Visualizing numbers

Recently I added another math curriculum to the list of math curricula; it's called RightStart Mathematics.
They are using a little different abacus from the usual to help children visualize numbers: the abacus has five beads one color, and the next five another color.

Their idea is that using such abacus, children will eventually learn to recognize the amounts on abacus with a single glance, without counting the beads. They will then form a mental image and learn to 'see' seven as five and two.

After checking their website, just a few days later, when checking dead links, I stumbled upon this same abacus in electronic form on another website.
(However you can only do a few things with the applet, so it's not nearly as good as a real physical abacus.)

This Ten frame tool uses a similar idea - kids are supposed to recognize amounts 0-10 visually, without counting.

Obviously another tool for this is just a die, or dice.

Great ideas, great tools.

Visualizing numbers

Recently I added another math curriculum to the list of math curricula; it's called RightStart Mathematics.
They are using a little different abacus from the usual to help children visualize numbers: the abacus has five beads one color, and the next five another color.

Their idea is that using such abacus, children will eventually learn to recognize the amounts on abacus with a single glance, without counting the beads. They will then form a mental image and learn to 'see' seven as five and two.

After checking their website, just a few days later, when checking dead links, I stumbled upon this same abacus in electronic form on another website.
(However you can only do a few things with the applet, so it's not nearly as good as a real physical abacus.)

This Ten frame tool uses a similar idea - kids are supposed to recognize amounts 0-10 visually, without counting.

Obviously another tool for this is just a die, or dice.

Great ideas, great tools.

Online Math Resources link list updated

I have checked my online math resources pages for dead links (again). It's just inevitable, it seems, that some links go bad after a while. Anyway, hopefully it's all now working.

I am continually AMAZED at the wealth of interactive mathematics things that one can find on the internet. I compiled this list almost three years ago, and it has grown little by little ever since. The quality of many of these math activities or tutorials is just great. Various people have done a lot when publishing these interactive (or otherwise) math things online.

It's like you have a big collection of virtual manipulatives for about any math subject you can think of, at your fingertips!

I have divided my list into 9 different pages. Go and check; I'm sure you'll find something interesting to benefit your math teaching. I've concentrated on interactive tutorials and other interactive math activities but the list has other resources as well, such as tutorials and software.

Math Help and Tutoring

Math history, problem solving, and test preparation

Place value, basic operations, times tables, factoring etc. (elementary math)

Clock, money, measuring, and elementary geometry

Fractions, Decimals, Percent, Integers (middle school math)

Geometry, Measuring, Coordinate plane (middle school math)

Algebra, Graphing, Calculus

Geometry, Trigonometry, Statistics (high school math)

Sites with lots of math games and interactive activities

Online Math Resources link list updated

I have checked my online math resources pages for dead links (again). It's just inevitable, it seems, that some links go bad after a while. Anyway, hopefully it's all now working.

I am continually AMAZED at the wealth of interactive mathematics things that one can find on the internet. I compiled this list almost three years ago, and it has grown little by little ever since. The quality of many of these math activities or tutorials is just great. Various people have done a lot when publishing these interactive (or otherwise) math things online.

It's like you have a big collection of virtual manipulatives for about any math subject you can think of, at your fingertips!

I have divided my list into 9 different pages. Go and check; I'm sure you'll find something interesting to benefit your math teaching. I've concentrated on interactive tutorials and other interactive math activities but the list has other resources as well, such as tutorials and software.

Math Help and Tutoring

Math history, problem solving, and test preparation

Place value, basic operations, times tables, factoring etc. (elementary math)

Clock, money, measuring, and elementary geometry

Fractions, Decimals, Percent, Integers (middle school math)

Geometry, Measuring, Coordinate plane (middle school math)

Algebra, Graphing, Calculus

Geometry, Trigonometry, Statistics (high school math)

Sites with lots of math games and interactive activities

Subtraction stories

This was just so cute.

Background: We've been looking at some subtraction stories with pictures in a book called "MIMOSA Moving Into Math" (you might find it at eBay) with my three-year old daughter. Well one evening here recently, *somehow* her mind started working, and suddenly, totally spontaneously, she came up with this story:

"Mommy, I have a subtraction story: There were three potties at first. But then mommy broke one (which is very true! - I did manage to break a potty a few days ago), and so now there are only two left."

Then she proceeded to help me with rinsing the dishes and the stories continued: "I have four spoons to rinse. I rinse one, and now there's three left to rinse!"

