Comments

I love it that you brought this up, because I had already been cruising the web for old slide rules.

I'm planning to:


a) learn how to use one myself (I think I was taught once, years ago)

b) teach Christopher


There are some amazing collections of slide rules out there.

-- CatherineJohnson - 28 Jun 2005

I haven't read this, but I love the title:

Modern Students Devour Old Math

I love it that everyone's supposed to learn Egyptian math, but no one's supposed to learn anything, ever, from modern history.

-- CatherineJohnson - 28 Jun 2005

Trailblazers has a whole long unit in which the students make their own abacus.

-- CatherineJohnson - 28 Jun 2005

Abacus, sí!

Slide rule, no!

-- CatherineJohnson - 28 Jun 2005

I love the photo!

-- CatherineJohnson - 28 Jun 2005

I remember being utterly fascinated by my dad's slide rule.

http://www.sphere.bc.ca/test/sliderule.html

-- CatherineJohnson - 28 Jun 2005

My Dad looked just like those two guys in the picture...

and the slide rule he had was like the yellow 'cheap (low cost) slide rule' on that website you posted!

-- CarolynJohnston - 28 Jun 2005

I found a quote I love in Sunday's New York Times. It isn't about math curriculum, but I think it fits perfectly. The quote is: "Technology is our servant, it's our valet,....but it's still about the content." Isn't that absolutely true about teaching math? We should be treating calculators as an aid, but not as the actual content.

BTW, I think we should start referring to these math programs as Math Appreciation. People might not understand the reference to fuzzy math, but most people have taken some type of appreciation course as an elective in college. It's a course that does a brief overview of everything in the subject for people who will not work in the field. Maybe then people would understand that we are not really teaching our children math, we are giving them a very brief overview.

About slide rules: we had to use them in Chemistry when I was in high school. Calculators had just come out, but they were too expensive for everyone to have. The teacher's position was that if everyone doesn't have one, no one can use them on the test. Not only does it help you understand things like logarithims better, but it gives you a real feel for significant figures and how accurate an answer really is. Now, a calculator and computer spits out numbers with many digits after the decimal point. And students think all the digits are significant.

-- AnneDwyer - 28 Jun 2005

Math Appreciation

I'll start phasing that one in ...

The only thing I don't like about it is that, for me, at least, constructivist math is almost the opposite of math 'appreciation.'

Often it seems like math denigration.

-- CatherineJohnson - 28 Jun 2005

Not only does it help you understand things like logarithims better, but it gives you a real feel for significant figures and how accurate an answer really is. Now, a calculator and computer spits out numbers with many digits after the decimal point. And students think all the digits are significant.

That's interesting.

I almost wish I could erase my brain for a couple of weeks, and start 'fresh'--and see what it would be like to have 'calculator knowledge' as my first & main knowledge of the operations.

I think this is another case of adults projecting their own needs and minds onto the minds of young children.

The adults advocating for calculator use are adults who learned the operations using paper and pencil first, then acquired calculators.

One of my friends has a very smart 12 year old son who cannot tell time from an analog clock.

Period.

He can't do it.

(I think this is related ... but I'm tired!)

The other analogy that occurs to me is that I often wish I could 'hear' American accents the way non-native speakers do.

I know what a French accent sounds like to Americans, but I don't know what I sound like to the French.

That reminds me: I haven't posted my French embassy story.

That will have to wait!

-- CatherineJohnson - 28 Jun 2005

OK, what I'm trying to say is that I don't know what it would be 'like' to KNOW multiplication and division as calculator operations first and foremost, and paper and pencil operations only secondarily.

-- CatherineJohnson - 28 Jun 2005

I was in college when we switched over from slide rules to calculators. I started out using slide rules but I recall learning about logarithms before using slide rules. I remember tables in the back of books and having races to see who could be the first to interpolate (in one's head) an answer from the table. How many kids do interpolation anymore? It's a great mental estimating process. So, for me, the slide rule didn't add anything. It might be fun to introduce one in a modern math class, but I wouldn't count on it for adding any understanding.

If you are looking for old slide rules, look for one of the circular ones. I remember when a professor pulled one out and all of us engineering students (it figures) turned green with envy. I still have my K+E slide rule somewhere and the book that came with it.

"...is Leinwand proposing that we make the same mistake with all the other types of computations?"

I understood logarithms before ever using a slide rule. However, those who promote calculator use in K-8 see it as a means of avoidance. They think that calculators will make math easier. They think that if you understand the concepts, then there is nothing to gain from doing the basic calculations by hand. They don't see any linkage between practice and understanding.

