hi Lynn -

Saxon integrates geometry into the study of algebra and trigonometry.

Apparently people felt that he slighted geometry in the earlier editions, but that he now gives sufficient attention to the subject.

Throughout Algebra 1 I was feeling that there wasn't sufficient attention to geometry, but I'm now beginning to think that's not so.

You really have to consider his high school books as a trilogy:

Algebra 1
Algebra 2
Advanced Mathematics

I'm almost positive that the 3 books together teach a full geometry course as Barry Garelick would define a full geometry course. (I mention Barry because geometry is his "beat" to some extent; he's very familiar with the SMSG books and he's tutored geometry.)

I've told Barry before that I thought Saxon slighted proofs, but I'm pretty sure I'm wrong about that, too. I'm 15 lessons into Algebra 2, and Saxon has begun to talk about proofs. I think he may have offered one informal proof of the Pythagorean theorem — and in fact he offers a proof of the Pythagorean theorem in the final "Investigation" of Saxon 8/7 iirc.

I'm fairly sure proofs are going to be taught in the 3rd book, ADVANCED MATHEMATICS.

-- CatherineJohnson - 22 Sep 2006

I'm very keen on the "integrated" approach because it gives you 3 years of distributed practice in the subejct of geometry.

Saxon explicitly says, eary on in Algebra 2, that you can use geometry & geometrical concepts to teach algebra.

For example, you can set up problems in which one angle of a triangle equals 2X + 30, another angle equals x + 10, and so on.

-- CatherineJohnson - 22 Sep 2006

My experience with afterschooling leads me to believe strongly in its value. And a perfectly good way to do this is to “teach the procedural knowledge first & attend to the conceptual knowledge as the procedural gels”.

(I've been lurking here, actually have become obsessed with this site & math education. This is my first post.)

Quick background of our situation: Last year in third grade my daughter struggled in math; homework was especially painful. The school threw a lot of “academic intervention” resources in her direction, we had a tutor and I spent a lot of time drilling and trying various worksheets. There was very little progress and we were floundering. Her last report card stated she was “meeting expectations” in math. Meanwhile, she was still using her fingers to perform single digit addition, she couldn’t tell you what the even numbers were, and word problems required extreme handholding. I also realized I had never really received any quantitative reports (also known as test scores) on her progress. (Did I miss something? Maybe.) And yet the school told me she was “meeting expectations’. Inspired by Catherine and other KTM info, I summerschooled her using Saxon third grade. Loved the curriculum. We have completed 130 out of 140 lessons. Both of us worked many hours. Now she is breezing through the first few weeks of math and I know she has mastered the key skills to handle fourth grade. (Thank you God!!) So, in our case “School then becomes practice for what you've taught”.

Now here’s a gem. During a meeting this week with her teacher, I questioned how my daughter could have been “meeting expectations” given how she couldn’t even add numbers with fluency. She explained how understanding the concept of addition was so important to doing well in math. Okayyyyyy, whatever. Still, my daughter couldn’t add!!! Also, when I asked about assessment, I was told that they did not want to “over-test” the kids.

BTW Catherine, I’m in lower Westchester County in one of the “good” school districts.

Tex

-- TexasDesert - 22 Sep 2006

I think most people do some afterschooling, conscious or not. My neighbor recently purchased an entire set of reference books because she was worried that she couldn't understand EM.

good point

That was one of my earliest purchases, the Math To Go series.

I remember strongly recommending them to Anne Dwyer, not realizing she has a degree in engineering!

She bought a couple of them (they're quite expensive) and was disappointed!

-- CatherineJohnson - 22 Sep 2006

Hi Tex!

Glad to hear from you!

(Will go read your comment now!)

-- CatherineJohnson - 22 Sep 2006

I also purchased the books (can never have enough books, right?) and found out that the salesperson had sold over 200 sets in our (very) small town and one other small town. There is a huge undercurrent of unease around here.

Interesting.

What are the books everyone is buying?

-- CatherineJohnson - 22 Sep 2006

I know I never would have jumped wholesale into Singapore Math if I hadn't started slow -- trying to reactively teach in response to homework issues, leading to buying traditional workbooks at local bookstores, and then realizing that for the amount of time I was already committing to this, I should just use a systematic, sequential curriculum.

Now that's interesting, too.

You found that just going with traditional workbooks - which I've found to be terrific - wasn't enough.

