Entries from RussianMathematics

I used Singapore math books 2B, 3A, 3B and half of 4A before having my daughter take the ITBS test. She completed the 2B placement exam but took 3 times as much time to complete it as was recommended. I thought better to start her slightly below her level to build confidence, learn the rod diagrams, and build speed and fluency with her facts and basic procedures. We also used Intensive Practice books 2B, 3A, 3B, and part of 4A (not every problem though) I made the decison to use Singapore because through my research 2 titles kept appearing over and over: Saxon and Singapore. Saxon is expensive and did not seem to be a good fit for my youngest daughter. Singapore seemed to be the best one to try first, since I wouldn't be out a lot of money if it flopped! Not very scientific or glamorous but the truth. Once I worked with the program and saw the children's response to it I was sold. I am average in my math ability and studied through Trig in college. I think at first Singapore can be intimidating, but after working with it, it is fairly straightforward. I used the Instructor Guide for 2B and have not really used it since. I try to work out all the rod diagrams, and boy am I getting good at them. Jenny, at the Singapore Forum board, is a great help if I am hopelessly stuck. All problems at this level can be solved without using algebra and Jenny is very helpful for teaching people how to set up the rod diagrams. (singaporemath.com) I also am learning much along with my daughters. I think Saxon is also a great program and a few of my homeschooling friends' kids are doing very well with it. I am going to look into the Russian Math program too.





Rod diagrams are another term for bar models! Honestly, the only thing I did with the Singapore program was to follow it. This is what a day at our kitchen table looked like: First a warm up. At first this consisted of basic facts practice. Usually a worksheet of facts isolated by family (ie: just 9's in multiplication) until enough families were learned to combine them. The text presented them this way as well. Eventually we did our multiplication and division randomly mixed and often multiplication facts presented as missing factors 9 X ___=72. Sometimes the children practiced on a hand held device called "Math Shark" or used flash cards. After the children mastered their multiplication and division facts the warm up was several problems from the series that were difficult for them. These problems came from prior days' instruction and I often changed the story slightly and always changed the numbers. We would repeat "types" of problems each day until these problems became routine and easy to solve. Also, once they learned to compute equivalent fractions and reduce fractions to lowest terms I would have them do a warm up of these types of problems until I saw mastery of the procedure. This part of our lesson took about 5-10 minutes. The second phase of our Kitchen Table Math consisted of 1 or 2 pages of Intensive Practice from a book one level below the text. For example we are working in book 4A but are working in Intensive Practice book 3B. I found this was a great way to provide extra review and also not overdosing on the topic currently being studied in the text. Also parts of IP are quite challenging and having extra skills did not hurt. This part took about 15 minutes. The third part was the actual lesson in the text. The children worked orally and on white boards. They completed most of the practice exercises. Sometimes if I saw they had mastery, they only completed a few. We also completed every word problem using bar modeling if appropriate. This took 10-20 minutes. The final section of our lesson consisted of the children completing the corresponding workbook page(s) independently usually taking 5-20 minutes. I reviewed their work and had the children correct errors immediately. That's it!

Indus Academy is an independent math academy started by math PhD usually trained in another country to teach their children here. I know there is one in Newton, Mass started by Russian mathematicians and they use Russian texts. In Russian.

One of Daniel's friends was going to this school so I asked to sit in on a class. They give a placement test...very challenging and it tests lots of different math concepts.

Just to give you an idea, I sat through one class on a Satuday morning. Every child except one was Asian or Indian. That morning, the teacher introduced the students to geometric and arithmetic series. (These kids are 3-5 grade.)

A geometric series is when you add 1 + r^2 + r^3+....+ r^n. So the teacher started by picking an r and generating the first couple of term. Then, he started asking students, "OK, now how would you generate the next term? How would you generate the 10th term? How would you generate the 19th term?" And then he led them through the process. In this way, the students start to think about advanced concepts in manipulating numbers.

