Comments

I remember beginning algebra containing drills (yes, the dreaded D-word) requiring us to translate English into algebraic expressions; e.g., adding two to a number and doubling it becomes 2(x+2). I don't recall any of the students in the class groaning; we enjoyed it. That type of drill is invaluable in understanding how to set up the story problems into the equations that Ben likes to solve. And nothing wrong with Ben liking to do "the easy stuff". I run into plenty of students who are unable to do the stuff that Ben finds easy.

-- BarryGarelick - 29 Jun 2005

Since Daniel has never been able to draw, we just skipped the bar models. On every word problems that he does, I require him to write the equation first, then solve the problem. Once kids get used to translating easy problems into equations, it is relatively easy to introduce an unknown. Because they understand how to form equations. BTW, Singapore math is a great program for this. All of their word problems lend themselves to this.

-- AnneDwyer - 29 Jun 2005

Carolyn, why do you think so many (maybe it's not 'so many,' I don't know) serious people think 8th grade is too early for algebra?

-- CatherineJohnson - 29 Jun 2005

Assuming I know what algebra is, and I think I do (I took both algebra courses in my own high school)....I don't see the argument against studying algebra in 8th grade.

When I say 'I don't see' I mean literally that: I don't grasp the argument.

It's not that I disagree with it; I don't know what the 'con' argument is.

-- CatherineJohnson - 29 Jun 2005

I remember beginning algebra containing drills (yes, the dreaded D-word) requiring us to translate English into algebraic expressions; e.g., adding two to a number and doubling it becomes 2(x+2).

Oh that's interesting!

You drilled algebra.

Well, Saxon does that, for sure.

Christopher yesterday had a beginner's algebra drill to be done in under 5 minutes.

The questions were all things like:

a + 12 = 20

a = _

I'm not sure if Saxon has the kind of drill you're talking about.

If not, I'll create some of my own.

-- CatherineJohnson - 29 Jun 2005

We're definitely sticking with the bar models, though.

Christopher is now doing 1 a day, starting with the 3A (1st semester 3rd grade) book.

Tomorrow he'll start doing 2 a day.

We're doing them because they made such a huge difference inside my own head. The only words I can use are the standard lame words everyone uses when they try to describe 'math appreciation' or 'math enrichment' or 'discovery': I felt as if my mathematics knowledge became 'deeper.'

This was a feeling, really.

I can't explain it any better than that, though I do have an example I'm going to post whenever I get around to constructing the bar model, printing it out, scanning it into my computer as a gif, reducing it in iPhoto, and attaching it to a KTM page....

The other thing is that I just don't think Christopher's spatial-visual WHATEVER is the greatest.

Nor is mine.

I just discovered: CHRISTOPHER CAN'T MEASURE.

I mentioned that he got a 4 on the TONYSS by the skin of his teeth; he just squeaked by.

Well, that turns out to be because he bombed on the measurement subtest, missing 4 out of 10 questions.

So I started paying attention to measurement, and sure enough: the child cannot measure line segments with a ruler.

Sigh.

I could be completely and totally wrong about this, but the spatial element in bar models feels 'right' to me, or right 'for' me, given how unbelievably verbal & symbolic my mind is.

Christopher seems to have the same kind of brain (and reading & language arts are his absolute strengths).

-- CatherineJohnson - 29 Jun 2005

I forgot to mention--Anne--if you have any interest in re-introducing bar models for your son, just get quadrille paper and have him follow the lines.

I never do a bar model on regular paper, and neither does Christopher.

Now the Singapore kids.....sheesh.

They do bar models from 3rd grade on with no guidelines at all. No quadrille paper (at least not in the workbooks) and no regular lined notebook paper.

-- CatherineJohnson - 29 Jun 2005

"...serious people think 8th grade is too early for algebra?"