I'm assuming that was because you were still having to do an awful lot of reactive teaching/reteaching.

Yes?

-- CatherineJohnson - 22 Sep 2006

The piecemeal approach becomes far more burdensome than just diving into the parallel universe. And the parallel universe has turned out to be fun!

Again, very interesting.

I don't have your experience with "piecemeal," because Christopher was so drastically far behind when I started - and because I "started at the top" with fractions - AND because I discovered Wayne Wickelgren's MATH COACH (with info on U.S. math performance) two weeks into the whole business.

Basically, I had two weeks of piecemeal teaching before I realized I was in serious trouble & would have to find a curriculum that would teach Christopher and support me in teaching.

Then not too long into using Saxon 6/5 I decided I needed to do all the lessons & problems along with Christopher.

I don't think I had to do that.

But I wanted to, and there's no question it helped.

-- CatherineJohnson - 22 Sep 2006

Tex!

I just read!

You're in Westchester!

Lower districts!

So you're a neighbor.

Boy, you're singing my song. They don't want to overdo the tests and understanding addition is what matters.

yup

That was me back at the very beginning of 4th grade. I was completely overrun, because I was running way late on my deadline for ANIMALS IN TRANSLATION and my entire professional world was as crazed & stress-filled as it's ever been, and I didn't know a thing about math ed.

Christopher had always done well in everything at that point.

It was the classic advantaged-child-with-strong-vocabulary-breezes-through-the-early-grades situation. (We also have, IMO, very strong teachers & teaching in those early grades.)

He was whizzing through school.

All of a sudden he starts coming home with scores of 76 on math tests.

I flagged down the math teacher in the hall and asked about the 76 and she said, with a big smile on her face, "Don't worry, it doesn't matter. He understands the concepts."

And I bought it!

She was a very nice person, I want to add, and she was dedicated to teaching well. She wasn't spinning me; she believed that he understood the concepts and that understanding was what mattered.

But as it turned out, neither was true.

-- CatherineJohnson - 22 Sep 2006

What are the books everyone is buying?

Volume Library from Southwestern Company.

They are the modern equivalent of Encyclopedias. But are more of a compendium of everything you might ever want to teach your kid from Kindergarten through High School. They are a terrific resource for parents, but a little dense for kids to use directly, especially in the early grades. It'd be hard to teach yourself an entirely new subject (like trigonometry) with them, but they are an excellent refresher if you've forgotten something your kid is now working on.

Plus, excellent traceable maps of everything you might ever do a school report on.

-- LynnGuelzow - 22 Sep 2006

Hi, Tex!

We've all lurked at one time.

What was the name of the program that your school is using?

-- LynnGuelzow - 22 Sep 2006

Here in Westchester we're getting killed by anti-testing sentiment.

It's absolutely true that a the kind of all-summative-assessment-all-the-time approach we've experienced in the middle school is terrible teaching.

But people - people meaning parents as well as school personnel - aren't drawing the right conclusion from this truism.

What we need is integrated formative assessment - assessment for learning not for stamping a letter grade on the permanent record.

-- CatherineJohnson - 22 Sep 2006

You found that just going with traditional workbooks - which I've found to be terrific - wasn't enough. I'm assuming that was because you were still having to do an awful lot of reactive teaching/reteaching.

I was doing a lot of reactive teaching and felt we were jumping around the workbooks trying to put out fires as they cropped up. We couldn't do anything in a coherent order because we were always spiraling along with EM.

What finally pushed me over the edge was the fear of gaps. I didn't know what wasn't being covered and I didn't know enough math ed to know what should be covered. I figured the only way to resolve this was to jump into an actual curriculum rather than reinvent the wheel.

-- LynnGuelzow - 22 Sep 2006

I don't have your experience with "piecemeal," because Christopher was so drastically far behind when I started - and because I "started at the top" with fractions - AND because I discovered Wayne Wickelgren's MATH COACH (with info on U.S. math performance) two weeks into the whole business.

I guess this is where I just plain got lucky. My son was in the 4th grade when they piloted EM and the 5th when they fully implemented it. So he had 4 years of traditional math. When we got to fractions in 5th grade, I could keep up because he already had a strong foundation. Piecemeal was working. After 5th grade, he moved to a magnet school that used traditional curriculum in combination with constructivist and so he did okay (with piecemeal help along the way).

