what it was?

The guy said that spiraling came from .... Piaget?

-- CatherineJohnson - 10 Jan 2006

Yes, Engelmann describes spiralling in Academic Child Abuse. Not sure if he gives the entire history.

The topic list seems to be alot more material from the Algebra I class than I remember taking in 8th grade.

-- KDeRosa - 10 Jan 2006

It was funny, though.

Ed was doing big-time Discovery Learning.

He had no memory of ever laying eyes on a negative exponent, and at least four of the answers in the Teacher's Edition were wrong.

So he was doing 'multiple solution math'; he was coming up with extensions of principles he already knew -- like you subtract the denominator exponent from the numerator exponent to get the answer -- and applying them to negative exponents.

Done that sort of thing myself a few times. I guess it's why educators can go nuts over discovery learning - it's the sort of thing you can do in maths if you know the basic concepts well enough and have a vague memory of doing something more difficult once. They just try to skip the part of doing heaps of maths and learning to automaticity the stuff you use to do the final derivation.

But perhaps you should be posting Chris's maths problems here each day, the moment he brings them home, for us to tackle? You can then explain to Chris afterwards. The distributed power of the internet appears to be better at coming up with right answers than the back of his textbook.

-- TracyW - 10 Jan 2006

The topic list seems to be alot more material from the Algebra I class than I remember taking in 8th grade.

I'm going to compare it to my algebra books here.

Plus, on top of the algebra, there's statistics, probability, 'representing data' and on and on (plus they have to learn some geometry this year, because it's on the state test).

-- CatherineJohnson - 10 Jan 2006

Done that sort of thing myself a few times. I guess it's why educators can go nuts over discovery learning - it's the sort of thing you can do in maths if you know the basic concepts well enough and have a vague memory of doing something more difficult once. They just try to skip the part of doing heaps of maths and learning to automaticity the stuff you use to do the final derivation.

I know — it's a Mad Dash.

I'm thrilled when I realize I know enough math to figure something out if I've forgotten it.....

That is NOT where Christopher is, needless to say.

When Ed showed him 'two-step equations' he couldn't begin to do them. They were completely foreign.

I think that a child who had learned to solve one-step equations to mastery would generalize.

I think.

(I'd love to know.)

-- CatherineJohnson - 10 Jan 2006

I'm thinking teachers & texts end up having to do way more teaching precisely because the child never learns to mastery and thus never, ever, generalizes anything.

In a way, they're 'giving' these children autism.

My favorite autism story is the little boy (this is a true story) who was painstakingly taught to butter his bread.

Then one day they gave him peanut butter and he had no idea what to do. They had to start all over.

That's the way it is with Christopher & equations.

We 'teach' him to solve a one-step equation; then, when we show him a two-step equation, he has no idea what to do.

-- CatherineJohnson - 10 Jan 2006

The distributed power of the internet appears to be better at coming up with right answers than the back of his textbook.

yeah, and it's not so distributed

everyone here gets the right answer instantly

pretty soon here i'm going to start sounding like those parents in the fuzzy math articles

I WANTED TO THROW THE BOOK AGAINST THE WALL SEVERAL TIMES, said Jack Stoner, father of Jillian.

-- CatherineJohnson - 10 Jan 2006

Constructivism in the Classroom (pdf file)

-- CatherineJohnson - 10 Jan 2006

For a student to excel in a class like this, by the end of fifth grade he should know fractions up, down, and sideways, how to manipulate decimals cold, and how to do long division skillfully. Then assuming the teacher is doing an adequate job teaching the material and the student is getting enough practice solving problems, the student might be able to keep up the pace. A lot of ifs.

-- KDeRosa - 10 Jan 2006

by the end of fifth grade he should know fractions up, down, and sideways

I bet there isn't one child in the entire class who knows fractions up, down, and sideways

decimals, either

Christopher does have long division down cold or close to.

I would have said he's not bad on decimals, but that's the chapter he got his D on...

I need to pull that test out and analyze it.

-- CatherineJohnson - 10 Jan 2006

assuming the teacher is doing an adequate job teaching the material and the student is getting enough practice solving problems

I think you could probably get kids through it in one piece with daily formative assessment and 30 problems homework each day (ranging from easy to intermediate to hard).

But I question whether you could get any conceptual knowledge at all — and when I say 'conceptual' I'm thinking about word problems.

Oh — here's a good one.

The teacher put a word problem on the 4-problem test today.