I said how about one with bananas. She said, "YES! There were ten bananas originally. Then, we ate one. And there are 1,2,3,4,5,6,7,8,9,10... (she had to count up).. NINE left for DADDY!"

(My husband does like bananas quite well.)

After dishes she still made some more with the plastic bears and such.

What a moment of joy for the teacher - makes me thankful!

Subtraction stories

This was just so cute.

Background: We've been looking at some subtraction stories with pictures in a book called "MIMOSA Moving Into Math" (you might find it at eBay) with my three-year old daughter. Well one evening here recently, *somehow* her mind started working, and suddenly, totally spontaneously, she came up with this story:

"Mommy, I have a subtraction story: There were three potties at first. But then mommy broke one (which is very true! - I did manage to break a potty a few days ago), and so now there are only two left."

Then she proceeded to help me with rinsing the dishes and the stories continued: "I have four spoons to rinse. I rinse one, and now there's three left to rinse!"

I said how about one with bananas. She said, "YES! There were ten bananas originally. Then, we ate one. And there are 1,2,3,4,5,6,7,8,9,10... (she had to count up).. NINE left for DADDY!"

(My husband does like bananas quite well.)

After dishes she still made some more with the plastic bears and such.

What a moment of joy for the teacher - makes me thankful!

Math teacher's toolkit (elementary)

I was not going to post during these days when everyone is busy... But I just got this nice weblink and want to pass it along:

Math Teacher's Toolkit

It has...
- Place Value Calculator
- Virtual Hundrets/tens/units Place Value Cards
- Hundred Square
- Times tables tester
- 12 Hour Clock
- 24 Hour Clock
- Sequences
- 10 Digit Number Line
and a few more interactive math teaching tools for elementary grades (using Macromedia Flash).

Math teacher's toolkit (elementary)

I was not going to post during these days when everyone is busy... But I just got this nice weblink and want to pass it along:

Math Teacher's Toolkit

It has...
- Place Value Calculator
- Virtual Hundrets/tens/units Place Value Cards
- Hundred Square
- Times tables tester
- 12 Hour Clock
- 24 Hour Clock
- Sequences
- 10 Digit Number Line
and a few more interactive math teaching tools for elementary grades (using Macromedia Flash).

Some fun optical illusions

Here's 8 optical illusions for some fun:
www.grand-illusions.com/opticalillusions

Nothing short of amazing... little black balls that are yet aren't there, moving dots, moving cogs, Dr Angry and Mr Calm faces, and a few more.

Some fun optical illusions

Here's 8 optical illusions for some fun:
www.grand-illusions.com/opticalillusions

Nothing short of amazing... little black balls that are yet aren't there, moving dots, moving cogs, Dr Angry and Mr Calm faces, and a few more.

Mathematics: The science of patterns

I just started reading the book Mathematics : The Science of Patterns by Keith Devlin. I'm planning on writing a review of it for my site. I can already tell it's a good book, and I've just read the prologue, titled "What is mathematics?"

The author says that yes, mathematics WAS the study of number - up till about 500 B.C. Then it became the study of number and shape. After invention of calculus, it became the study of number, shape, motion, change and space.

But nowadays you can't say that any more, because the field of mathematics has expanded SO much: in 1900 it would have taken about eighty books to write all the world's mathematical knowledge. Devlin estimates that nowadays (or when he wrote the book about 10 years ago) it would take maybe 100,000 volumes to contain all knonw mathematics!

You know, your average person doesn't realize that. I got a glimpse of it while in university, when I saw all these lists of subcategories of mathematics, and when I studied a little bit of something like Abstract Algebra or Complex Analysis or Measure Theory - and all we did was touch the surface of such topics.

And so today, mathematics is often defined as the study of patterns - numerical patterns, patterns of shape, patterns of motion, and so on.

Devlin also touched on beauty in mathematics in his prologue - and I agree: mathematics is beautiful. That, too, I fear, most folks don't realize or understand. It's because you need to be able to understand the math first - then it kind of forms a picture in your mind of a structure that is beautiful, harmonious. I can't explain it.

So I want to recommend the book Mathematics : The Science of Patterns for all of you even before I've written complete review on it. It's written for a layman, and is about the essential history of mathematics.

Categories: philosophy, math

Mathematics: The science of patterns

I just started reading the book Mathematics : The Science of Patterns by Keith Devlin. I'm planning on writing a review of it for my site. I can already tell it's a good book, and I've just read the prologue, titled "What is mathematics?"