And picture the juxtapostion: these kids are sitting through Everyday Math which is at least a year behind their grade level and learning advanced mathematical concepts on the weekends.

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MathAndLanguagePart2 22 Aug 2005 - 18:26 CatherineJohnson


Carolyn M is back!

So, while I have her attention, I'm posting my new Russian Math question.

This is from a chapter section called:

The Quotient of Two Numbers

1. A quotient greater than 1. ... In this case the quotient shows how many times larger the dividend is than the divisor. The number 8 is 1.6 times bigger than the number 5.

2. A quotient equal to 1. ... In this case, the divisor and the dividend are equal numbers.

3. A quotient less than 1. In this case, the quotient shows what part of the divisor the dividend is. And so, the number 6 is 3/4 of the number 8.

This was, believe it or not, the first time I had ever seen a quotient greater than one defined as how many times larger the dividend is than the divisor. Wonderful!

However, I was then surprised to find that a quotient less than one was not defined as how many times smaller the dividend is than the divisor.

Here's my question.

Is it incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor?

Or is it that the authors view the definition they give (what part of the divisor the dividend is) as more important or more fundamental or more profound?

If it is incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor, why is this incorrect?

Thanks!

Russian Math for everyone

While I'm on the subject, I'll add that the Russian Math book is pure pleasure. It's incredible. I think every 6th grade child in the country should study this book.

And every parent.

update

When I say this is the first time I've ever read that 'the quotient shows how many times larger the dividend is than the divisor,' I don't mean that I've never heard expressions like '6 is 2 times larger than 3.' I have!

What I mean is that I've never seen a direct, explicit, comparative distinction drawn between 'a quotient larger than 1' and 'a quotient smaller than 1.' I've never seen (I don't think) 'a quotient larger than 1' defined as having a separate & distinct meaning.

This is a standard technique throughout RUSSIAN MATH, which I find incredibly powerful, and which I haven't noticed in U.S. textbooks (or, I think, in the Singapore series).

The RUSSIAN MATH book constantly teaches through difference, which in this case means 'disaggregating' a concept that normally stays aggregated in a U.S. textbook.

In a U.S. textbook--at least in the ones I've used--a quotient is a quotient.

In a 6th grade Russian textbook, all of a sudden a quotient is 3 different things. It's a remarkable book. Incredible.


Another thing: I've never seen a 'quotient larger than one' defined as anything other than 'how many times the divisor goes into the dividend' or 'how many of the divisor are in the dividend.'

Last, but not least, I don't think I ever was made conscious of 'decimal' or 'fractional' phrases like '1.6 times larger.' 'Two times larger,' sure. '1 '1/2 times larger,' yes.

But '1.6 times larger'--never.

This is important, because it looks like we're hardwired to understand 'friendly fractions.' Five year olds know what one-half is.^* Five year olds do not know what 6-tenths are, and they certainly do not know what 1 and 6 tenths are!

Morever, our brains do not automatically or easily generalize from 'one half' to '1.6.'

This is another reason to object to the exclusive focus on friendly fractions in constructivist curricula. Friendly fractions are the fractions children know without having to be taught.


^*Hunting, Robert P. (1999). Rational-number learning in the early years: what is possible?. In J. V. Copley. (Ed.), "Mathematics in the early years", (pp 80-87). Reston, VA: NCTM.


What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...)

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
the language of numbers is not language 7-3-05

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WilliamKSmithCalculus 16 Sep 2005 - 12:16 CatherineJohnson


Here's another recommendation from Barry Garelick:

Calculus with Analytic Geometry by William K. Smith (also available at Amazon)

I've already ordered my copy.

Have I mentioned I'm planning to take calculus?

Well, I am. I'm planning to take calculus.

But first I have to 're-take' algebra & geometry. Then trig, which I've never studied.

You folks here at ktm are helping me so much. Even though I'm a writer, I can't locate the words to describe what you've given me. The reason I can't 'locate the words,' of course, is that I don't actually know what I'm learning from ktm. I study & absorb what people say, but then forget the source of my new knowledge once it's been assimilated into my store of old knowledge. I'm left with the hazy feeling that 'I'm learning a huge amount from the Commenters at ktm.'