This after not teaching or assigning any word problems in the two Lessons she was testing.

The incredible thing is that Ed had Christopher do some word problems this weekend (because I insisted!) — and she put the same word problem on the test that Ed had Christopher do.

That was another discovery moment....never having drawn a bar model in his life, Ed drew an awkward, beginner bar model to explain the problem.

Christopher has actually done half of the third grade workbook's bar models, and he got it.

Today, when he saw the same problem on the test with different figures, he drew a bar model.

Then he did the correct operation.

-- CatherineJohnson - 10 Jan 2006

The problem was something like, 'A room is 20 feet wide, and floor boards are 2 1/2 feet wide. How many floor boards to you need?'

When Christopher tried to do this problem with his dad he didn't know whether to multiply or divide. He wanted to multiply. (I'm almost positive that if both numbers had been whole numbers he would have known what to do.)

When he drew a bar model representing the 20 feet, he saw that he needed to divide.

This was in the middle of a test.

I think that's incredible.

He said when he got the test he got really scared. He started thinking, 'I can't get an F. I can't get an F.'

Then he looked at the test and thought, 'Oh, I know how to do this.'

We keep hammering away at that.

The way to not be terrified of a test is to go in knowing the material.

-- CatherineJohnson - 10 Jan 2006

We're going to be doing bar models this summer.

-- CatherineJohnson - 10 Jan 2006

I'll get around to posting the KUMON word problems.

I think you could probably get some conceptual meaning inside kids' heads in a course this fast with superb teaching and 'embedded' KUMON-type word problems....

-- CatherineJohnson - 10 Jan 2006

Directly taught conceptual knowledge is overrated. You only really understand something conceptually after doing a bazillion (a technical math term) problems. Directly taught conceptual knowledge might help lessen the amount of time it takes to get this eureka moment, but you still have to put your time in solving problems. There are no shortcuts.

-- KDeRosa - 10 Jan 2006

When Christopher tried to do this problem with his dad he didn't know whether to multiply or divide. He wanted to multiply.

Well, that's still pretty cool, because he did need to multiply: 2.5 x what >= 20 ?

Keep on trucking! I mean, bar modeling!

(I am a faithful believer in Number Lines.)

-- BeckyC - 10 Jan 2006

When Christopher tried to do this problem with his dad he didn't know whether to multiply or divide. He wanted to multiply.

I know you don't have time to teach him this in addition to all the re-teaching you are already doing, but...

Dimensional analysis tells you whether to multiply or divide.

You start out with the unit of feet (The floor is 20 feet wide). You want to get to the unit boards ("How many boards?" is the question you want to answer). So, you need to multiply by a conversion factor that has boards in the numerator and feet in the denominator:

(20 feet) * (1 board/2.5 feet) = 8 boards

The dimensions guide you through it.

-- DanK - 10 Jan 2006

I want to be a contrarian here. I haven't read Engelmann. I see a lot that's good in the way that Saxon does spiraling. I'm one of those people who doesn't always pick up a new topic right away. My experience is that I often failed to really get it until we had moved on to the next topic--or the next one--where we actually started putting the original topic to use. Then, I'd finally see it. I think Saxon-style spiraling facilitates this.

I acknowledge that it also means that stuff can continue to get included in problem sets long after it is fully understood. Still, the spiraling works against students forgetting/losing a concept over time.

I agree with the argument that this complicated expression with exponents is hardly pre-algebra. I also agree with the other thread where you argue that they are trying to pack too much stuff into pre-algebra. Even more, I agree that grades 6, 7, and 8 seem to overlap so much to seem to be repetitions of one another. I just don't think that spiraling is the problem. If the middle school curriculum had progressed out of a spiraled elementary curriculum and carried forward sensibly, I think grades 6, 7, and 8 could build on one another; and you wouldn't have to cover every sub-topic from scratch again every year.

Also, I think a lot of algebra should be started in fourth and fifth grades. Use letters as variables in formulating multiplication/division fact family problems, for example. Use variables (and UNITS) when setting up and solving story problems at any level.

-- DanK - 10 Jan 2006

Directly taught conceptual knowledge is overrated.

It's way overrated — and it's much easier to acquire than procedural knowledge.

I'll get the KUMON pages posted.

KUMON has you do a gazillion (technical math term) fraction divisions and then gives you 3 very simple, straightforward, and obvious word problems requiring fraction division — all in a row.