The author says that yes, mathematics WAS the study of number - up till about 500 B.C. Then it became the study of number and shape. After invention of calculus, it became the study of number, shape, motion, change and space.

But nowadays you can't say that any more, because the field of mathematics has expanded SO much: in 1900 it would have taken about eighty books to write all the world's mathematical knowledge. Devlin estimates that nowadays (or when he wrote the book about 10 years ago) it would take maybe 100,000 volumes to contain all knonw mathematics!

You know, your average person doesn't realize that. I got a glimpse of it while in university, when I saw all these lists of subcategories of mathematics, and when I studied a little bit of something like Abstract Algebra or Complex Analysis or Measure Theory - and all we did was touch the surface of such topics.

And so today, mathematics is often defined as the study of patterns - numerical patterns, patterns of shape, patterns of motion, and so on.

Devlin also touched on beauty in mathematics in his prologue - and I agree: mathematics is beautiful. That, too, I fear, most folks don't realize or understand. It's because you need to be able to understand the math first - then it kind of forms a picture in your mind of a structure that is beautiful, harmonious. I can't explain it.

So I want to recommend the book Mathematics : The Science of Patterns for all of you even before I've written complete review on it. It's written for a layman, and is about the essential history of mathematics.

Categories: philosophy, math

Geometry course for middle school

I was going to write about Euclid but this came up. Someone submitted a review of a math program called RightStart mathematics to my site. While browsing that site, I found out they offer a separate geometry program which looked really good:

See, many students have great problems with high school geometry. I've written an article Why is high school geometry so difficult? about that already.

One of the remedies for that is to do things right BEFORE high school: teach true geometry and DRAW - don't spend all your time calculating areas and volumes from grades 3 till 8.

So now I have stumbled upon a geometry program for middle grades that seems to be just that: emphasis is on drawing. Check it out:
RightStart Geometry program.

I haven't seen all of it, but based on the example pages, it looked good. It's $57 for the whole package.

Categories: geometry

Geometry course for middle school

I was going to write about Euclid but this came up. Someone submitted a review of a math program called RightStart mathematics to my site. While browsing that site, I found out they offer a separate geometry program which looked really good:

See, many students have great problems with high school geometry. I've written an article Why is high school geometry so difficult? about that already.

One of the remedies for that is to do things right BEFORE high school: teach true geometry and DRAW - don't spend all your time calculating areas and volumes from grades 3 till 8.

So now I have stumbled upon a geometry program for middle grades that seems to be just that: emphasis is on drawing. Check it out:
RightStart Geometry program.

I haven't seen all of it, but based on the example pages, it looked good. It's $57 for the whole package.

Categories: geometry

Pi's digits

Here's the answer as to how many decimal places Pi's decimal expansion has been calculated:

1,241,100,000,000 decimal digits

It's more than 1 trillion digits!

This was done in 2002 by Yasumasa Kanada and 9 other individuals in Tokyo.

See www.super-computing.org/pi_current.html for more details on this project.

Okay, and if you'd like to start memorizing it, here's a link to Pi trainer :)

Pi's digits

Here's the answer as to how many decimal places Pi's decimal expansion has been calculated:

1,241,100,000,000 decimal digits

It's more than 1 trillion digits!

This was done in 2002 by Yasumasa Kanada and 9 other individuals in Tokyo.

See www.super-computing.org/pi_current.html for more details on this project.

Okay, and if you'd like to start memorizing it, here's a link to Pi trainer :)

Lesson plan/article about Fibonacci numbers and golden ratio

I have combined the recent blogposts about Fibonacci numbers and golden ratio into one article on my site that you can use as a teaching guide or just to educate yourself:

Fibonacci numbers and golden section - lesson plan

I hope you enjoy my posts... you're welcome to leave commens about the blog - just mention in your comments it's about the blog.

Lesson plan/article about Fibonacci numbers and golden ratio

I have combined the recent blogposts about Fibonacci numbers and golden ratio into one article on my site that you can use as a teaching guide or just to educate yourself:

Fibonacci numbers and golden section - lesson plan

I hope you enjoy my posts... you're welcome to leave commens about the blog - just mention in your comments it's about the blog.

Hands-on with pi

Pi seems to be such a simple thing - it's just two letters, and after all, you just peek into any math book to find out its value. Everybody knows about it, "Pi? Ah, yes, they taught us that in school, so it's something pretty simple."