I can be quite un-serious, but I see it as a practical issue. I don't think it much matters whether you get to algebra in 8th or 9th grade. HOWEVER, algebra in 8th grade seems to be the key to get into the real math courses in high school. Those who get into algebra in 9th grade probably have had more watered down math and get a watered down algebra course and track in high school. The probability is much lower that they will be prepared for a technical degree in college. It doesn't have to be this way. You really don't need to take Calculus in your senior year, but you do need to be in the more rigorous math phase or track.

-- SteveH - 29 Jun 2005

HOWEVER, algebra in 8th grade seems to be the key to get into the real math courses in high school.

Well, that's my concern to a t.

But Carolyn & Bernie both feel there is absolutely a maturity issue with calculus (I'm sure she'll write about it at some point).

When she & Bernie both started telling me that, I thought, 'So how come I'm breaking my back here to get Christopher on track to take calculus in high school when he won't be ready anywhay?"

I've seen other mathematicians say that high school calculus is worthless.

OTOH, our school has an AP statistics class that I'm positive a high schooler can handle; it also has an AP economics class.

Plus our high school has better offerings period, so if Christopher is done with Algebra 1 at the end of 8th grade that opens up a slot for one more high-quality high school course.

Still, I'm curious about algebra-in-8th & calculus-in-12th.

-- CatherineJohnson - 29 Jun 2005

My other issue is simply wanting Christopher to be on track with his peers in high-achieving countries.

Christopher, at this point, does not seem to be strongly talented in math.

I want to see him doing the same level of work as bright-but-not-mathematically-gifted kids do in Singapore.

-- CatherineJohnson - 29 Jun 2005

I'll bet there's a lot of individual variance with kids and their readiness for algebra in 8th grade (or calculus in 12th grade). Some kids probably need the slower pace -- but many can probably do algebra in 8th grade,

My oldest stepson did calculus in 11th grade. For him, that was definitely too soon; waiting another year would have been best. When I was in high school, AP calculus was new, and only the really advanced seniors took it.

I should be more careful when i say things like "11th grade is too soon for calculus". I don't really know if that's true in general; I only have one data point. But in my stepson's case, the blockage was, I think, really developmental. He just wasn't ready to learn calculus yet.

I imagine these 'experts' have seen similar things happen to other kids sometimes. It's painful to watch kids struggle with a topic when you have reason to suspect that if they'd waited a year, they wouldn't have had to struggle.

Really, we need more varied approaches: the kids who are ready to move faster should move faster. And kids who need a traditional math program (as my son manifestly did) should be allowed to have it.

-- CarolynJohnston - 29 Jun 2005

We had algebra in the last month of 8th grade, but it certainly could have been introduced sooner. If you have a good curriculum (big if in the US), there's no reason why it can't be taught in the 8th grade. But if some kids are slower to grasp the symbol manipulation, taking algebra in the 9th grade is not the end of the world.

-- BarryGarelick - 29 Jun 2005

Barry, do you have thoughts on calculus in high school?

-- CatherineJohnson - 29 Jun 2005

Is the Pope Jewish? Of course I have thoughts on calculus in high school. If a student has the potential, go for it. If a student has been struggling through math the whole time, give the kid a break. I REALLY think there are students who can get into college without having taken calculus in high school. If they want to go into a technical field, but they are struggling a bit in high school, let them take calculus in college. It's taught better (unless it's reform calculus, but that's another topic), and the student has had more time to absorb the basics. There's something called mathematical maturity that has to happen. So while I think it's great that calculus is offered in high school, be realistic about what your child is capable of. And be realistic about how well your high school teaches it. This may be difficult to assess.