When I was in college, calculators allowed the professors to increase the difficulty of the problems we had to do. Five page hand calculated assignments turned into 40 page calculator assignments using much more sophisticated theories. Then again, we already knew how to add, subtract, multiply and divide.

Calculator use in grades K-8 is all about avoidance. On top of it all, they don't replace the time used for mastering the paper and pencil basics with any more complex or sophisticated problems.

-- SteveH - 28 Jun 2005

Slide rules forced you to do mental math and know about place value. If you multiplied 52.5 x 602, you pretty much had to figure out how many places the product would have; the slide rule wouldn't tell you that. But I never learned the slide rule. I do remember the log tables in the back of the book and interpolating.

I remember learning logarithms in Miss Beck's algebra class. It was the same time I was reading The Grapes of Wrath. My father was telling me that what we take for granted in terms of human rights was fought for and to keep that in mind as I read the book. I did, but also as I did logarithms I had the same notion that this idea of logarithms that we take for granted in an algebra class was revolutionary at the time, and people fought and died for it sort of like the Oklahomans on their way to California. I told my brother this theory and he laughed. He said the revolutions in math were very quiet.

-- BarryGarelick - 28 Jun 2005

MoreOrLessPenAndPaper

Question: Is 'less pen and paper' the real goal? Witness the work to be assigned in Passport to Mathematics, Book 1, by McDougal Littell, 1999, page 13.

(This is going to be long because I have a point to make.)

Example 2

The list shows the whole numbers from 1 through 72.

1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72

a. Copy the list and circle all the multiples of 4. b. Color all the multiples of 6 blue. c. Describe the numbers that are circled and blue.

Solution
a. The multiples of 4 consist of the 4th, 8th, and 12th columns.
b. The multiples of 6 consist of the 6th and 12th columns.
c. The numbers that are circled and blue are in the 12th column. They are all multiples of 12. So, in general you can csay that if a nuymber is both a multiple of 4 and 6, then it is a multiple of 12.

Study Tip
The pattern shown by a list can depend on how the list is written. In Example 2, the pattern would not be as clear if you used 10 columns instead of 12.

My observation: This example is obviously a study or review of multiples and finding common multiples. It is Lesson 1.3 entitled "Making a list". (Lesson 1.1 was entitled "Looking for a Pattern") Here are the goals for this Lesson 1.3:

Goal 1: How to solve problems by making a list. Goal 2: How to use lists to help you solve problems.

(No mention of multiples in the goals. No mention of looking for patterns.)

Now my question: How much pen and paper is used in this assignment? I am aware that there is something 'fun' about seeing patterns in math. It can also be enlightening to see these patterns. But to have students copy the entire list before finding the pattern -- ?? What a laborious, time consuming task!

How much more time could students spend on finding common multiples for other pairs of numbers in the time wasted just making this list? Lots of time here that could have been used on speed drills, mental math reviews, other computations, etc.

So to answer my question: No, apparently pen and paper work isn't bad afterall. It's only bad if I use pen and paper for traditional math algorithms and drills. A "new math" discovery-type lesson can use all the pen and paper that it wants.

Also, note that Study Tip. Keep that in mind as I give the Extra Example suggest in the T.E.

Extra Example 2

Copy the list of 72 numbers in Example 2
a. Draw a square around all the multiples of 5.
c. Describe the numbers that are in squares and colored yellow.

In light of the Study Tip, does anyone find it odd that this extra example uses the same list structure?

(This list is but one type of list that students are making in this lesson.)

What would you rather your students spend their math time on making lists or actually finding common multiples and doing computations?

One more point: Did any one read the two goals? Any thoughts?

-- InterestedTeacher

Hi there and welcome back!

This makes me think of the chapter Ben and I encountered last night in the Prentice Hall pre-algebra book. It was on 'using 'problem-solving strategies' to solve word problems. The problems had directions like 'use the problem solving strategy in example 1 to solve these problems'. The problem solving strategies went like this:

1. read the problem carefully.
2. decide how to set up the problem.
3. solve the problem.

Don't get me wrong -- in general I like Prentice Hall. But this sort of thing is not useful at all: it's just hot air. And the stated goals of the 'lists' problem are hot air, too.

That 'circling multiples' sort of exercise has a place, I believe. And the place is on the day that the notion of least common multiples (LCMs) is introduced. Do it once or twice, with a couple of different examples, and you've got the idea of an LCM. You would never want to use that strategy to actually compute an LCM.