BUT have you ever given pi some more thought? And, if you let your students give pi some more thought, it might just make math a little more interesting and 'lively' for them, too.

Think thought about it - how come the ratio of any circle's circumference to its diameter can be an irrational number? After all, this is not something that would naturally come to mind if you've never heard of irrationals before... If you start measuring lots of circles and calculating ratios, yes, you would soon notice that in every circle the ratio is the same. You would notice it's a little over 3. But would you just GUESS on your own that this ratio isn't a rational number?

If you have the time (and homeschoolers might), give your student a project to find what is the ratio of a circle's circumference to its diameter by measuring. Tell them they can play an ancient mathematician now and let them work at it - don't even tell them it's called pi and hide all math books for a while...

Some kids might just love such a hands-on activity.

See if they get so far as to realize it's the same constant in all circles. And congrats if they get 3.14 or even get it between 3.1 and 3.2.

During all the measuring your student is not likely to think that this number would be irrational - he probably hasn't even heard the word before!

And mathematicians didn't find that out real soon either. Ancient cultures were aware of the fact that the ratio of circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for pi as 377/120 or 3 1/8 or 256/81.

Archimedes found out that this ratio was between 223/71 and 22/7. Or, to be exact, he was finding an approximation for the area of the circle via this simple method: he drew a regular polygon inside the circle and outside the circle, and calculated the areas of the polygons-- obviously the area of the circle would have to be between those two numbers.

Archimedes wasn't afraid of calculations and used a polygon with 96 sides...! You might try giving 8th graders this as a project using just hexagons or octagons.

It makes for a nice exercise in geometry, plus it should be interesting since it ties in with history. Here's a nice slideshow about Archimedes' method.

Over time, mathematicians were able to calculate Pi more accurately - find better rational approximations to it. I'm sure they started to guess that maybe Pi is irrational - in fact, my guess is they suspected it to be so for a long time.

Proof of pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have been sure of that - but the quest for finding pi's decimals still continues.

Guess how many decimals they have found thus far?

Answer will be posted here in a coming blogpost. I'll leave you hanging - you can do the same in school. Why hand them all the answers on a plate?

P.S. Here's a link about history of pi if you want to learn more.

Hands-on with pi

Pi seems to be such a simple thing - it's just two letters, and after all, you just peek into any math book to find out its value. Everybody knows about it, "Pi? Ah, yes, they taught us that in school, so it's something pretty simple."

BUT have you ever given pi some more thought? And, if you let your students give pi some more thought, it might just make math a little more interesting and 'lively' for them, too.

Think thought about it - how come the ratio of any circle's circumference to its diameter can be an irrational number? After all, this is not something that would naturally come to mind if you've never heard of irrationals before... If you start measuring lots of circles and calculating ratios, yes, you would soon notice that in every circle the ratio is the same. You would notice it's a little over 3. But would you just GUESS on your own that this ratio isn't a rational number?

If you have the time (and homeschoolers might), give your student a project to find what is the ratio of a circle's circumference to its diameter by measuring. Tell them they can play an ancient mathematician now and let them work at it - don't even tell them it's called pi and hide all math books for a while...

Some kids might just love such a hands-on activity.

See if they get so far as to realize it's the same constant in all circles. And congrats if they get 3.14 or even get it between 3.1 and 3.2.

During all the measuring your student is not likely to think that this number would be irrational - he probably hasn't even heard the word before!

And mathematicians didn't find that out real soon either. Ancient cultures were aware of the fact that the ratio of circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for pi as 377/120 or 3 1/8 or 256/81.

Archimedes found out that this ratio was between 223/71 and 22/7. Or, to be exact, he was finding an approximation for the area of the circle via this simple method: he drew a regular polygon inside the circle and outside the circle, and calculated the areas of the polygons-- obviously the area of the circle would have to be between those two numbers.

Archimedes wasn't afraid of calculations and used a polygon with 96 sides...! You might try giving 8th graders this as a project using just hexagons or octagons.

It makes for a nice exercise in geometry, plus it should be interesting since it ties in with history. Here's a nice slideshow about Archimedes' method.

Over time, mathematicians were able to calculate Pi more accurately - find better rational approximations to it. I'm sure they started to guess that maybe Pi is irrational - in fact, my guess is they suspected it to be so for a long time.

Proof of pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have been sure of that - but the quest for finding pi's decimals still continues.

Guess how many decimals they have found thus far?

