Math readers

I've had some fun building this page about math readers, or "living math" books for my main site. The list is not terribly long yet, but it will grow, I'm sure.

I discovered that Amazon has all kinds of new "widgets" available to promote their products. What do you think of this? It is a slideshow featuring MathStart readers by Stuart J. Murphy. Hover your mouse over it:



Math story books offer children an interesting way to learn math concepts, to get interested in mathematics, and to explore some fascinating topics outside of the main curriculum. It is one way to bring math to "life". Kids are almost sure to enjoy it.

On my page, I list for example Cryptoclub, a fascinating storybook that teaches how to encrypt and decrypt secrete messages, or The Adventures of Penrose - THE MATHEMATICAL CAT, in which you will encounter many fascinating mathematical topics from fractals to tessellations, or many books for little kids.

I was thinking putting all of that on a blogpost, too, but it's just "duplicate content". Maybe I'll feature a few of them here soon.

I realize not everyone can buy many math readers on top of their regular school books, but try find some of these books in your local library!

Carnival of math

You might not have heard about it, but there exists a blog carnival for math, too. I submitted my rainbow entry into the latest one.

Not all of the entries there are higher math, by the way, such as MathMom's Calculator rant or Puzzler puzzled from JD2718.

If interested, go check it out: Carnival of math, edition 17!

Humorous short history of mathematics

Enjoy:

A Very Short History of Mathematics

This is how it starts:

MATHEMATICS is very much older than History, which begins* in +1066, as is well known; for the first mathematician of any note was a Greek named Zeno, who was born in -494, just 1,559 years earlier. Zeno is memorable for proving three theorems: (i) that motion is impossible; (ii) that Achilles can never catch the tortoise (he failed to notice that this follows from his first theorem); and (iii) that half the time may be equal to double the time. This was not considered a very good start by the other Greeks, so they turned their attention to Geometry.

continue here...

Hat Tip to Let's Play Math.

Tips for teaching integers

The main struggle with integers comes, not with the numbers themselves, but with some of the operations. There seem to be so many little rules to remember (though less than with fractions).

Some good real-life MODELS for integers are:

- temperature in a thermometer

- altitude vs. sea depth

- earning money vs. being in debt.

When first teaching integer operations, tie them in with one of these models.

I'll take for example the temperature.

Assuming n is a positive integer, the simple rules governing this situation are:

* x + n   means the temperature is x° and RISES by n degrees.
* x − n   means the temperature is x° and DROPS by n degrees.

It's all about MOVEMENT — moving either "up" or "down" the thermometer n degrees.

For example:

• 6 − 7 means: temperature is first 6° and drops 7 degrees.


• (-6) − 7 means: temperature is first -6° and drops 7 degrees (it's even colder!).


• (-2) + 5 means: temperature is first -2° and rises 5 degrees.


• 4 + 5 means: temperature is first 4° and rises 5 degrees.

These simple situations handle adding or subtracting a positive integer. Practice those first, until kids are familiar with these cases.

The remaining cases to handle ar adding or subtracting a negative integer:

• (-2) + (-5) would mean: temperature is first -2° and you "add" more negatives so it gets even colder.

The last case is least intuitive one:

• 1 − (-5) or subtracting a negative integer. I personally just remember the little rule of "two negatives turns into a positive".
  
  Some people explain it this way. In (-7) − (-3) you can think that you have 7 negatives at first, and you "take away" three of those negatives, leaving -4.
  
  This rule of "two negatives makes a positive" might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property).

See also an excellent treatise of integers vs. submarine depth at Text Savvy. Excerpt:

"When you add or subtract with integers, you are NOT combining collections or extracting from collections; you are moving in certain directions."

The "collections" idea does work nicely, for ADDITION:

• 7 + (-4) means you have a collection of 7 red balls and 4 blue balls. A pair of one red and one blue ball "cancels" or becomes zero. Total therefore will be 3 red balls.


• (-3) + (-9) means you have 3 blue balls and 9 blue balls more. Total 12 blue balls, or -12.