The first time I did them, I could practically feel my brain changing.

Doing simple word problems that perfectly illustrated the procedures I'd just been practicing gave me a mini-Eureka moment even though I know how (and usually why) to divide a fraction by a fraction.

-- CatherineJohnson - 10 Jan 2006

"Then he looked at the test and thought, 'Oh, I know how to do this.'"

Congratulations on a huge victory. A few more of those in a row and he should start going into the test with the expectation that he already knows the material. That attitude is probably worth a letter grade by itself.

-- DougSundseth - 10 Jan 2006

After my last comment, I realized that it might be useful to discuss why attitude is so important on a math test.

If you go in thinking that you can solve every problem, you'll still run across the occasional problem that seems impossible at first glance. But, since you know you can figure it out, you'll dink around until you see some little string hanging off that you can pull and unravel the whole thing.

OTOH, if you're convinced that there are parts you can't figure out, you'll stare at the problem in despair and never really try to solve it. Despair is time-consuming and unproductive.

Of course, you still have to remember to do the easy stuff first, so you don't waste all your time dinking around and lose the easy points. Save the tricky problems for the second or third pass. Heck, your subconscious might have figured some of them out by the time you get back to them. If that happens, they're just more easy problems and leave more time for the hard parts.

Math is sometimes hard for everybody, but if you're convinced that it's not impossible, it isn't.

-- DougSundseth - 10 Jan 2006

Becky

I am a faithful believer in Number Lines.

Absolutely. I've got Doug's printed out and socked into the Big Practice Book I made up.

Supposedly, humans have a kind of number line in our brains.

If that's true, it's a darn good reason to keep directing kids back to number lines.

-- CatherineJohnson - 10 Jan 2006

Dan

That's an interesting way to go about the floor boards problem....I hadn't thought of that.

I've only just this minute learned unit conversions.....I'll have to try this.

Also, on spiraling, yes, definitely, I like the Saxon approach.

I've had the experience many times of 'getting' something after I already got it, supposedly.

I'm griping about spiralling in the sense algebra seems to be spiralling, which is to throw everything at the kids, have them not learn it, then throw it at them again the next year and the next.

-- CatherineJohnson - 10 Jan 2006

Dan

I just don't think that spiraling is the problem. If the middle school curriculum had progressed out of a spiraled elementary curriculum and carried forward sensibly, I think grades 6, 7, and 8 could build on one another; and you wouldn't have to cover every sub-topic from scratch again every year.

Well I think Singapore Math considers itself a spiral curriculum, doesn't it?

(I can't quite remember at the moment....)

My sense of Singapore Math is that they teach the same topics year after year, but each year's teaching is much more advanced.

I'd say that Singapore Math has a philosophy of teaching to mastery with spiralling.

Didn't Dan Willingham cite research showing that if you study the same material for 3 years running you remember it forever?

I think he did.

(Lots of stuff to look up.....)

-- CatherineJohnson - 10 Jan 2006

Doug

If you go in thinking that you can solve every problem, you'll still run across the occasional problem that seems impossible at first glance. But, since you know you can figure it out, you'll dink around until you see some little string hanging off that you can pull and unravel the whole thing.

OTOH, if you're convinced that there are parts you can't figure out, you'll stare at the problem in despair and never really try to solve it. Despair is time-consuming and unproductive.

This gets us back to our lopped-off Why Do People Have Math Anxiety? discussion.

I don't think we talked about temperament & math.

Christopher is a fairly anxious kid, and Ed was, too. Ed told me he always tested worse than his abilities, because he was anxious every time he took a test. He wouldn't sleep at all the night before; he'd be completely cranked up; etc.

I'm the opposite. I always kind of liked taking tests, probably because the heightened demands are like ritalin. John Ratey always said this about ADHD-types taking risks. To a person with ADHD, a risk is organizing & focusing. I don't know that I have ADHD, but I'm close enough that I probably enjoy the pressure of a test for the 'focusing effect.'

Your comment made me realize the (obvious) fact that some kids are going to be more bullet-proof than others when it comes to math (especially when it comes to bad math textbooks and bad mad courses).

I'm going to focus on this from now on.

The cure for anxiety is PRACTICE.

otoh, I'm not sure what I can tell him about the test-taking situation itself.....

-- CatherineJohnson - 10 Jan 2006

Save the tricky problems for the second or third pass. Heck, your subconscious might have figured some of them out by the time you get back to them.

That happens to me all the time.