Answer will be posted here in a coming blogpost. I'll leave you hanging - you can do the same in school. Why hand them all the answers on a plate?

P.S. Here's a link about history of pi if you want to learn more.

How do we know this is true? What is the proof?

Someone emailed me just recently with this question:

"How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?"
(The person who emailed me this question is from Japan, based on the email address.)

I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi.

You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever.

I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational.

Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number.

It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi and it's irrational. And what does that mean, irrational? they ask. And they're told it means the decimal expansion continues forever without repeating.

School mathematics does a lot of announcing facts. I hope it doesn't kill in the students that small wondering, questioning, interested voice that wants to know more:
But why? How do we know that? Is there a proof?

Do you know?

Now that I have you interested, I'll probably disappoint you by telling that in the case of pi, the proofs about its irrationality require understanding of calculus and "advanced math" - math advanced beyond basic algebra. But here's one anyway:
Proof that Pi is irrational

How do we know this is true? What is the proof?

Someone emailed me just recently with this question:

"How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?"
(The person who emailed me this question is from Japan, based on the email address.)

I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi.

You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever.

I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational.

Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number.

It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi and it's irrational. And what does that mean, irrational? they ask. And they're told it means the decimal expansion continues forever without repeating.

School mathematics does a lot of announcing facts. I hope it doesn't kill in the students that small wondering, questioning, interested voice that wants to know more:
But why? How do we know that? Is there a proof?

Do you know?

Now that I have you interested, I'll probably disappoint you by telling that in the case of pi, the proofs about its irrationality require understanding of calculus and "advanced math" - math advanced beyond basic algebra. But here's one anyway:
Proof that Pi is irrational

The Golden Section

Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?

The discussion below covers the basics of golden section. I'll try to keep it short.

---

Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.

Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).

This is how we get this golden section or golden cut:


Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.

S:L = L:W

From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).

And the golden ratio is the ratio Phi:1.

Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2

Golden rectangle

Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.

here is one golden rectangle

Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.

And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.

• The Golden section in architecture


• Dividing the line in golden section using compass and ruler


• Phi in pentagons and pentagrams


• Where Fibonacci numbers are found in nature


• Where golden section is found in nature

P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.

The Golden Section

Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?

The discussion below covers the basics of golden section. I'll try to keep it short.

---

Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.

Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).

This is how we get this golden section or golden cut:


Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.

S:L = L:W

From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).

And the golden ratio is the ratio Phi:1.

Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2

Golden rectangle

Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.

here is one golden rectangle

Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.

And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.

• The Golden section in architecture


• Dividing the line in golden section using compass and ruler


• Phi in pentagons and pentagrams


• Where Fibonacci numbers are found in nature


• Where golden section is found in nature

P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.

Homeschooled math whiz wins competition

Michael Viscardi, 16 years, from San Diego, has won the 2005 - 06 Siemens Westinghouse Competition in Math, Science and Technology. The judges said they can't find the limits of his understanding!

He's been homeschooled since fifth grade but he takes math classes at the University of California three times a week. His father is a software engineer and his mother, who stays at home, has a Ph.D. in neuroscience. I read somewhere that he also plays piano and violin and has won music competitions too. Quite a whiz!

His competition entry was about a 19th century math problem, which he solved in a new way and was able to obtain several new results in his research. He says he liked the problem because it used complex analysis... That rang a bell for me, as I liked complex analysis, too, during my university studies and my master's thesis used it.

Read more...

See here his picture and read even more.

Now, he probably would have been taking extra math classes even if he had been in public school, but homeschooling is surely a great option for gifted students.

Homeschooled math whiz wins competition

Michael Viscardi, 16 years, from San Diego, has won the 2005 - 06 Siemens Westinghouse Competition in Math, Science and Technology. The judges said they can't find the limits of his understanding!

He's been homeschooled since fifth grade but he takes math classes at the University of California three times a week. His father is a software engineer and his mother, who stays at home, has a Ph.D. in neuroscience. I read somewhere that he also plays piano and violin and has won music competitions too. Quite a whiz!

His competition entry was about a 19th century math problem, which he solved in a new way and was able to obtain several new results in his research. He says he liked the problem because it used complex analysis... That rang a bell for me, as I liked complex analysis, too, during my university studies and my master's thesis used it.

Read more...

See here his picture and read even more.

Now, he probably would have been taking extra math classes even if he had been in public school, but homeschooling is surely a great option for gifted students.