However, this "moving" idea is exactly how I have always intuitively done simple integer problems — except adding (negative) + (negative) and subtracting a negative, which I change to adding a positive.

Some books might present the rules for adding integers this way:

• If you add two integers that have the same sign, add the absolute values and put the same sign as what the numbers had.


• If you add two integers that have a different sign, subtract their absolute values and the answer will have the same sign as the number wiht greater absolute value.

Then they instruct to change the subtraction to addition; for example 5 − 7 becomes 5 + (-7) and (-4) − 2 becomes (-4) + (-2), and 5 − (-3) becomes 5 + 3.

While these are technically totally correct, I find it SO much easier to use the "moving" idea for most quick integer calculations. It is easier to start that way, and then learn these other rules to be used with more complex expressions, such as when adding many integers, or with negative and positive decimals.

Please feel free to download a simple integer addition and subtraction worksheet that practices some of these ideas (from Math Mammoth Grade 6-B Worksheets collection).

You will also benefit greatly from reading my previous article on teaching integers, which goes through all four integer operations.

Number rainbows to learn subtraction facts

I thought some of you (those who teach second grade) might enjoy my NUMBER RAINBOWS. The idea is that you connect two numbers with an arc if they add up to the particular number, such as 13.



Then, the child can use it as a reference when subtracting from 13 or when doing subtraction drill. You could first drill subtraction facts WITH the rainbow (such as 13 − 4, 13 − 7 etc.) and then without.



You would also ask the child to reproduce the rainbow - and color it, of course! These make for quite pretty math facts practice, don't you think!

I'm going to add these to my Add & Subtract 2-A book.

Math Manipulatives at my Amazon Store

I don't get around doing much editing of my Amazon Store, but today I was prompted because someone asked me to write a guide for manipulative use. So I'm doing a writeup on them, but also went to Amazon to see what they had.

So I added various ones that I thought might be useful to my Amazon store:

I was disappointed that there aren't any really cheap abaci there. The one abacus with 5 and 5 alternate color beads was quite pricey. Most of them had 10 beads the same color.

Other ones I included were:
- base ten blocks
- cuisenaire rods
- fraction circles or tiles
- scales, thermometers, measuring cups

Math Manipulatives at Amazon store

Prof. Lynn Arthur Steen and reform mathematics

Dave Marain at MathNotations has conducted an online interview with Prof.
Steen, one of the principal architects of the original NCTM Standards
and one of the most highly respected voices in reform mathematics
today.

This is how Dave describes the interview:

His replies to my questions are thoughtful, honest and
provocative.

Regardless of whether one agrees or disagrees with Prof.
Steen's views, we need to open up this kind of dialog in order to end
the Math Wars and move on in the best interests of our children.

In this first part of the interview, prof. Steen talks about for example the incoherency of math standards in various states learning basic arithmetic facts.

9-11 song

Please enjoy "America Cried" (a 9-11 memorial song) by Rockabilly.US

Math Mammoth placement advice

For those of you who are considering buying my Blue Series books, I've created a document that should help in the "placement".

For each of the books about addition, subtraction, multiplication or division I ask you several questions concerning what the child can do or understand.

Answering those you should be able to decide where the particular book is needful or not.

See more: Math Mammoth placement advice.

It does not as of yet contain any tests, but maybe in the future.

Does the child need to add completing the ten?

Question:

...have an other question about using your worksheets for my daughter. She is six years old - home schooled.
Currently, we are using Addition & Subtraction 2

She is able to add and subtract any numbers from 1 to 100 but when I try to explain "complete the ten" concept, she doesn't like thinking about it. She would rather solve 9+6 by counting on the finger 6 after 9 ... . She complains that I have to first subtract then add to make 10.

My question is: Should I let her complete the exercises without making her think in this manner?

It is important that she understands the IDEA in completing the ten. I gather that she does indeed understand the idea, but doesn't want to do it, since you'd have to both subtract and add, right? She just likes to do one operation, not two?