Plus, I would give them a worksheet with the list of numbers already on it! Making them copy it, what a waste of time.

-- CarolynJohnston - 28 Jun 2005

So to answer my question: No, apparently pen and paper work isn't bad afterall. It's only bad if I use pen and paper for traditional math algorithms and drills. A "new math" discovery-type lesson can use all the pen and paper that it wants.

Good point.

Thank you.

-- CatherineJohnson - 28 Jun 2005

"But to have students copy the entire list before finding the pattern -- ?? What a laborious, time consuming task! "

Some people take this pen/paper idea too literally. It depends on whether it is wasting time or not. Pen/paper versus calculators is not the problem; avoidance and wasted time are.

"It's only bad if I use pen and paper for traditional math algorithms and drills. A "new math" discovery-type lesson can use all the pen and paper that it wants."

Exactly!

Using calculators avoids having students practice and master the basic arithmetic skills because they think it isn't important. If you had a computer screen which acted like flash cards, that might work just fine. However, I like workbooks and the sites that print out any type of worksheet you want. It is quick and easy to hand out timed worksheets to the kids each day. No need for computers. Ten minutes a day can make an enormous difference.

"Goal 1: How to solve problems by making a list."

Lists! A list of what? If the kids were required to learn their times tables backwards and forwards, then the idea of multiples and factors would be trivial.

"Goal 2: How to use lists to help you solve problems."

Rather than learning from the teacher what they need to know and why (and then practicing), they have students use some vague list methodology to construct an answer.

-- SteveH - 28 Jun 2005

"New Math" is big on patterns.

I agree with Catherine. If you want students to recognize patterns and multiples, print out copies (worksheets)of these lists of numbers, in 12 columns, in 10 columns, etc. Laminate them and have students do this assignment of finding the multiples of 4 and of 6, or of 2 and of 5. Then clean them off and use them over. Students don't need to make the lists to learn from them.

However, when your goal is to NOT use the time for teacher-directed instruction, you scramble for all sorts of ways to fill that time with 'meaningful' (??) things which the students can construct in order to find the patterns, and making a list is 'construction'.

-- CarolynMorgan - 28 Jun 2005

However, when your goal is to NOT use the time for teacher-directed instruction, you scramble for all sorts of ways to fill that time with 'meaningful' (??) things which the students can construct in order to find the patterns, and making a list is 'construction'.

That's interesting.

You're saying that when a teacher is prohibited from doing much if any direct instruction, a 'time hole' opens up that has to be filled somehow?

Can you tell us more about this, about how it plays out in the classroom?

-- CatherineJohnson - 29 Jun 2005

I mentioned in an email to JoAnne Cobasko of SOCMM today that last year I was not only writing out all of Christopher's Saxon problems myself (Saxon doesn't publish workbooks), I was actually writing down each digit of his answers as he arrived at them.

I was his 'transcriber' or 'handwriter.'

I did this because we were trying to get through huge quantities of material, and because he was folding under the strain of trying to write & think & learn new things every single day all at the same time.

I was quite worried that I was bypassing a critical 'in the hand' aspect of learning, which Carolyn talks about.

But it worked out OK, as far as I can tell. (His teacher explained why this was an OK strategy shortly before the end of the year, but I didn't follow the explanation.)

One thing: I knew he'd be writing his own problems at school, so it wasn't as if I had completely severed 'hand' & 'motor skills' from 'math.'

Also, I figured he would start grabbing the pencil out of my hand and doing it himself once it got slower for me to do it than for him to do it.

That's exactly what happened.

Last but not least, I made him do all his many Saxon tests himself, to see if he could. If he'd been bombing the tests, I would have stopped doing his writing for him.

So...once again, I'm feeling my way.

I think Carolyn is absolutely right that we have some kind of 'motor knowledge' of math, or 'hand knowledge'....Temple says the exact same thing about her students who never learned to draw by hand, and have only used CAD.

I don't know why it was 'OK' to do things this way, but it was.

-- CatherineJohnson - 29 Jun 2005

"Motor knowledge, Hand knowledge' -- I like that. I think you're on to something. It's absolutely true.

I've tutored students after school off and on. I send them to the board to work -- not pen and paper, but the same idea and for some things I think it's even better. Students make great strides right away doing practice (drill!!) in this manner. There is indeed something that comes from the mind directing the 'hand' whether it's on paper of a board.