On another topic, Catherine evinced surprise that I mentioned "drilling algebra". I'm surprised that she's surprised. Of course you drill basic skills. There are many operations with symbols that have to become automatic. You don't get there by mastering the concept. You get there by mastering the skill. Case in point: Adding algebraic fractions like 3t/(x+2) + (7y)/2m. The explanation that you have to find a common denominator just like you do in arithmetic is fine, but until the student sits down and does it, they will probably be caught off guard. In the above problem they have to multiply 7y by (x+2). This means looking at x+2 as a single entity. Students will go blank and say "What do I do now?" You have to remind them that 7y is multiplied through the terms x and 2. After a few of these problems, suddenly the distributive property that they've been hearing about (I hope they've been hearing about it) actually has a use. So the drill actually helps them apply other concepts learned earlier to more advanced skills, and so on and so forth.

Yes, Saxon drills algebra.

-- BarryGarelick - 29 Jun 2005

I REALLY think there are students who can get into college without having taken calculus in high school. If they want to go into a technical field, but they are struggling a bit in high school, let them take calculus in college.

Getting into college isn't what I'm concerned about...

I don't know what I'm concerned about, frankly.

Oh!

Yes, I do.

(I'm flagging here.)

I'm concerned that high school calculus courses may not be worth the time & effort.....and I'm also concerned whether either Christopher or I (or Christopher's dad) will have a clue about this when the time comes.

As to drill and algebra....I have no memory of doing anything like drills in algebra.

Of course, I don't remember a lot of drills in anything else, either!

This is more of my shallow, inflexible knowledge, I realize.

I have a perception that WORKSHEETS are for MATH FACTS, and MATH FACTS are TIMES TABLES, not ALGEBRA....

It's true!

This is a perfect example of inflexible knowledge.

And there's more where that came from.

-- CatherineJohnson - 29 Jun 2005

I'm not saying that math facts are algebra. I'm saying that there are certain operations and skills in algebra that should be drilled. Adding fractions is one; factoring another. Completing the square, another. Knowing how to complete the square to solve a quadratic leads the students to "discovering" the quadratic formula if done correctly.

Perhaps there's some confusion over my use of the word "drill". I use the word without embarrassment or apology, which will not gain me very many friends at NCTM, I realize. Algebra homework for me when I was in high school consisted sometiems of 20 to 30 problems. Some problem sets might be factoring, others adding fractions, whatever. I don't mean drill in the sense of the teacher drilling kids on math facts.

You said: "I'm concerned that high school calculus courses may not be worth the time & effort.....and I'm also concerned whether either Christopher or I (or Christopher's dad) will have a clue about this when the time comes."

Very important. See how he's doing in math. I think it's important that students have two years in algebra at a minimum. Anything beyond that is great if they can handle it. If they are struggling the whole time, I don't think calculus in high school is worth the time and effort. If they have a good aptitude for math, then it is, but there should be some appraisal of how good the calculus high school class really is. If the teachers can't teach it well, then it would be a grand waste of time. Plus, the text is important. Many high schools use the AP text by Finney, DeMatta^? (Sp?), which I find attrocious. ATTROCIOUS!

-- BarryGarelick - 30 Jun 2005

I'm not saying that math facts are algebra. I'm saying that there are certain operations and skills in algebra that should be drilled.

Right!

What I'm trying to say is that I have an inflexible category called 'drill,' which I don't readily expand to include 'practicing algebra.'

For me, 'drill' means 'math facts.'

I was stunned--not to put too fine a point on it--to discover that Saxon Math has fast fact worksheets on reducing fractions!

-- CatherineJohnson - 30 Jun 2005

If they have a good aptitude for math, then it is, but there should be some appraisal of how good the calculus high school class really is.

Well, I hope we're all still around to give each other advice on this when the time comes.

Also, I'm utterly confused about aptitude for math.

I don't think of Christopher as having aptitude for math, but, on the other hand, .... I didn't think of myself as having particular aptitude for math, either, and it turns out that I love the subject.

I still don't know if I have aptitude, but I kind of hate the fact that I made decisions about coursework way back when based on the idea that I didn't have aptitude.

With math, aptitude seems to be an especially opaque concept.

(Wickelgren has great stuff on this, which I'll get around to posting at some point.)

I'm being long-winded here.