The adding though, is really easy, because you add 10 + 5 or 10 + 7 or something to ten. If she's counting with fingers, she's not yet seeing the easiness of this adding. It is also faster to do 9 + 6 by first subtracting one from 6 (getting 5), and then adding 10 + 5, because you can do 10 + 5 really easy: it is one ten and five, or we call that fifteen.

Of course she must understand place value to realize how easy that addition is.

Now, the goal is ALSO to eventually memorize the answers. You might let her know that she can't always count with fingers to find 9 + 6 but a quicker method is needed.

You can let her do some exercises "her way" for now as long as she also is able to understand and do the "complete the ten".

But come back to this idea how completing the ten makes the whole calculation easier and quicker. She WILL need to understand the idea when studying CARRYING in addition, (I mean to totally understand how it works. I realize kids can learn carrying mechanically as well.)

So you can say 1-3 months later reprint some pages and try again.

Review of Flatland: The Movie

I recently had the opportunity to watch Flatland: The Movie. It is a short, educational 35-minute animated film based on an old novel called Flatland.

I thought the movie was quite interesting and entertaining, and my husband felt the same way. We both liked watching it. The animations are excellent, the "voice acting" of the cast is very good, and the whole thing is very well made — thus very enjoyable to watch.

The storyline includes a world set in two-dimensional plane called Flatland. Its inhabitants are simple geometric shapes such as triangles, squares, pentagons, hexagons, and so on. Circles are the evil bosses of Flatland.

Arthur hears Spherius in his living room.

The main setting includes Arthur the Square who is parenting his oh-so-curious granddaughter Hex. Arthur goes to work in a "squaricle" in some office, and has (of course) a circle as a boss.

Hex starts contemplating about the third dimension, but since that is "heresy" and unimaginable in Flatland, Hex's investigations eventually get her and Arthur into trouble. But there is a "savior" — a sphere called Spherius from Spaceland who wants to educate the citizens of Flatland about the existence of the 3-dimensional world.

The interactions between these shapes express plenty of irony that makes the film fun to watch for older teens and adults — and can maybe make them think a little deeper about the prevailing "doctrines" that they are fed everywhere. Just like the shapes in the Flatland had never considered the possibility of a third dimension and rejected it as "heresy", even so we ourselves might be in a similar position (concerning something else), without realizing it.

I wouldn't recommend this movie for early elementary school or younger children because of the strong emotions expressed by the evil circles. Also, young children might not catch on to some of the main themes of the movie, the irony, nor the mathematical ideas. I couldn't put an exact age limit to this though; is up to the parents to decide, of course.

From a mathematical point of view, the movie just deals with a few concepts (after all, it is only a half-hour long). The whole theme revolves around the inability of 2-dimensional figures to grasp 3 dimensions. This is presented really well in different parts o the story. In his dream, Arthur explores all the dimensions up to 3:

• point - 0 dimensions (the King of the Pointland was really funny!)


• line - 1 dimension


• square - 2 dimensions


• cube - 3 dimensions

Also, I really liked the extra interview on the DVD which has a math professor discussing the intriguing possibility of a fourth dimension, and some neat illustrations.

Apart from the mathematical content, one could analyze this movie's themes (accepting new truth vs. sticking to the current ideology) for an English class project as well.

The website FlatlandTheMovie.com has photos, info on cast & crew, news, the trailer, and a store.

You can also see the official trailer below:



(Note: If you've read the book, you might wonder if the movie goes into gender issues and how women's position in the society is presented. This movie totally omits that part of the book. For example, Arthur's wife is a square just like Arthur is.)

Download issue

Update: The problem below has been fixed. It had to do with my server's configuration.


For my recent customers:

There is some kind of unexpected glitch in Kagi's download script. I'm working on the issue, hopefully it all gets fixed soon.

That script has been working just perfect for all these years. But not since August 30 and I'm not yet sure what happened or what is causing the problem.

Meanwhile, the download links will give a 404 error, BUT not to worry. I will email the files to you directly so you will get the book(s). If you have questions, just email me.

Maria Miller