I now use it as a conscious strategy on KUMON problems (I'm on fraction sheets that are pretty challenging now...)

-- CatherineJohnson - 10 Jan 2006

Challenging to do fast, that is.

-- CatherineJohnson - 10 Jan 2006

Also, I think a lot of algebra should be started in fourth and fifth grades. Use letters as variables in formulating multiplication/division fact family problems, for example. Use variables (and UNITS) when setting up and solving story problems at any level.

Absolutely.

Singapore teaches algebra — in the sense of finding an unknown — in 3rd grade. (Don't know about 2nd....)

-- CatherineJohnson - 10 Jan 2006

Saxon may spiral but the difference is what we've discussed many times. It is largely coherent. It teaches the right things at the right time. Not always in my opinion, but many, many times. Singapore is similar that way, also.

Saxon pushes for mastery of that one little skill it is teaching at the moment. Should you not get it you will be faced with it in the mixed practice on the next 150 some-odd chapters, so there is no way to get away from it.

I believe we've discussed how Singapore brings back skills through its word problems. Singapore seems to stay longer in a general area, bouncing around different related skills and working the different inverses. There's a real thoroughness to the Singapore approach.

I read somewhere where someone said that Saxon reviews too much, but it is reviewing the new skills taught in earlier chapters. There's a far greater chance of learning to mastery if you have to deal with it over and over again for the rest of the year.

-- SusanS - 10 Jan 2006

It is largely coherent. It teaches the right things at the right time. Not always in my opinion, but many, many times. Singapore is similar that way, also.

Right.

Exactly.

AND provides the practice anyone learning anything so desperately needs.

-- CatherineJohnson - 10 Jan 2006

I read somewhere where someone said that Saxon reviews too much, but it is reviewing the new skills taught in earlier chapters. There's a far greater chance of learning to mastery if you have to deal with it over and over again for the rest of the year.

I suspect people are startled by the amount of review at the beginning of each year.

However, according to Carolyn Morgan Saxon just does the standard amount, which I think is 3 weeks' worth of review. (IIRC)

-- CatherineJohnson - 10 Jan 2006

I was definitely startled by the amount of review at the beginning of each year.

At curriculum night for parents of third graders, the teachers explained that parents could expect their kids' math grades to drop in January. That's when they would begin seeing new material, as opposed to review. That was Saxon. I like Saxon's spiraling that continues to include problems for topics that were recently covered. I don't like the amount of review at the beginning of the year. It's too much.

When we got to Saxon 54, there was a textbook. So, we could look ahead to see when the review would run out. It was way deep into the school year.

Incidentally, those third grade teachers also warned us parents that the kids would begin the year with lousy scores in grammar. Apparently that's a real step up from second grade. They said that those grades would improve as the kids got the hang of it. Of course that would be about that time that math would get hard.

-- DanK - 11 Jan 2006

According to Engelmann, when teaching to mastery you only need five lessons worth of review at the beginning of a new school year. When taught to mastery kids will retain what they've learned between school years.

-- KDeRosa - 11 Jan 2006

I'm definitely coming at Saxon from the point of view of teaching to a severe LD kid, for sure. We're working about 2-3 years behind his grade, so my take is probably not typical.

Also, since this is the first text (6/5) that I've adhered to religiously, much of the review in the beginning was actually new material for him. He'd either never seen it or didn't really understand it.

I have skipped a few chapters because he was solid on the skills presented. (we can skip anything that involves rounding or place value or telling time. Forever.) Saxon seems an easy curriculum to skip certain review sections. It's easy to check back if there's a problem (The problems in the back of each chapter have the chapter that it came from if you need to look at it again, at least in the homschooling editions)

-- SusanS - 11 Jan 2006

Dan K

the teachers explained that parents could expect their kids' math grades to drop in January. That's when they would begin seeing new material, as opposed to review

Oh my goodness!

I had no idea it was that much.

Wow.

-- CatherineJohnson - 11 Jan 2006

Ken

According to Engelmann, when teaching to mastery you only need five lessons worth of review at the beginning of a new school year. When taught to mastery kids will retain what they've learned between school years.

wow

thanks for that

I believe it

I have managed to teach at least a couple of things to mastery (and so has the school) — and the 5-lesson review sounds right.

But wow.

Putting those two figures side-by-side.....yikes.

3 months of review versus 1 week of review

yet another cost of not teaching to mastery

-- CatherineJohnson - 11 Jan 2006

I managed to completely screw up my Saxon program last summer.