Fibonacci numbers part 2

In a previous post, we studied about Fibonacci numbers :
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(add two consecutive numbers to get the next one).

I noted that those numbers appear in nature in many places, and asked if your child/student know about that.

My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

---

Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.)
It's not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

(Oh I will some day write more about irrational numbers; I think the concepts of rational versus irrational numbers is very fascinating - they're kind of elusive and as if they weren't from this world)

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.

So far so good... I hope I'm not confusing you. I think I will stop here with pure math and continue some other time - yes, there's still more to come! This Phi is an exciting number in many mathematical senses that you may not know about, but you might have already heard about Phi as the GOLDEN RATIO... (to be continued)

Fibonacci numbers part 2

In a previous post, we studied about Fibonacci numbers :
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(add two consecutive numbers to get the next one).

I noted that those numbers appear in nature in many places, and asked if your child/student know about that.

My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

---

Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.)
It's not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

(Oh I will some day write more about irrational numbers; I think the concepts of rational versus irrational numbers is very fascinating - they're kind of elusive and as if they weren't from this world)

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.

So far so good... I hope I'm not confusing you. I think I will stop here with pure math and continue some other time - yes, there's still more to come! This Phi is an exciting number in many mathematical senses that you may not know about, but you might have already heard about Phi as the GOLDEN RATIO... (to be continued)

Goals in math teaching

With my daughter, we've been concentrating on simple addition problems and the topic of tens and ones. Right now we are working on these goals:

• Understand subtraction concept - she partially does
• Tens and ones - she partially understands. (I realize mastering place value well takes often years.)
• Understand something + 1 type problems - she partially does
• Memorize answers to double 2, double 3, etc doubling facts

And, we're heading towards these goals that are further away, yes, but I am keeping them constantly in mind:

• Memorizing all basic addition facts
• Learn to use those to do subraction problems

Having definite goals is important. Good math teachers know where students are supposed to be heading, and also how to get there.

Make sure you know (and understand!) what the final and intermediate goals are. If you are simply following the math book page by page, it's like walking on a road while looking down at the road - not knowing exactly where the road is leading.

When you have the final goals down pat, you can often jump back and forth in your math book, skip pages or provide extra practice according to need.

Of course to do this, the teacher needs to understand the math beforehand and know how it's structured, how one concept leads to another. Soon I will be posting a review on a book that can help homeschoolers to do just that.

---

I also often have my dd fill in 10x10 grid with numbers from 1 to 100. The other day, after filling it, I asked her to color yellow all whole tens, and then color pink all those that end in five. Okay, after those were done, she asked to continue coloring, and before I knew it, the little artist in her took over and the whole grid turned into a rainbow!
Well I thought you might enjoy it too so here it is:



P.S. You can make similar number grid worksheets (either partially filled in or not at all) for counting and skip-counting at www.homeschoolmath.net/worksheets/number-charts.php

Goals in math teaching

With my daughter, we've been concentrating on simple addition problems and the topic of tens and ones. Right now we are working on these goals:

• Understand subtraction concept - she partially does
• Tens and ones - she partially understands. (I realize mastering place value well takes often years.)
• Understand something + 1 type problems - she partially does
• Memorize answers to double 2, double 3, etc doubling facts

And, we're heading towards these goals that are further away, yes, but I am keeping them constantly in mind:

• Memorizing all basic addition facts
• Learn to use those to do subraction problems

Having definite goals is important. Good math teachers know where students are supposed to be heading, and also how to get there.

Make sure you know (and understand!) what the final and intermediate goals are. If you are simply following the math book page by page, it's like walking on a road while looking down at the road - not knowing exactly where the road is leading.

When you have the final goals down pat, you can often jump back and forth in your math book, skip pages or provide extra practice according to need.

Of course to do this, the teacher needs to understand the math beforehand and know how it's structured, how one concept leads to another. Soon I will be posting a review on a book that can help homeschoolers to do just that.

---

I also often have my dd fill in 10x10 grid with numbers from 1 to 100. The other day, after filling it, I asked her to color yellow all whole tens, and then color pink all those that end in five. Okay, after those were done, she asked to continue coloring, and before I knew it, the little artist in her took over and the whole grid turned into a rainbow!
Well I thought you might enjoy it too so here it is:



P.S. You can make similar number grid worksheets (either partially filled in or not at all) for counting and skip-counting at www.homeschoolmath.net/worksheets/number-charts.php

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.

