05 Jul 2005 - 19:34

Christopher on Singapore Math

Christopher managed to bargain me down today.

Instead of doing:

• Megawords 2, Worksheet 10-J
• Saxon Math 8/7 Lesson 11 Mixed Practice
• Saxon Math 8/7 Lesson 12 Warm Up
• Saxon Math 8/7 Lesson 12 Lesson
• Saxon Math 8/7 Lesson 12 Lesson Practice
• Math Olympiads: 1 problem

he's doing:

• Saxon Math 8/7 Lesson 12 Mental Math
• Primary Mathematics 3A Workbook, problems 8, 9, & 10

So maybe he has a future as an agent.


He just looked up from his bar modeling and said, 'I like the problems in Singapore Math.'

I said, 'You do?'

'Yeah.'

'How come?'

'They're not stupid.'

No idea what that means.

update

Christopher got all 3 of his bar model problems right today. (ummm....no, he didn't. He flubbed the arithmetic on the first one, but he got the bar model almost exactly right.)

I checked his answers & models, and when we got to the 3rd problem, he said confidently, 'This one's a two-parter.'

I was happy to hear that.

I think this signals a new category inside his mind.

• one-part problems
• two-part problems

He can tell the difference!

what bar models do for your brain

I'm trying to figure out how to write about bar models and what I think they do for my 'math brain.'

It's incredibly difficult to articulate, and will involve printing out sample bar models, scanning them back into iPhoto, and reducing the image size...so it will be awhile.

But I'll get there.

For the time being, I'll say that I could do the 3-variable problem from Primary 6 that Carolyn posted using algebra.

But I couldn't do it using a bar model.

There's a reason for that, but I'm going to need visuals to express it.

OTOH, once I'd done the problem algebraically, I realized how to interpret the (correct) bar model I'd drawn--thanks to the Math Olympiads problems I did this weekend.

So today's hypothesis is that the perfect 'problem-solving' curriculum for me would be an amalgam of PRIMARY MATHEMATICS & MATH OLYMPIADS.

math-heads & word-heads

Carolyn has mentioned that mathematicians think facility with geometry may be a good indicator of mathematical talent.

I wouldn't be remotely surprised to find out that's true, if only because of the connection between spatial-visual ability & maths. (I've decided I like 'maths' better than 'math.' fyi)

I don't remember having trouble with any of the high school math I took. (Maths!) It may have been an easy curriculum, I don't know.

But I do remember having lots of fun with algebra. The X's and the Y's and all the neatly stacked-up linear equations....it all just felt right.

I could still solve a two-variable equation 30 years later, without even having to think about it.

This has made me wonder if there is something 'word-like' about standard algebra.

Temple, btw, absolutely could not learn algebra.

She's a brilliant person, but algebra was out.

'I couldn't make a mental picture of it,' she told me. 'It was too abstract.'

I have to remember to ask how she did with geometry the next time we talk.

Back to main page.

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Algebra was a nightmare for me in high school. I was all set in college until I found out that I couldn't get my undergrad diploma without it. (One should never switch from a BFA to a BA unless they're clear about such things.) My girlfriend (who was planning on being a math teacher) tutored me every day. I was highly motivated, obviously. She was very kind and just said something about me not having my high school algebra skills down and that we'd just go back and get them. Looking back, I realize that she probably had to go back and re-teach me fractions as well. (I pretty much stared out windows during those years, and then there was that "new math." That was the final nail in the coffin.) Anyway, due to my extremely intensive attention to detail, I actually managed to make A's by the middle of the course (my first in math since grade school), and for a second, I actually saw the beauty of it all. For a second, anyway.

Geometry, totally got that. Loved it. It's not as reliant on previous skills. I had a chance.

Did I mention that I am completely and utterly right-brained. (The other side is dead in the water, including any little math synapses over there.) I'm all visual. I quietly assumed that I had an LD, that is, until I became acquainted with what a real LD looks like.