I started Christopher with Saxon 7/6, moving through each and every lesson, no exceptions made. It seemed way too easy, but it was the next book in the series.

Then Carolyn sent me the link for the placement test, and Christopher placed into 8/7.

Since he was going to be in Phase 4 in the fall, I figured I better get him into 8/7 immediately.

So then, like an idiot, I marched him lesson-by-lesson through 8/7 — and we ended up doing maybe the first 15 to 20 lessons in both books.

WE DID ONLY THE REVIEW SECTION OF 7/6 AND 8/7, AND BOTH BOOKS WERE REVIEWING THE SAME SKILLS.

He didn't learn one new thing the whole summer, practically, and what he did learn that was new he didn't remotely practice to mastery.

-- CatherineJohnson - 11 Jan 2006

The ENTIRE Phase 4 class will have to be reviewed next year.

That's what the 7th graders are doing this year, reviewing last year's class.

Only with the 7th graders they call it 'algebra' starting in January.

-- CatherineJohnson - 11 Jan 2006

from Student-Program Alignment and Teaching to Mastery pp. 16-17:

Rule 2: At the beginning of the school year, place continuing students who have been taught to mastery no more than 5 lessons from their last lesson of the preceding year.

If something is thoroughly learned and applied, it will be retained by lower performers as well as by higher performers.

The conventional wisdom, in contrast, holds that lower performers “have it one day and forget it the next.” And whatever they have, “they completely lose over the summer.” Again, this expectation results largely from the kind of instruction students have received. Even after teachers have learned to teach students to mastery, however, they often retain their expectations about how much lower performers will retain.

In the first ASAP schools we worked with in Utah, teachers routinely placed continuing students at the beginning of the school year 80 to 100 lessons behind the last lesson they had completed the preceding spring. [ED: That's about half a year of lessons] Teachers had been told the ASAP policy for placing students at the beginning of the school year: Go back no more than five lessons in the program sequence and bring students to a high level of mastery on the material. This firming is to take no more than five school days. After the review, students should be well prepared to pick up in the program where they had finished in the spring.

The teachers were openly skeptical about this procedure, and they ignored it. They argued that, over the summer, students forget much of what they had learned. We told them that learning didn’t work that way. We pointed out that there is a lot of literature on learning and retention that shows that even if something that had been thoroughly learned and had not been practiced for years, there would be great “savings” in the amount of time needed to reteach this material to mastery.

Therefore, if appropriate placement for students in the fall (based on error performance) is 80 lessons behind where they finished in the spring, the only possible conclusion is that they had never learned the material in the spring.

For several years, the teachers resisted following the fall-placement rules and continued to use their traditional practices. To correct this situation, we documented the mastery of all students several weeks before the end of the school year. We staged “show off ” lessons that were observed. The observations confirmed what students did know, and in some cases, identified some things they had not adequately mastered. Before the end of the school year, students were placed according to the rules about first-time-correct percentages so they were firm in everything that had been presented in the program sequence.

At the beginning of the next school year, we controlled the placement of students to make sure that teachers were placing students no more than 5 lessons behind where they had left off in the spring. Students performed as predicted. After possibly one or two lessons, they clearly performed as well as they had in the spring. The response of the teachers was overwhelmingly one of disbelief and revelation. Most of them said something like, “I’m amazed. They actually retained what they had learned.” The magnitude of their surprise suggests how strong the belief was that students could not possibly retain the information over the summer.

This strong belief had been supported by what they had observed in the past, which was based on spring placements that were far beyond what students had actually mastered.

-- KDeRosa - 11 Jan 2006

The conventional wisdom, in contrast, holds that lower performers “have it one day and forget it the next.” And whatever they have, “they completely lose over the summer.”

That is absolutely fascinating.

I've heard that ALL MY ADULT LIFE: kids with l.d. have it one day and lose it the next. (Though I think that could be true with autism....you can see regression in Direct Instruction ABA programs...unless we 'normals' don't know what mastery is for an autistic child, which is possible.)

-- CatherineJohnson - 11 Jan 2006

We pointed out that there is a lot of literature on learning and retention that shows that even if something that had been thoroughly learned and had not been practiced for years, there would be great “savings” in the amount of time needed to reteach this material to mastery.

Absolutely true.