So I'm going to start taking notes. God is in the details.

thank you!

integers! integers!

So Christopher's math class started integers on Monday—a topic he knows virtually nothing about—and he's having a test tomorrow. He is way not prepared, so I'm busy today writing an Integer Lesson. Probably won't be posting much (though I may have a couple of things from Barry.)

I'm taking a moment to make one more plug for Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, though.

I could probably add & subtract integers in my sleep. (Though I did have to do some review last year when I first re-encountered the topic, which I take as a sign that my knowledge was more procedural than conceptual.)

But last night, after working with Christopher for awhile, who was semi-lost (I don't think he could pass a test at this point) the Math Fog rolled in.

This is the good thing about working with people who know less math than you do. Concepts and procedures you thought you understood turn out to be not quite so clear. I assume that's what Bernie meant when he said the other day that he'd realized there were aspects of reciprocals he hadn't thought about (if I've got that wrong, Bernie, I'll change it!) Carolyn has said something similar at times. I'll be asking her about some elementary concept that, for her, is as simple as breathing in and out, and suddenly she'll see why Ben--or anyone else--might get confused.

lost in translation

This is another one of those constructivist insights that's been lost in translation.

For me, and I think for most teachers & writers, teaching or writing about a subject always forces you to understand it far better than you did.

Radical constructivists conclude from this that children should explain all of their answers in words.

I'm pretty sure that's wrong, because math is not language. Math is math. A child who can explain his answer by showing the mathematical steps he took to find it has produced a proper mathematical explanation as far as I'm concerned. (Russian Math & the Chinese teachers in Liping Ma all offer mathematical explanations & demonstrations.)

But what really bothers me about the 'explain your answer in words' business is that it puts the onus on the child to teach himself. The teacher doesn't have to work and fight and struggle to find the right words; the child does. I know that's wrong.

While I'm on the subject, why don't I just go ahead and take umbrage at the suggestion that a child is capable of explaining math in words?

Writing is hard. Writing well is extremely hard. Finding the words to explain any mathematical concept well is a vast and ambitious undertaking in itself, not a toss-off in the middle of a homework assignment or state assessment. (I'm seriously against the extended response (pdf file)requirement that's taken over IL state rubrics. At least, for the time being I am. [update 5-14-06 sorry, link no longer works])

back to Russian Math

I shouldn't be putting words in people's mouths, so if I've misunderstood Bernie or Carolyn I'll issue a CORRECTION.

In the meantime, why don't I just return to quoting myself.

It's true for me that when I work with a child for awhile, I realize I don't understand things as well as I thought (or hoped).

After Christopher went to bed, I got out Mathematics 6 and turned to the section on adding & subtracting integers.

The first thing that struck me was the fact that this topic appears at the very end of the book. Prentice Hall Pre -Algebra^* opens with integers, and I question that. I question it not based on any profound grasp of pre-algebra as a coherent whole. I question it on grounds that Nurk & Telgmaa are geniuses, and they put adding & subtracting integers last, not first.

I'm sure they have their reasons. (I intend to figure out what their reasons were.)

Reading through Nurk & Telgmaa's discussion, I realized why I was confused. I think I realized why Christopher was confused, too. I hope so.

We were both, I believe, stumbling over this type of problem:

5 - (-7) = ?

Both Saxon Math 8/7 & Russian Math teach addition & subtraction of integers using the number line. Saxon's lessons were particularly strong, I thought.

But when I tried to untangle myself by resorting to the number line, I got stuck.

Start at zero, move five to the right, then.......then what?

What was my next move? My very next move, without renaming or re-expressing - (- 7) as + 7 ?

I was stuck.

Reading through Mathematics 6 I realized that the problem is something Wayne Wickelgren & his daughter Ingrid have raised: the same letter or sign has been made to stand for two different things.