My daughter was a different story. She is 5 years younger and got EM right from the start in K. So I knew what was coming and thought I could compensate on my own. We did okay until 3rd grade when we really hit the spiral wall. We struggled through 3rd grade and finally in 4th grade we dove into Singapore.

-- LynnGuelzow - 22 Sep 2006

I was doing a lot of reactive teaching and felt we were jumping around the workbooks trying to put out fires as they cropped up. We couldn't do anything in a coherent order because we were always spiraling along with EM.

That was my life last year. It was a nightmare. Putting out fires - except they never got put out. They just kept GROWING.

-- CatherineJohnson - 22 Sep 2006

What finally pushed me over the edge was the fear of gaps. I didn't know what wasn't being covered and I didn't know enough math ed to know what should be covered. I figured the only way to resolve this was to jump into an actual curriculum rather than reinvent the wheel.

Very interesting.

I've got to get all these comments compiled into a front page post.

I agree completely, though as I say I didn't have a long enough experience with reactive teaching early on to come to this conclusion "the hard way."

I must say, though, that after 6th grade in Ms. K's Phase 4, I relearned the lesson of teaching your own coherent curriculum at home. Christopher at this point has got to have so many gaps there's no way I'm going to figure them all out.

So forget it.

He's doing every word & every problem of Saxon Algebra 1/2. End of story.

I don't remotely have the expertise to identify all the weak spots & gaps in his math knowledge. They're going to be everywhere.

-- CatherineJohnson - 22 Sep 2006

When we got to fractions in 5th grade, I could keep up because he already had a strong foundation. Piecemeal was working.

Well I think that must have been the traditional concept of "helping with homework." Parents who were pretty good at math did a certain amount of filling in - explaining things, connecting lessons with other lessons, etc.

The school curriculum was OK; there were opportunities to practice to something near mastery; the procedures & "math facts" were taught.....the school handled the fundamentals.

Becky C always says that - and now Ed is saying it, too.

There's been a role reversal.

We're handling the fundamentals while the schools are getting fancy.

-- CatherineJohnson - 22 Sep 2006

With Christopher, now, I shudder to think of "gaps."

I have virtually no idea what he knows & doesn't know, or of how well he knows the material he does know.

I'm stunned every time he seems to know a concept or procedure well.

-- CatherineJohnson - 22 Sep 2006

Of course, that's because Christopher

-- CatherineJohnson - 22 Sep 2006

My daughter was a different story. She is 5 years younger and got EM right from the start in K. So I knew what was coming and thought I could compensate on my own. We did okay until 3rd grade when we really hit the spiral wall. We struggled through 3rd grade and finally in 4th grade we dove into Singapore.

whoa

And not a moment too soon.

-- CatherineJohnson - 22 Sep 2006

Lynn - How well versed in math were you when you started with Singapore Math?

-- CatherineJohnson - 22 Sep 2006

That's another thing (someone else may have said this already) - I think Linda Moran is certainly correct when she says that teaching math isn't easy.

But otoh how much math do K-6 teachers know?

although....as I think about it.....most K-6 teachers have it all over parents when it comes to experience with kids learning math

Any decent K-6 teacher who's been on the job for awhile knows two things:

• the "totality" of K-6 math as a whole
• how little kids learn, where they get derailed, etc.

Most K-6 teachers, no matter how much they do or do not know about math per se, have a level of pedagogical content knowledge I would have killed for going in.

-- CatherineJohnson - 22 Sep 2006

Ken

Brilliant.

That's exactly right.

That's a HUGE concept in special ed, where it's normally called "priming."

The one stroke of luck we had this year is that Christopher has math as his final class of the day, immediately after study hall.

Normally, having math at the end of the day would be a disaster - that was part of the problem in 4th grade, when he fell off the math cliff.

But he's using his study hall to preview the lesson Ms. K is (presumably) going to touch.

It's a HUGE help, and he's mature enough to understand that this year.

-- CatherineJohnson - 22 Sep 2006

"What was the name of the program that your school is using?"

SRA Math Explorations & Applications

From what I’ve read it’s not the worst program.

But last year there seemed to be a lot of supplementing with different types of worksheets. From my perspective there was a lot of skipping around, covering a lot of topics, “mile wide & inch deep”. I felt my daughter was exposed to many concepts, but did not master many.

Interesting note: This year our school is pilot testing two “promising, research based” math programs. I have gotta find out more.

Tex

-- TexasDesert - 22 Sep 2006

SRA Math was the curriculum our district was using before the switch to Math Trailblazers.