Thinking "abstractly", or in a linear fashion is like waking up a groggy, old friend in my brain. If I can finally get him up we can actually do it, but I'd much rather just live in my visual brain. Unless you're a genius, I don't think it's always best to do that, at least not from my experience.

It was, however, a very important life lesson for me. The college algebra teacher that I had was what I would call "typical." He wrote furiously on the board and talked mathematically at all times as though everyone was in love with numbers. I remember the familiar chill everytime I walked through the door and sat down. (of course, I was also envisioning not graduating with my peers and having to come back and take summer biology with a lab, no less.)

My friend, on the other hand, had the gift of the teacher, as well as the gift of the mathematician. That made the difference for me.

I cringed when I read "A Difficult Child" by Ralph Raimi. I was that girl and let me tell you, it is hell. You feel you're stupid and you just need to get by, that's all. By 7th grade I didn't know what anyone was talking about, but I was too embarrassed to ask for help (in case I didn't understand the explanations.) Her immaturity and teenage angst are a major part of the derailment of her math education. I dodged, cheated, and lied my way through a lot of high school math until it all caught up with me around junior year. My parents were horrified. They had no idea. I was told I was lazy. I just wanted it to go away, like that poor girl.

I'm just glad I got to find out what the real problem was.

-- SusanS - 05 Jul 2005

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Catherine, I'd be really interested to find out whether Temple could connect with bar models and, through bar models, with algebra.

-- CarolynJohnston - 05 Jul 2005

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Catherine, I'd be really interested to find out whether Temple could connect with bar models and, through bar models, with algebra.

I know; me, too.

I MUST find a time & place to find this out.

Having known her for so long, I have the strongest sense that she MIGHT (might!) just be able to do two and three-variable algebra problems via the bar models.

But I don't trust my judgment.

Have you ever looked at the 'Math-U-See' books?

I'm curious about them, and I think they could be another possibility.

-- CatherineJohnson - 06 Jul 2005

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Geometry, totally got that. Loved it. It's not as reliant on previous skills. I had a chance.

I find this stuff utterly fascinating.

I'm massively a 'word person,' and algebra was fun.

I think that's actually the right word: fun.

When I first started doing bar models, I found them fantastically difficult (not all of them, but whenever I had to do a more advanced problem using a bar model).

I still don't find them easy for more challenging problems.

Bar models now 'pop' into my mind's eye for simpler problems, but not for more challenging problems.

At first I thought this was just because the 'X's' and 'Y's' of algebra were so ingrained; I was having to fight an overlearned impulse to set up equations.

But now I'm not so sure.

Bar models are now pretty ingrained, too; as I say, they spring up unbidden.

And I still find X's and Y's far more 'natural' and 'obvious.'

What's interesting is that I have very little conceptual understanding of what I'm doing when I solve an algebra problem using equations.

I do understand conceptually what I'm doing when I solve a problem using a bar models. (At least, I think I do.)

That doesn't make the bar models any easier.

-- CatherineJohnson - 06 Jul 2005

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Her immaturity and teenage angst are a major part of the derailment of her math education. I dodged, cheated, and lied my way through a lot of high school math until it all caught up with me around junior year. My parents were horrified.

This is amazing.

Can you tell us more?

When you say that logical, linear thinking doesn't work for you...what does that mean?

What is it like?

What's an example of not being able to think abstractly?

I always felt that I had a 'school brain,' and it's obvious, from reading your comment, that I do.

The best thing anyone ever said to me was Carolyn's comment: 'You were the kind of kid who could learn anything they threw at you.'

I had never put it quite that way, but it's perfect.

I was the kind of student who could learn well with a bad curriculum, a bad teacher, a bad head cold--whatever!

I'm sure I had plenty of good teachers, and probably some decent textbooks, but at some basic level I was bad-teacher-proof: they could just throw stuff at me, I'd catch it.