I've had primary students use a box of sand and write their problems with their finger in the sand. This too is effective, but not always convenient.

-- CarolynMorgan - 29 Jun 2005

Embarassing fact: I learned the math that later became my thesis topic in a very short period of time, basically by copying the book verbatim.

That was what I had to do to get it into my brain. There weren't any problems to do -- it wasn't the kind of math book that gives you problems -- so I couldn't learn the math in the usual way, but I couldn't just read it either. I had to do something about writing the symbols out, in order to get them into my brain, so I copied them. My doctoral advisor was surprised at how fast I learned the stuff.

I have a sort of kinesthetic brain. I can't visualize things -- like a cup of coffee -- without imagining that I'm touching them. The imagined touch makes my visual imagination more vivid.

-- CarolynJohnston - 29 Jun 2005

What Carolyn just said is important. We all have to get it into our brains somehow. I look for an example and try to replicate it in the problem sets. It's sort of like someone moving your hands to help you draw, or when your parents sat you on their lab and "let you drive the car" with their hands on yours the whole time. I've talked to mathematicians about this, and they say just about everyone, no matter how advanced, has to go through a kind of "rote learning" phase in order to get the gist of it. Once they have it, then they can think with it, but not until then. The constructivists want you to discover your way to this point. Like Carolyn, in order to understand epsilon-delta proofs, I copied the various proofs down and then replicated them by changing the variables slightly until I got the hang of it.

So, this leads me to the point that text books that have worked examples are very good for students. When students are working the problem sets they can go back to the examples and "get the feel" for how the problem is worked. In some books I've seen the problems even have references to the example number in the book so students know where to look. When tutoring students I point out to them how this works and encourage them to read and work through each example. Reading a math book is a skill to be learned. As Carolyn pointed out you don't just read it and absorb it. In her case she copied things down. In others it may be a combination of that and something else, but the pencil and paper aspect is part of reading a math book. You don't read a math book without pencil and paper.

-- BarryGarelick - 29 Jun 2005

I do that with technical papers. It's easy to read the paper and get the gist of it. But, if I have to implement the theory in an exact computer program, then I get out my paper and pencil and copy the equations over, line by line. It forces me to slow down and really understand what I am doing. I have created boxes and boxes of these scratchings over the years. Some of these are useful years later because they fill in the gaps in the technical paper.

-- SteveH - 29 Jun 2005

I've tutored students after school off and on. I send them to the board to work -- not pen and paper, but the same idea and for some things I think it's even better. Students make great strides right away doing practice (drill!!) in this manner. There is indeed something that comes from the mind directing the 'hand' whether it's on paper of a board.

Are you saying that you prefer the blackboard to pencil paper?

I have to say that, intuitively, that KIND of makes sense...because the blackboard has lots of 'drag' on the chalk...(I'm finding that my Pilot Point pens practically slip off the page--they move too fast, because the ball & ink are slippery. Pencil is much better, because it grabs the page. I'm going back to the old Flair felt-tips for that reason.)

Chalk has a lot of drag, and it's huge, which I think is a big advantage.

I know you ALWAYS start out writing very big letters when teaching kids to read (there's real research on font size & early reading, I believe); the same must hold true of math!

-- CatherineJohnson - 29 Jun 2005

btw, there are special needs companies that sell table-top blackboards. They're like a blackboard-tabletop easel.

I don't think they were hideously expensive, but on the other hand everything marketed to special ed parents IS hideously expensive, so I don't know.

-- CatherineJohnson - 29 Jun 2005

We've got to corral some LD experts to come write for us or at least comment, because I'm SURE we're reinventing the wheel--!

-- CatherineJohnson - 29 Jun 2005

I learned the math that later became my thesis topic in a very short period of time, basically by copying the book verbatim.

I absolutely believe that.

It reminds me of classical training in painting, where students spent years precisely copying the Masters.

Golly, where was it I read recently about a program where kids copy out sentences from really good children's literature?

It might have been on KTM. (My source memory is shot.)

-- CatherineJohnson - 29 Jun 2005

I remember one professor who would never answer my questions when I went to see him during office hours. He would just say, "Go the the board". "OK, write this down." We students would have to derive or answer our own questions under his guidance. He could have had us do this on paper at his desk, but being up at the board is quite different.

-- SteveH - 29 Jun 2005

I have a sort of kinesthetic brain. I can't visualize things -- like a cup of coffee -- without imagining that I'm touching them. The imagined touch makes my visual imagination more vivid.