It's definitely a mile wide & an inch deep, and it seems to have enough constructivist elements embedded in the text to make it very, very tough.

I couldn't even begin to teach fractions using the TRAILBLAZERS book; it was very, very tough.

In the 6th grade book they actually ask you to figure out the equation for a line without having taught you anything at all about slope, linear equations, etc., etc.

SRA seems to be one of the main books districts turn to when they're trying to give kids a more traditional math text.

But it is he** to teach from.

Mrs. Panitz, who is a terrific teacher, just rolled her eyes when I mentioned trying to teach from SRA. Apparently the teachers pretty universally felt that way about SRA Math; certainly every teacher I ever talked to thought the book was a nightmare.

That's one reason I credit our K-5 teachers with superb math teaching skills.

Christopher's knowledge up 'til Grade 4 is rock solid, and his teachers all managed to do that using a textbook they saw as impossible to teach from.

(His knowledge from Grade 4 might have been rock solid, too, if he'd had a different teacher. I've mentioned before that he had a new inexperienced teacher who wasn't given tenure.)

-- CatherineJohnson - 22 Sep 2006

I admit that my math skills were rusty when I started this project. But having learned quite a bit of math (before abandoning it in favor of the humanities midway through college), it comes back quickly. That is probably the best thing about mastering something, you can dredge in back up to passable levels if you really want to.

I agree on the pedagogical stuff. Although I'm not sure how much it would really have helped me. Knowing my daughter and knowing how to read the signs of the train about to leave the tracks -- eyes begin to blink and wander, etc. -- is probably more important than anything else. When I see a probably surface, we just slow down, back up, try a different way of explaining. MUCH easier to do when you only have one kid to worry about.

For what its worth, my husband has an engineering degree and then moved into medicine. He is very helpful if I get myself into a knot that I can't untie with my own math skills. But he tends to give the kids far more information than they can handle.

I became very interested in game theory and economics a few years ago and will play around with that with the kids -- but none of them are ready for the heavy lifting of the math involved in that.

-- LynnGuelzow - 22 Sep 2006

But having learned quite a bit of math (before abandoning it in favor of the humanities midway through college), it comes back quickly.

yeah, you had a much better base to start from

that's good


That is probably the best thing about mastering something, you can dredge in back up to passable levels if you really want to.

Absolutely.

And you can do so quickly.

-- CatherineJohnson - 22 Sep 2006

Although I'm not sure how much it would really have helped me.

It would have been a big help to me.

That was my revelation about Christopher hosing the measurement section of the TONYSS in 4th grade. I had no idea measurement was so hard for kids to learn until classroom teachers left comments filling me in.

If I'd known that measurement is a challenging subject that takes a long time for kids to master, I would have been paying attention!

-- CatherineJohnson - 22 Sep 2006

btw, the teachers who filled me in on this left comments here

one was Carolyn Morgan & there was a fantastic Comment left by a high school teacher whose name I've forgotten about how his h.s. kids couldn't measure anything

That was an eye-opener & incredibly helpful.

-- CatherineJohnson - 22 Sep 2006

game theory!

cool!

I can't wait to start learning all that stuff myself

it's going to be awhile

-- CatherineJohnson - 22 Sep 2006

The special ed classes in Megan's school are piloting SRA Real Math this year. It's too early to tell, but her math homework looks more like math than EM did, and there seems to be more opportunities for practice. So far, the topics in Ch. 1 seem pretty coherent and thus I have an easier time supplementing them the materials I have at home. I think I'll ALWAYS need to supplement and afterschool some topics, though I know KUMON will help with overlearning of some topics.

-- KathyIggy - 22 Sep 2006

My problem with SRA was that I did notice a lot of bizarre jumping around and problems with skills that had never been taught. I was never sure, though, if the teachers just hadn't taught the concept or the curriculum was off.

Several times my LD son would come home with a sheet that had skills that he could basically do, and then one or two that were way beyond him. I looked through his workbooks and often found the same thing.

For example, one time he came home with a sheet and the last problem was asking for a specific circumference of a circle. I had never seen pi or anything on any of his earlier sheets. Nothing in his workbook. His teachers still hadn't gotten him off telling time and single-digit addition. Luckily, I had taught him pi and he knew what to do.

I'll be curious to see if any of you notice that.

-- SusanS - 22 Sep 2006

For me, and for most if not all of our teachrs here, SRA was very, very tough.