There are two 'minus signs' in 5 - (-7). One means 'opposite,' and the other means 'subtract.'

One means 'perform an operation' and the other doesn't (I don't think. Is 'taking the opposite of a number' considered an operation? I don't know.)

In any case, for both Christopher and me, 'subtract' and 'take the opposite of' are two different things.

Mathematics 6 has a formal demonstration of the fact that:

5 - 7 = 5 + ( -7 )

This is something I think I figured out on my own many, many years ago. I've been using it ever since to de-confuse myself when dealing with long lines of integers to add & subtract. At some point, if I'm getting confused about whether I can or can't use the commutative or associative properties, I just turn the whole thing into addition.

Reading Mathematics 6 I realized that's what needed to happen with 5 - ( -7):

5 - ( - 7) = 5 + [ - ( - 7) ]

Voila!

Christopher and I both understand that 'the opposite of the opposite' is the number you started with originally; the opposite of the opposite of 7 is 7. (This wasn't an especially hard idea for Christopher, but the number line really nails it down.)

Once you convert '5 minus negative 7' to '5 + the opposite of the opposite of 7' it's in a form Christopher understands, and can do.

AND it's in a form you can perform on the number line, if you like or just want to check.

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

Once you've converted a 'double negative' subtraction problem into addition, you no longer have an anomaly, The One Subtraction Problem That Cannot Be Done On A Number Line.


We'll see how it goes. This morning I had Christopher quickly rewrite 12 subtraction problems as addition problems. (I haven't explained to him why a subtraction problem can be rewritten as an addition problem, and I don't know whether I'll get to that today. I haven't closely studied Mathematics 6's presentation to see whether I can introduce it 18 hours before the test.

Fortunately, Ed had already introduced the idea that 'subtraction is addition' last night, when he used the addition-of-debt-to-debt (a concept that is not foreign to our household) to show Christopher that:

- 7 - 7 = - 14

I think he had a lesson in Saxon on subtracting a positive from a negative being the same thing as adding a negative to a negative, so he probably had some knowledge to build on before Ed gave him the add-one-debt-to-another example.

It's the minus-minus issue that's throwing him.

I hope.

one last thing

Looking at this, it strikes me I'm also going to have to create some problems that I ask Christopher to 'simplify'—'simplify' defined broadly as 'write it in the simplest possible correct way that will allow you to recognize what the computation is and do it.'

For instance:

-7 + 5

He probably needs some practice rewriting this as 5 - 7.

I'll see.

I'm also going to try to put together an incredibly simple 2 - 1 type problem that he can always solve quickly when he gets jumbled up. Something like this:

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

He hasn't learned the Polya line about how 'For each complicated problem you can't do, there is a simple problem you also can't do.' I realize it's not clear that you can explicitly teach problem solving, but I'm going to have to try. He's got to learn the strategy of creating a super-simple version of a hard problem in order to see how to deal with the hard problem SOON.






^*new title: Prentice Hall Mathematics: Explorations & Applications

keywords: subtraction negative minus absolute value subtraction is addition integers extended response

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RussianMathQuestion 04 Oct 2005 - 15:54 CatherineJohnson




from:
Mathematics 6
page 240, #905

Can the sum a + b be less than a? (Provide examples.)

The answer is yes, obviously, but I'm wondering whether I've got the rule right.

I come up with yes, when b < 0.

I'm starting to look forward to getting back into proofs.

I think.

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RussianMathProblem961 05 Oct 2005 - 17:19 CatherineJohnson


961.

A point with a coordinate of -3 moves along the number line in the following manner: First, it goes 5 units in the positive direction; Then it goes 7 units in the negative direction followed by 10 units in the positive direction and 8 units in the negative direction; Then it goes 3 more units in the negative direction and, lastly, 13 units in the positive direction.