Fibonacci numbers

The solution to the little thinking exercise of previous week is here:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
-> Add two consecutive numbers from the sequence to get the next one following them

This sequence is called Fibonacci numbers. And it isn't just any ole sequence of any ole numbers... it has some amazing properties, plus it's found in nature in many places.

For example, Fibonacci numbers are found in

* Petals on flowers
* Seed heads
* Pine cones
* Leaf arrangements
* Vegetables and Fruit

The links go to a magnificent site about Fibonacci numbers with tons of information and pictures. Go click the links and see for yourself! It's very informative and well done, plus I don't have those photos.

Then come back. I have a question for you:

Should your child or student learn about this? Is this important to know? Well, you think about it - in the next post we will study just a tiny bit more about Fibonacci numbers.

Fibonacci numbers

The solution to the little thinking exercise of previous week is here:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
-> Add two consecutive numbers from the sequence to get the next one following them

This sequence is called Fibonacci numbers. And it isn't just any ole sequence of any ole numbers... it has some amazing properties, plus it's found in nature in many places.

For example, Fibonacci numbers are found in

* Petals on flowers
* Seed heads
* Pine cones
* Leaf arrangements
* Vegetables and Fruit

The links go to a magnificent site about Fibonacci numbers with tons of information and pictures. Go click the links and see for yourself! It's very informative and well done, plus I don't have those photos.

Then come back. I have a question for you:

Should your child or student learn about this? Is this important to know? Well, you think about it - in the next post we will study just a tiny bit more about Fibonacci numbers.

Too busy for a friend

I'm sorry this is pretty much off topic (though it mentions a math teacher!) but I just got it this morning in my email and it is a good story about something important.

---

One day a teacher asked her students to list the names of the other students in the room on two sheets of paper, leaving a space between each name.

Then she told them to think of the nicest thing they could say about each of their classmates and write it down.

It took the remainder of the class period to finish their assignment, and as the students left the room, each one handed in the papers.

That Saturday, the teacher wrote down the name of each student on a separate sheet of paper, and listed what everyone else had said about that individual.

On Monday she gave each student his or her list. Before long, the entire class was smiling. "Really?" she heard whispered "I never knew that I meant anything to anyone!" and, "I didn't know others liked me so much." were most of the comments.

No one ever mentioned those papers in class again. She never knew if they discussed them after class or with their parents, but it didn't matter. The exercise had accomplished its purpose. The students were happy with themselves and one another. That group of students moved on.

Several years later, one of the students was killed in Viet Nam and his teacher attended the funeral of that special student. She had never seen a serviceman in a military coffin before. He looked so handsome, so mature.

The church was packed with his friends. One by one those who loved him took a last walk by the coffin. The teacher was the last one to bless the coffin.

As she stood there, one of the soldiers who acted as pallbearer came up to her. "Were you Mark's math teacher?" he asked. She nodded: "yes." Then he said: "Mark talked about you a lot."

After the funeral, most of Mark's former classmates went together to a luncheon. Mark's mother and father were there, obviously waiting to speak with his teacher.

"We want to show you something," his father said, taking a wallet out of his pocket. "They found this on Mark when he was killed. We thought you might recognize it."

Opening the billfold, he carefully removed two worn pieces of notebook paper that had obviously been taped, folded and refolded many times. The teacher knew without looking that the papers were the ones on which she had listed all the good things each of Mark's classmates had said about him.

"Thank you so much for doing that," Mark's mother said. "As you can see, Mark treasured it."

All of Mark's former classmates started to gather around. Charlie smiled rather sheepishly and said, "I still have my list. It's in the top drawer of my desk at home."

Chuck's wife said, "Chuck asked me to put his in our wedding album."
"I have mine too," Marilyn said. "It's in my diary."

Then Vicki, another classmate, reached into her pocketbook, took out her wallet and showed her worn and frazzled list to the group. "I carry this with me at all times," Vicki said and without batting an eyelash, she continued: "I think we all saved our lists."

That's when the teacher finally sat down and cried. She cried for Mark and for all his friends who would never see him again.

The density of people in society is so thick that we forget that life will end one day. And we don't know when that one day will be.

So please, tell the people you love and care for, that they are special and important. Tell them, before it is too late.

And One Way To Accomplish This Is: Forward this message on. If you do not send it, you will have, once again passed up the wonderful opportunity to do something nice and beautiful.

If you've received this, it is because someone cares for you and it means there is probably at least someone for whom you care.