I don't mean to be discouraging — it's not precisely a constructivist text, and the topics are serious.

But it is a VERY difficult text to teach (and learn) from.

iirc Ms. Duque, Christopher's brilliant 5th grade teacher, held this view, too.

You're just going to need to monitor what's going on....

-- CatherineJohnson - 23 Sep 2006

I still remember the day I opened the book to a problem asking the student to find the equation of a line.

NOTHING had been taught about finding the equations of a line - including what a linear equation in two variables was in the first place, iirc

This was in the 6th grade book

-- CatherineJohnson - 23 Sep 2006

Mondo page splatter, too

-- CatherineJohnson - 23 Sep 2006

Hi! Linda Moran here. I was not clear, and I'll clarify here. I think parents should definitely help if they can, but they need to be careful not to correct what the school is doing. That only becomes frustrating for the student. Also, I'm trying to say, and obviously badly, if you have no love for math yourself, you could make matters worse for your kid. I'm just saying be careful and have a positive attitude. Parents definitely do not have to have a math degree to instruct their kids. Just a little enthusiasm, willingness to learn, and a respect for understanding-based mathematics, which is what their schools are trying to do, albeit badly. Also, if parents want to stick with teaching algorithms, and to steer clear of word problems, that's okay, but they need to understand there are several right ways of solving the problem. Rote memorized math procedures is a bad idea. They need to convey the beauty and logic of math, where there is little need for memorization. Does this clear it up, or am I still being clear as mud? More power to all of you who teach your kids! Linda

-- KtmGuest - 28 Nov 2006

I think parents should definitely help if they can, but they need to be careful not to correct what the school is doing.

Linda:

Sorry. Totally disagree. I can't wait for the schools to get it right, and chances are good they won't. Particularly with TERC. What ARE you talking about when you tell us to wait for the schools to stop flailing around with TERC? What's happening to our kids in the meantime when the school is figuring out how to make up for a bad curriculum? And what do you mean "rote memorized math procedures is a bad idea". That phrase pushes a button with me as I'm sure it does a lot of the readership here. Teaching students the standard algorithms, generally comes with some explanation of why they work. Once you've demonstrated how and why a procedure works, it should become algorithmic, hence the name algorithm. Otherwise you ask the kid to go back to first principles every time, and I guarantee you that that will obscure the beauty and logic of math quicker than learning any efficient algorithmic process.

My daughter had Everyday Mathematics in elementary school. In sixth grade I tutored her and a friend using Singapore. My goal was to ensure they learned multiplication and division of fractions, because EM surely wasn't doing them any favors. I don't consider this "correcting" what the school was doing, I consider this teaching them what they were not being taught.

-- BarryGarelick - 28 Nov 2006

Hi, Linda!

-- CatherineJohnson - 28 Nov 2006

Hey - I love your "Beyond TERC" listserv - somehow I didn't even know you had it!

Everyone - here's the address: beyond TERC

So helpful!

-- CatherineJohnson - 28 Nov 2006

Linda - Have you read Willingham??

Check out his Inflexible Knowledge piece. That has influenced all of us - probably no one around here believes that rote learning actually exists ("rote learning" is what a parrot does or, sometimes, autistic children do).

Also, it's probably fair to say that many or most of us have come to the conclusion that procedural knowledge can and in many cases should come first - that conceptual understanding is an "emergent property."

I'll try to pull Schmid's statement on that for you.

I didn't believe this myself for quite awhile, btw.

Carolyn had always made this claim pretty strongly; she always said, "Teach the procedures first; the understanding follows" (paraphrasing - I think I've got it right).

I trusted Carolyn, and had to assume she was correct, and yet that idea didn't jibe with my own experience.

However, as I've gone further in teaching myself math, I've seen what she's talking about.

Often I've developed understanding of a procedure only after using that procedure for a time.

Saxon says the same thing, btw.

It's counterintuitive (at least it was for me) but my own experience, at least, has born it out....

-- CatherineJohnson - 28 Nov 2006

Inflexible Knowledge: The First Step to Expertise

-- CatherineJohnson - 28 Nov 2006

btw, I know we have a lot of readers who don't Comment - so I don't mean to be putting words into people's mouths -

I'll just speak for myself!

-- CatherineJohnson - 28 Nov 2006

Here it is:

Schmid on procedural learning

"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."

-- CatherineJohnson - 28 Nov 2006