That's fascinating!

My friend Debbie was asking me one time what you meant by kinesthetic brain--I'll let her know!

-- CatherineJohnson - 29 Jun 2005

Goal 1: How to solve problems by making a list. Goal 2: How to use lists to help you solve problems.

(No mention of multiples in the goals. No mention of looking for patterns.)

That drives me nuts.

-- CatherineJohnson - 29 Jun 2005

Plus, I would give them a worksheet with the list of numbers already on it! Making them copy it, what a waste of time.

This is the kind of homework assignment I ruthlessly deleted from Phase 3 last year, not that we had that many of them.

But we'd have 'fun' assignments where the kid had to solve some kind of problem, then check a chart on another part of the page to find out which letter the numerical answer subscribed to, then find that letter on a zoo of animals, each represented by a letter, then write the letter somewhere else....

Forget it.

This could take hours, and it wasn't math.

It was apparently supposed to be fun.

That was my favorite Christopher saying:

They don't get it! When you make math fun, it's even more boring!

-- CatherineJohnson - 29 Jun 2005

Like Carolyn, in order to understand epsilon-delta proofs, I copied the various proofs down and then replicated them by changing the variables slightly until I got the hang of it.

That's a great tip.

And yes, worked examples are critically important.

As are answers in the back of the book.

-- CatherineJohnson - 29 Jun 2005

Reading a math book is a skill to be learned.

Excellent point!

I am a very fast reader at this point, and it was a big deal for me to learn to slow down as much as you have to when 'reading' math.

I don't really read math now, either.

I work the problems and examples.

The RUSSIAN MATH book is unbelievably pleasurable, btw, and is a brilliant, brilliant example of what (I think) an elementary mathematics textbook should be.

It has about 2 paragraphs of prose for every 100 problems to be worked.

(OK, that's a joke, but not by much.)

There are 1118 problems in the book.

I'm on 229.

As Carolyn pointed out you don't just read it and absorb it. In her case she copied things down. In others it may be a combination of that and something else, but the pencil and paper aspect is part of reading a math book. You don't read a math book without pencil and paper.

I don't read any book I'm serious about without pencil and paper.

I highlight things, write in margins, and write inside the covers and on frontpages.

That's the only way to read when you're serious.

-- CatherineJohnson - 29 Jun 2005

I don't really read math now, either.

I work the problems and examples.

This is the only way to learn to do math! You go back and get what you need to get out of the text in order to do the problems.

This is why I think the most critically important aspect of a math text is that the problem sets be extensive and well designed,

Catherine, a blog post that's all about how you READ would be incredibly valuable for the rest of us!

I'm glad to hear you mark up your books -- I was taught never to do that, and it's a big restriction. I just started doing it a little -- in pencil (I hate writing with a pencil).

-- CarolynJohnston - 29 Jun 2005

This is why I think the most critically important aspect of a math text is that the problem sets be extensive and well designed

boy, I think it's worth it spending the thirty bucks for the Russian Math text just to see the way they put problem sets together.

UNBELIEVABLE.

I can feel my brain expanding practically from one question to the next.

It is just extraordinary.

At the moment, I'm pretty sure I'm going to have Christopher work through Russian Math next summer, between 6th and 7th grades.

-- CatherineJohnson - 29 Jun 2005

'how I read'--

It's funny; that's one of the things I probably have no idea how I do.

Reading really is my Big Fat Talent in life.

I never quite realized that until just recently, working on the bliki with you, looking at recentered SAT scores, etc.

This is bad behavior (at least I'm not posting this on the front page) but .... my 'recentered' SAT verbal score is 790. (Original was 720.)

Of course math was only 620--and is STILL 620, after recentering.

I was so disappointed when I found that out.

I now have a secret goal of re-taking the SAT's!

-- CatherineJohnson - 29 Jun 2005

I don't read any book I'm serious about without pencil and paper.

oops, I misspoke.

I don't use 'paper'; I write in the book.

I go to town with my books; they're written all over the place.

-- CatherineJohnson - 29 Jun 2005

I used to use zillions of different colors & stuff, too.

Then I got sick of that for some reason (probably declining front lobe filtering ability) so now I'm using pencil.

-- CatherineJohnson - 29 Jun 2005

What do you write? Do you just write whatever comes to mind? Examples, please.

-- CarolynJohnston - 30 Jun 2005

I very frequently simply re-write what the author has written, in a shorter phrase, maybe.