I could learn, because I have....a linear brain? an abstract brain?...I have no idea what it is that gives me an advantage in school settings.

I'm curious to know why I hit the wall with calculus.

When I got to Wellesley I signed up at once for a calculus course for non-majors, flunked the first test, and quit.

One of my goals now is to take & master calculus....but I'd also like to know what happened way back when.

Obviously I was under-prepared, but what about calculus exactly suddenly pulled the rug out from under me?

Or what about maths per se did me in?

-- CatherineJohnson - 06 Jul 2005

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Catherine, you just seem unintimidated by anything. We should all be that way. Very inspiring.

There is a great book (another one!) called Organizing for the Creative Mind which, to me, is just a nice way of saying, Organizing for the Total Slob. Creative sounds much better. Anyway, it goes into right brain/left brain stuff in such a way that makes you realize that you haven't lost your mind and that it really is okay. Those left-brainers aren't as superior as they think they are with their neat closets and drawers. Hooks and the backs of chairs work perfectly fine and I can see everything, too, a critical part of left-brainedness. Like I said, it's quite funny and horrifying at the same time.

The point I was actually getting to, though, was that the book mentions that no matter what brain is dominant you need the skills from the other side for success. The book mentions (and I'm not sure how they arrived at this, but it's interesting) that CEO's are really right-brained people with excellent left-brained skills. You must work well in both brains. I have to kick my left brain and say, "Get up fool, we have children to raise!"

Here's an intersting part--The left-brain thought sequence (according to the book): analysis->action->feeling.

The right-brain thought sequence: feeling->action->analysis

Or you can look at the toothpaste tube test. If you grab the tube (I guess the old fashioned ones) and squeeze from the middle or anywhere you want because it doesn't matter after all, you are probably right-brained. If you meticulously squeezed from the bottom (rolling it up in the old days) and replace the cap every time, you are probably a left-brained person.

Packing for a trip is another way to determine what you are. Lefties carefully pack in as few things as possible. Righies have a million little bags designated for various things.

-- SusanS - 06 Jul 2005

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I'm also pretty sure that I'm the ADHD parent. A lot of things became clear after my child's diagnosis.

-- SusanS - 06 Jul 2005

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Susan, I have a little trouble with the left-brain right-brain differentiation. It's useful to a degree, but as a determiner of what you are, I don't think it's either psychologically or physically accurate.

I've often been told I'm a left-brainer. It may look that way, but the truth is that I paint and do math and do programming and do music and write and draw, and I'm a completely unorganized person by nature who has to overcompensate in order to be functional.

I'll whine to anyone who will listen about how it stinks to have such trouble organizing myself, and how much I dislike not knowing left from right and having to use a map to find my way around my own block. I wish I were smarter, and not "geometrically challenged", and not ADD (inattentive type). But it seems the right/left-brain dichotomy isn't quite the right description, and it doesn't really offer much help when it's time to get something going that is difficult for you. (Also it makes it sound as though "left-brained types" are uncreative -- is that really true?)

The only book I've found that ever really helped me get my life under control was "Getting Things Done." I'll check into "organizing for the creative mind", too.

-- CarolynJohnston - 06 Jul 2005

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Carolyn,

I understand what you're saying about the simplicity of it. I have an engineer brother who thinks very much like that left-brained sequence I wrote above. But he is incredibly creative and an inventor. However, when we talk to each other we need a translator.

I know plenty of people who don't fit neatly into either slot, as well. It really is just a light humorous look at the subject with some helpful hints. It helped me with some of the internal dialogue that goes on with the chronically disorganized (which I'm a member of, also) particularly about how "time" plays into it. Of course I lost the book for a while so I couldn't remember some of the better tips.

I'll definitely check into Getting Things Done. Is it pointed towards the unorganized type person?

-- SusanS - 06 Jul 2005

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Here's an intersting part--The left-brain thought sequence (according to the book): analysis->action->feeling.