What I'm saying is that, having read Wickelgren, and having collected a few personal narratives of late-blooming math interest, I refuse absolutely to define Christopher's 'math future' now, or maybe even in high school....

-- CatherineJohnson - 30 Jun 2005

Aptitude and interest aren't the same thing.

I have an aptitude for singing, but minimal interest in it. I have an interest in painting, but minimal aptitude for it.

The thing with math is that there are so many other areas that use math at a fairly sophisticated level. For example, most people in my line of work don't have math Ph.D.s; their Ph.D.s are in physics or electrical engineering or photogrammetry. But they need to know the math required to do their jobs, which is pretty sophisticated stuff.

You just don't want your kid shut out of any of those fields because they can't do the math.

-- CarolynJohnston - 30 Jun 2005

Have your child take the courses that are appropriate. If a student is struggling hard in algebra and a parent asks me what math course he needs to take next, I'll tell them he needs more work in algebra or he/she's going to have a tough time in the other courses. In terms of your kids,you're giving them proper structure/foundation upon which to build. But as you know, sometimes that doesn't happen. A friend of mine (who, ironically, is very much involved in the whole math ed issue), found that her son was not doing well at algebra. She tried various avenues, and finally had to face the fact that his preparation in middle school had been poor. So now she's getting him help in those areas. What I'm saying is be realistic about what your child can do. I tend to think there's a very big push for kids to take calculus in high school, and my feeling is if they need more time to mature mathematically, college might be a better place for it. You won't be shutting doors if your child takes calculus in college instead of high school.

-- BarryGarelick - 30 Jun 2005

I tend to think there's a very big push for kids to take calculus in high school, and my feeling is if they need more time to mature mathematically, college might be a better place for it.

Right, exactly.

And then I started hearing from 'real' people, meaning people like you and Carolyn!--that calculus in high school may not be a realistic or even a good goal (I've seen at least one of the math warriors say that hs calculus courses are bad, aside from the issue of mathematical maturity.)

I definitely don't have a fixation on calculus-in-high-school, so that isn't a burning goal for me.

I do have a fixation on not wasting any more of Christopher's young life than we absolutely have to given all the other crud that's included in schools these days. (Proud parent of a DARE student talking here.)

Our high school seems to have some terrific courses, so I don't want him taking algebra in 9th grade if we can possibly avoid it. If he clears up the 9th grade algebra slot in 8th grade, but doesn't take calculus, he'll be able to take something else that's good.

-- CatherineJohnson - 01 Jul 2005

At some point in high school, a second year of algebra might be good, but again, see how he does. FYI, the goal of the California framework for math was to have students take algebra in the 8th grade. That hasn't happened yet. Which gets back to your question; is it better in 8th or 9th? The answer is the classic "It depends". If the student has the right background, they can take it in the 8th grade, no problem. In Russia, they're exposed to algebra as early as 7th. (Maybe earlier; not sure). A good foundation can work what we think are miracles and what in Russia is standard.

-- BarryGarelick - 02 Jul 2005

A good foundation can work what we think are miracles and what in Russia is standard.

You've GOT to get Mathematics 6 by Turk & ... (can't remember the other last name at the moment.)

I'm working through the whole book, and it is pure pleasure.

I'm so glad you said Russia is the standard--that's been exactly my sense since working with the book.

I had pretty much made up my mind to have Christopher do Russian Math next summer.

(btw, how do we know that Russia is the standard?)

-- CatherineJohnson - 02 Jul 2005

Reading & doing Russian Math makes Singapore Math look like a shiny fake (and I don't mean that to sound as harsh as it does).

But working with Russian Math I have, on each and every page, a feeling of, 'OK, this is the real thing.'

Here's a better way of putting my Singapore Math feelings, when working with the Russian text: Singapore Math is a fantastic overachiever.

Russian Math is 'natural born.'

And by the way, I like over-achievers.

-- CatherineJohnson - 02 Jul 2005