Question: What is the final location of the point on the number line?

source:
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, page 255

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RussianMathProblem976 09 Oct 2005 - 22:14 CatherineJohnson



976.
A rectangular park is 400 m longer than it is wide. The ratio of the length of the park to its width is 5:3. How long will it take someone walking at a rate of 2 km/h to go around the park?

question

How would a child work this problem without using algebra?

Here's the way it's done using a bar model, but I'm not seeing how you would do this without simple arithmetic.

(I hate the way this looks. I need Quark.)

Anyway, using the bar model you see that you have a bunch of equal units, and that two of these units equal 400.

Therefore, each individual unit equals 200.

So the width has to equal 3 x 200, or 600 and the length has to equal 5 x 200 or 10,000.

ratio problem

Actually, now that I think about it, at this point in RUSSIAN MATH, kids have learned ratios & proportions, so they could just solve it that way.

width = w
length = l = w + 400

5/3 = l/w

5/3 = w + 400/w



What I can't remember is whether the book has taught kids to use two variables...

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FinishedRussianMath 18 Oct 2005 - 18:35 CatherineJohnson



started: June 6, 2005
finished: 10:05 pm, October 14, 2005
number of chapters: 6
number of lessons: 59
number of pages: 300
number of problems: roughly 10,000
number of gratuitous graphics: 0



It just struck me.

This is probably the book that got me hooked on math.

Not Saxon Math 6/5, not Primary Mathematics, not even Challenging Word Problems 3.

Mathematics 6

just one thing left to do

revisit the chainring problem

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RussianMathProblem624 30 Dec 2005 - 15:23 CatherineJohnson



One think I love about Russian Math is that problems are numbered sequentially from beginning to end. The authors don't start over again from 1 with each chapter. This means you've left the realm of the friendly numbers by page 22. (First problem on page 23: #102.)

300 pages
1118 problems

A lot of those 1118 problems include as many as 9 to 12 separate problems. I figure there are at least 10,000 problems in the book, every one of which I've done. (And I still didn't get the answer to Carla and her 1/3 money expenditure at the amusement park. Is that a diagnostic?)

All math books should do this.

The sense of progress you feel working your way through the book is tremendous.

Plus you get a lot of practice doing percent problems, as in: what percent of the book do I have left if I'm up to problem number 507?

Seriously.



what's the answer to this problem?

769. An old brainteaser by Leonty Magnitsky: If a man drinks a barrel of water by himself in 14 days and the same barrel with his wife in 10 days, how many days would it take his wife to drink the barrel by herself?



my next question

How do you check your answer?

I have an answer, but I've 'checked' it in what seems like a circular way.....that is, I've checked it in such a fashion that I've simply confirmed that I did my calculations correctly.



incorporating 'historic' story problems into contemporary texts

Another wonderful aspect of MATHEMATICS 6: scattered throughout the book are 'brainteasers' written long ago by 'Central Asian scholars' and the like. I perked up every time I came across one of these.

Mathematics 6 teaches students something about the history of mathematics—and, crucially, about the fact that mathematics has a history—through the problem sets, not through dorky 4-color sidebars that interrupt the text and bring progress to a screeching halt. (Mathematics 6 has no colors. The text and sparse illustrations are black and white. Good.)

This is living history. Working a problem somebody thought up in 973 that's still hard today, you feel yourself connected to people living nearly a thousand years ago. It's magic.

624. An old brainteaser by the Central Asian scholar Biruni (973—1048):
If 10 dirhams (a unit of currency) earn 5 dirhams of profit in two months, how much profit will 8 dirhams earn in 3 months?

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RussianMathTableOfContents 15 Mar 2006 - 18:56 CatherineJohnson



I've just discovered that the Table of Contents for Mathematics 6 ('Russian Math') is posted online! (pdf file)

You can see how beautiful the book is graphically. It's the exact opposite of page splatter.

The book is so lovely it made working through 12 trillion fraction problems a pleasure.




speaking of 12 trillion fraction problems

So check this out.

Here we have Chapter 3, Fractions, Decimals, and Percents.





Look at Lesson 3.3.

Multiplying Fractions

Lesson 3.3 is 8 pages long.