The right-brain thought sequence: feeling->action->analysis

Oh that's fun!

A fantastic--and quick--book about left brain-right brain (John recommended this) is: The Right Mind: Making Sense of the Hemispheres by Robert Ornstein.

Wonderful!

I'll have to get out my copy and post stuff.

Elkhonon Goldberg has a different idea that may make sense of some of the objections Carolyn has raised.

He thinks that the left brain handles expertise.

When we become fluent in anything, it jumps from right to left.

When you're still struggling to learn, you're using your right brain.

This makes sense of why we'd always see language in the left brain: everyone's good at language (just about).

I don't know how well that hypothesis is holding up, but he made a good argument for it (and he had some brain scan data, as I recall, showing this jump in learner's brains).

I remember thinking at the time that this made sense of the kinds of conundrums Carolyn mentioned.

-- CatherineJohnson - 06 Jul 2005

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Oh, this is a facinating thread. Some of you are writing about me -- and you don't know me.! Right brained, trouble keeping most things straight and neat (Why file it away some place where I will later not be able to find it? I can see it right out here on the countertop!), no clue how to organize. And I'm married to Mr. Organization himself.

Organizing doesn't come natural to me -- I have to really work hard at it. Really hard.

For example, I'm embarrassed to say how long it took me to figure out how to organize my files for my school subjects. You mean you actually separate things when you file them away? Into different chapters? Oh,OK. I got that. But years later, after not being able to find things easily, it finally occurred to me that I could organize things from Chapter 1 into sub-files: Tests; Test Reviews; Quizzes; Quiz Reviews; Worksheets; Experiment workshsets; Group Activities; Homework assignments, etc. You laugh because it just seems natural that a teacher would know how to do that.

You know, a person who is disorganized would do it a better way if she knew how. I just didn't know how. And still don't. It's "unnatural" to me.

My Husband is so organized that if I ask him to cut a piece of wood for me to use for something, he has to spend 2 hours getting things set up properly and his space organized properly. It will be measured perfectly, squared up at the corners, sanded, and will be a fine masterpiece. All I wanted was a piece of board!

I know we are wired differently. I'm the talker, he's a man of few (or fewer) words.

He reads the instruction, every word of them, and follows them to the max.

I hate to read instructions, and will do so only if forced to (Wake up, left brain!). I'd much rather just look at the thing and figure it out by myself.

When it comes to math, I must have had excellent early teachers in basic math, because I understood fractions and never forgot them. Algebra was not at all difficult for me, and geometry was easier still. There were friends who I had to tutor in geometry because they couldn't understand it. I could just see it.

Then I hit a wall after that in Trig. I never did understand it. I've often wondered why. I felt so dumb and was embarrassed to be in the class with the 'smart' kids and not to be able to get it. It scared me away from trying any more math ever again.

I'm like Catherine. I never understood how I would ever use algebra -- it was just like solving a puzzle. I loved manipulating the letters and numbers around. It made sense without understanding how I would use it. I enjoyed it.

Then when our son took algebra, I decided to review it. There were word problems and I actuallly saw how it was used. Then for the first time I really understood its purpose.

Language arts was my real strength. I loved to read, enjoyed diagraming sentences, flourished in writing -- it all came so easy.

So when I first heard of "left-brain, right-brain" thinking, I just figured I was right-brain all the way. It's interesting to me that I now am a math teacher, and am good at helping children who don't understand, helping them "get it". I remember it feels to be totally lost. And I'm determined that no child will ever feel that way in my class. If he doesn't understand what I've taught, I'll find another way to explain it so that he can understand.

Carolyn, I've told you before how inspiring your "journey through math" story is. Maybe some day I'll try the next higher math again.

-- CarolynMorgan - 07 Jul 2005

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Another thing of interest to me is different learning styles. (No, I'm am NOT a constructivist.) I know that students, people learn differently. I know that there are students who learn better by looking and reading. Others can't grasp what they read, but can remember anything they hear. Others, have to be moving and feeling, toughing. Some need to talk their way through their work to learn it.