-- CatherineJohnson - 11 Jan 2006

"We pointed out that there is a lot of literature on learning and retention that shows that even if something that had been thoroughly learned and had not been practiced for years, there would be great “savings” in the amount of time needed to reteach this material to mastery."

BINGO!! Needless to say (and yet I say it anyway)teaching to mastery is the key. Perhaps teachers need to be "taught to mastery" that teaching to mastery is essential with the fundamentals.

-- KtmGuest - 11 Jan 2006

BINGO!! Needless to say (and yet I say it anyway)teaching to mastery is the key. Perhaps teachers need to be "taught to mastery" that teaching to mastery is essential with the fundamentals.

Teach to mastery & formative assessment (along with 'world-class curriculum,' which is code for NO CONSTRUCTIVISM) are my themes now.

But this stuff gives me WAY BETTER ammo.

-- CatherineJohnson - 11 Jan 2006

I've heard that ALL MY ADULT LIFE: kids with l.d. have it one day and lose it the next

I do this with physical movements - I can have learnt something physical one moment and lose it when I immediately try to repeat. Mastery to the point where I don't forget can take a very long time.

And then it can all collapse anyway when I am trying to do too much at once.

Plus sometimes when I was younger I chose not to do something for purposes of my own.

-- TracyW - 11 Jan 2006

And then it can all collapse anyway when I am trying to do too much at once.

wow!

I've been hesitating to recommend this book to you (UNLESS I ALREADY DID RECOMMEND IT AND JUST FORGOT, WHICH IS ALWAYS A POSSIBILITY).....but you might want to take a look at Where Is the Mango Princess?

It's a wonderful book about, and I know this will sound weird, traumatic brain injury.

I've been hesitating about recommending it because of your brother; I don't know if it would be a good thing for you to read or not.

The book is written by a woman whose husband suffered a TBI.

He's in pretty good shape apparently — considered 'high functioning.'

To me, however, he sounded like he was pretty roughed up.

That's why this may be the LAST book on earth you should take a look at. (Years ago I went into a major depression after a developmental pediatrician told me that the 'Rain Man' was an example of a high-functioning autistic person. Jimmy was 4 at the time, and the message was: Rain Man is the best it gets. I was devastated.)

Anyway....I thought of the book just now, because she describes exactly this phenomenon in people with TBI.

Any time they're stressed or tired, their symptoms just skyrocket.

I remember that this woman's husband, who was initially paralyzed on one side of his body, would suddenly be more or less re-paralyzed again any time he was overwhelmed.

He could be walking down the street and his left arm would curl up the way it had been curled up just after the injury.

-- CatherineJohnson - 11 Jan 2006

This makes me think that what's happened for all of us is that we've jumbled up 'brain injury' & things of that nature with 'bad teaching.'

It does seem to be the case that under certain circumstances some (or all) people can 'lose' material anyone would swear they had learned to mastery.

Probably most of us have a sense of this, or have experienced it ourselves. (Like not being able to remember names after age 40, for instance.)

So we have a kind of 'folk psychology' telling us it's possible, common, and normal to lose material we know well.

But in fact that's not the case.

Under normal circumstances, material you've learned to mastery stays put.

-- CatherineJohnson - 11 Jan 2006

I've been hesitating about recommending it because of your brother; I don't know if it would be a good thing for you to read or not. The book is written by a woman whose husband suffered a TBI. He's in pretty good shape apparently — considered 'high functioning.' To me, however, he sounded like he was pretty roughed up.

From your description of the book and the comments on it at Amazon, my brother is already functioning much better than this woman's husband. As I said, we are ridiculously lucky. His worst problems have been physical and the fatigue. Even before he could talk, he was showing intelligence and concern for other people. And he's responded amazingly calmly to all his life plans being turned upside down.

Furthermore, when my brother was in a coma we read a lot of books on TBI, and the worst ones were not the case histories but the ones that listed ALL of the bad symptoms that could happen.

And I think we had a far easier situation in coping with a TBI. I think there's a big difference between a spouse and a blood family member - a spouse has fallen in love with someone because of their personality, while my brother held a place in my parents', our other brother's and my affections just because he is our brother/son. It's just so much rougher on a spouse. And, my parents are still healthy and working and doing yoga and my brothers and I had all left home and were working, so we have lots more spare resources than a mum with young kids who just lost the major wage-earner.

Anyway, will order the book next time I do an order from Amazon. (One of the major disadvantages of being in NZ).

-- TracyW - 11 Jan 2006