Of those 8 pages, 2 1/2 are lesson.

5 1/2 pages are wall-to-wall multiplication problems. Hard ones.

Problems like this one:





I was sweating my way through 5 1/2 pages of this stuff when I looked ahead to see what came next and discovered that my reward for getting through 5 1/2 pages of huge fraction multiplication problems was going to be:

Lesson 3.4 More Multilplying Fractions

One page of instruction, 6 pages of fractions.


-- CatherineJohnson - 15 Mar 2006

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RussianMathLessonFourPointFour 15 Mar 2006 - 18:56 CatherineJohnson

4.7 Circles

The circumference of a circle divides any plane into 2 sections. The portion of the plane on the circle's circumference and inside of it forms a disk (fig. 4.9). The center, radius, and diameter of the circle are simultaneously the center, radius, and diameter of the disk. The distance from the center to any point on the disk cannot be greater than the radius.

The problem set begins like this:

A.

522. Read the text to make sure you understand:

       1. How points on the circumference of the circle are located in relation to the center;

       2. What radius and diameter means, and how they are related to each other;

       3. How circumference differs from a circle.

I love it that the first 'problem' is: read the text.

Since I was slavishly obedient while working through the book, I ended up reading the whole thing twice.

    
                4.9                                      4.10

523. Draw a circle and measure its radius and diameter.

524. Find the diameter of a circle if its radius is:
        4 cm.; 6.8 cm.; 42 cm.; 71 cm.; 0.6 m.; 30.84 m.; 1/4 m.; 2/5 m.

525. Find the radius of a circle if its diameter is:
       6.8 cm.;18 cm.;53 cm.;71/2 cm.;1 m.;27/10 m.;42.6 m.;61/2 m.

526. Draw a circle with a radius of 5.2 cm.
       Identify its center and show two of its diameters.

527. From fig. 4.10, identify which points belong to:
       a) the circumference;
       b) the circle; and
       c) the region outside the disk.

Lesson 4.9 Area of a Circle

In Lesson 4.9 we learn that:

Stringent mathematical reasoning has proven that the area of a circle is p times the area of a square whose side is equal to the radius of the circle. If we denote the radius of the circle by the letter r, then the area of a square whose side is equal to the radius is r². It follows that the formula for the area of a circle is calculated by the formula (fig. 4.14):

A=pr2

How often does one come across an expression like "stringent mathematical reasoning has proven" in U.S. math books?



p.s.

Remember what Carolyn says about Russian mathematicians.

They have chops.


-- CatherineJohnson - 15 Mar 2006

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KiselevGeometry 30 Oct 2006 - 19:22 CatherineJohnson




I've just discovered that Singapore Math is carrying Kiselev's Geometry Book I. Planimetry. According to them it's "the most famous Russian textbook":

In Russia, everyone knows it by this nickname: "Kiselev's Geometry." It is by far the most famous Russian textbook. It has been published over 40 times in dozens of millions of copies, and lived through many epochs, wars, reforms and revolutions - and not only in education.

For nearly two decades the book had been the standard geometry text for all schools in the Soviet Union, serving the students of the age corresponding to the US grades 7-9. Now it is translated into English and adapted by a professor of mathematics from UC Berkeley to fit common guidelines for a high-school level course in plane geometry. The book is equally suitable for homeschooling, grade school or college classes, teachers' professional development, or independent study.

Reviewers, editors, and users of former editions praise the exceptional clarity of exposition, and an excellent collection of problems, counted in hundreds, and with the difficulty level varying from reasonably easy to reasonably hard. Unlike many other textbooks in the same subject, "Kiselev's Geometry" is good not only for being studying geometry, but also for having learned it.

UPDATE 10-14-2006: Linda P has found a terrific review of the book here, as well as the company that sells the book for $29.95. (They have Ron Aharoni's book, too!)








excerpts from Kiselv's Geometry

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)



-- CatherineJohnson - 09 Oct 2006

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LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

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