I have students every year who do not process well what they hear. My instructions or words go right over their heads. In fact they are often not even aware that I've spoken. However, if I say their name, to get their attention, and repeat it, they "hear me" this time.

My son was that way. Also, he was (is) definitely ADD. I learned however, how to handle his lack of "hearing my words". Here is a typical scenario:

"Son, I'm going to leave. If Mrs. So-and-So calls, tell her that I'll be back in half an hour." (He is working intently at the computer.)

"Okay, Mom, I will."

At first I would just leave, but I learned to do the following:

"Okay, son. Now, what did I just say?"

"I don't know, mom. I'm sorry. Tell me again."

Some of this was (and still is) probably his ADD.

But my husband, oh, my. He really doesn't process what he hears, (but easily processes everything he takes in through the "eye-gate".) Verbal directions are almost useless. Written directions are necessary. He is one of the most pure verbal learners I've ever seen.

Dr. Cynthia Tobias has some very interesting books on "The Way We Learn". She describes four ways people process information: concrete, abstract, random, sequential, and discusses the different ways these four can be combined and how they are manifested in people. Has anyone read any of her books?

She says it's not enough just to identify what type of learner you are and how you best take in and process information. Everyone must work at learning to process information through other "gates", and it is very important for us to train our children to learn to process information through other "gates" (quotes are mine).

--Main.CarolynMorgan - 07 Jul 2005

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I'm like Catherine. I never understood how I would ever use algebra -- it was just like solving a puzzle. I loved manipulating the letters and numbers around. It made sense without understanding how I would use it. I enjoyed it.

I know!

It was a blast!

This is why so much of what people say and believe about math ed just doesn't ADD UP! (I hate puns, but since that one just popped out, I'll leave it there.)

Supposedly if you don't acquire conceptual knowledge of elementary arithmetic, you're doomed for algebra.

But I wasn't doomed at all.

I sailed through my algebra courses, and had fun.

This is why i really wish I could go back in time and figure out what happened with calculus at Wellesley.

-- CatherineJohnson - 07 Jul 2005

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I'll look up the Cynthia Tobias books--

-- CatherineJohnson - 07 Jul 2005

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This is why I think some IQ tests, like the Woodcock-Johnson, can be very enlightening. You can see places where a person can be in the gifted range and then somewhere else where the score is in low-average. The overall score might suggest a good, solid above-average intellect when, in fact, what you may have is a superior intellect with a deficit. I find that an important distinction for teachers to know.

As far as learning paths go, I remember in one of my other lives as an aerobic instructor having students who would not move even with explicit instructions unless they visually saw me do it. Others could not mimic what I was doing until they actually heard me describe it. Only when they heard the words could they "see" me. I often entertained myself watching who could follow best with what technique.

-- SusanS - 07 Jul 2005

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Susan spoke of being gifted at teaching and gifted in mathematics. I love that thought. Indeed, math teachers must be both. I think some of our teachers and professors have the gift of mathematics but not much of a gift of teaching.

My engineer, architect husband is NOT a teacher. He knows all kinds of "stuff" for his field. And he knows math. But he can't explain something so I can understand. He invariably jumps over 3 or 4 steps.

When I don't follow, he repeats the same explanation, skipping the same steps. I'm not talking about math here. I'm talking about trying to tell someone how something works or how to do something.

A good math teacher can immediately determine what a student is NOT getting, figure out what steps to go back over, or realize what to add to get the student from "here to there".

Now, on the other hand immagine having the gift of teaching and insufficient knowledge or understanding of mathematics.

I had one of those twice for middle school math -- an ART TEACHER, who was good at art. My dear daddy helped me every night. Then, I'd go back to school the next morning and teach my friends how to do the problems. My daddy helped all of us get A's.

-- CarolynMorgan - 07 Jul 2005