Basically I'm making a kind of skeleton outline of the points I want to remember

But I also write anything else that I want to remember, like associations to what the author has said.

If I think something is wrong, I'll write 'No!'

If I think something is super-right I'll write 'Yes!'

Let's see...

• I 'copy,' in probably the exact same sense you copied reading your math text
• I argue & evaluate
• I MAKE CONNECTIONS (YAY! as Christopher would say)
• I create my own PERSONAL index. All the white pages at the beginning and end of the book (if I need them) are covered in the page numbers that are important to me, along with a summary of what's there -- some of the pages have asterisks by them
• I also highlight the bibliography & footnotes

-- CatherineJohnson - 30 Jun 2005

Inside the front cover of RUSSIAN MATH, here are some typical things:

• page 18 'nice way of explaining prime factorization (this is noted because I want to remember the Russian way of doing prime factorization, and will want to review this page, but also because I'll probably want to write about why their way is good for teaching

• page 6 "teaching factors & multiples together" I may have blogged about this already. I constantly confuse factors and multiples, which I realize must sound simply bizarre to math people. The Russian way of teaching them reminds me of de Saussure's dictum that all meaning comes from difference, because the Russians teach factors & multiples together, as OPPOSITES....giving me a kind of...I visualize a strong board or sturdy stick holding these two collapsing definitions apart inside my brain

• page 37 "easily transition from factor sequence (skip-counting) to equivalent fractions -- I'd have to look that up to see the details, but there, again, I'm making note of what was, for me, a fantastically good way of organizing content so that I SAW it, understood it, could do it on the problem set, and so on

-- CatherineJohnson - 30 Jun 2005

I write in my math books, also. Particularly when I figure out what it is the author's saying and why I wasn't getting it. I write down the clarification so when I come back to it later (years later sometimes) I don't have to go through the whole thing all over again.

Like Carolyn, the problems are very important, and they cause you to read the text as it should be read. Constructivists will read this thread and say we are learning in a constructivist style. No. Sorry.

I discovered this technique when I decided to go through my elementary analysis text again two years ago. (Sets, Sequences and Mappings by Anderson and Hall). The sections were long, and I was reading the whole thing and then doing the problems until I realized I should be doing the problems along with the reading of the text. Then it all made sense.

-- BarryGarelick - 30 Jun 2005

The sections were long, and I was reading the whole thing and then doing the problems until I realized I should be doing the problems along with the reading of the text. Then it all made sense.

OH, that is a fantastically good tip.

You know--I have intuitively done that with Saxon, and so has my neighbor.

As we go through a Saxon lesson, all of the problems are fully worked out.

But I make Christopher work them out if he possibly can--and Saxon teaching is so precise that very, very often he can.

So I turn it into direct instruction in the sense of 'scripted questioning.'

But it's the same thing you're talking about: he does problems as he reads about them.

-- CatherineJohnson - 30 Jun 2005

Catherine asks, "Are you saying that you prefer the blackboard to pencil and paper?"

For one-on-one tutoring, "Yes, yes, yes!!"

For group review and drill, "Yes!"

First, the tutoring: I've noticed that students do much better, learn much faster, seem to gain understanding much quicker. I never really understood why -- it worked and I kept doing it. Then, our Learning Center Director tells us that new studies and new research show that every time the hand (or foot for that matter) crosses the midline of the body, something important happens in the brain. I need to run up to the school or talk to her to get information to explain this properly. But I'm going to take a stab at it and ask that you give me a chance to reference it and get back to you.

Apparently, the brain cells really fire away and brain activity picks up every time the hand crosses the midline. There are special drills that our L.C.teachers have their students do just to be sure that the hand crosses that imaginary center line of the body. The brain becomes actively involved as the student is working.

Perhaps this is why board work helps my students so much with concepts they've covered before but have never grasped or been able to reason through.

Now, for the group review: I've seen "working at the board" just do wonders to help students nail down procedures or recall. (Those not at the board are working in a spiral notebook at their desks on the same problems.) And sometimes students who have lost their way through a multistep problem can see the missing steps in someone's board work, and it helps them recall.

Group board work also helps me "see" 3-4 students at a time and I can then zero in on areas where students are still struggling.

It helps me "assess" students' needs (assess is a big word right now in education) both as a class and as individuals.

So, "yes" there are times that I definitely prefer board work to paper and pencil. Students love it and beg to get to do it. If they're excited about doing it, all the better.

-- CarolynMorgan - 30 Jun 2005