If you're "too busy" to take those few minutes right now to forward this message on, would this be the VERY first time you didn't do that little thing that would make a difference in your relationships?

The more people that you send this to, the better you'll be at reaching out to those you care about.

Remember, you reap what you sow. What you put into the lives of others comes back into your own.

May Your Day Be Blessed As Special As You Are

Too busy for a friend

I'm sorry this is pretty much off topic (though it mentions a math teacher!) but I just got it this morning in my email and it is a good story about something important.

---

One day a teacher asked her students to list the names of the other students in the room on two sheets of paper, leaving a space between each name.

Then she told them to think of the nicest thing they could say about each of their classmates and write it down.

It took the remainder of the class period to finish their assignment, and as the students left the room, each one handed in the papers.

That Saturday, the teacher wrote down the name of each student on a separate sheet of paper, and listed what everyone else had said about that individual.

On Monday she gave each student his or her list. Before long, the entire class was smiling. "Really?" she heard whispered "I never knew that I meant anything to anyone!" and, "I didn't know others liked me so much." were most of the comments.

No one ever mentioned those papers in class again. She never knew if they discussed them after class or with their parents, but it didn't matter. The exercise had accomplished its purpose. The students were happy with themselves and one another. That group of students moved on.

Several years later, one of the students was killed in Viet Nam and his teacher attended the funeral of that special student. She had never seen a serviceman in a military coffin before. He looked so handsome, so mature.

The church was packed with his friends. One by one those who loved him took a last walk by the coffin. The teacher was the last one to bless the coffin.

As she stood there, one of the soldiers who acted as pallbearer came up to her. "Were you Mark's math teacher?" he asked. She nodded: "yes." Then he said: "Mark talked about you a lot."

After the funeral, most of Mark's former classmates went together to a luncheon. Mark's mother and father were there, obviously waiting to speak with his teacher.

"We want to show you something," his father said, taking a wallet out of his pocket. "They found this on Mark when he was killed. We thought you might recognize it."

Opening the billfold, he carefully removed two worn pieces of notebook paper that had obviously been taped, folded and refolded many times. The teacher knew without looking that the papers were the ones on which she had listed all the good things each of Mark's classmates had said about him.

"Thank you so much for doing that," Mark's mother said. "As you can see, Mark treasured it."

All of Mark's former classmates started to gather around. Charlie smiled rather sheepishly and said, "I still have my list. It's in the top drawer of my desk at home."

Chuck's wife said, "Chuck asked me to put his in our wedding album."
"I have mine too," Marilyn said. "It's in my diary."

Then Vicki, another classmate, reached into her pocketbook, took out her wallet and showed her worn and frazzled list to the group. "I carry this with me at all times," Vicki said and without batting an eyelash, she continued: "I think we all saved our lists."

That's when the teacher finally sat down and cried. She cried for Mark and for all his friends who would never see him again.

The density of people in society is so thick that we forget that life will end one day. And we don't know when that one day will be.

So please, tell the people you love and care for, that they are special and important. Tell them, before it is too late.

And One Way To Accomplish This Is: Forward this message on. If you do not send it, you will have, once again passed up the wonderful opportunity to do something nice and beautiful.

If you've received this, it is because someone cares for you and it means there is probably at least someone for whom you care.

If you're "too busy" to take those few minutes right now to forward this message on, would this be the VERY first time you didn't do that little thing that would make a difference in your relationships?

The more people that you send this to, the better you'll be at reaching out to those you care about.

Remember, you reap what you sow. What you put into the lives of others comes back into your own.

May Your Day Be Blessed As Special As You Are

US students lag behind in math

Well this is not real uplifting, nor is it really news (we knew it before), but here is another study showing how US students score way below average among industralized nations
.
NEW STUDY FINDS U.S. MATH STUDENTS CONSISTENTLY BEHIND THEIR PEERS AROUND THE WORLD - FINDINGS CHALLENGE CONVENTIONAL WISDOM ABOUT U.S. MATH SUCCESS IN EARLY GRADES.

US students lag behind in math

Well this is not real uplifting, nor is it really news (we knew it before), but here is another study showing how US students score way below average among industralized nations
.
NEW STUDY FINDS U.S. MATH STUDENTS CONSISTENTLY BEHIND THEIR PEERS AROUND THE WORLD - FINDINGS CHALLENGE CONVENTIONAL WISDOM ABOUT U.S. MATH SUCCESS IN EARLY GRADES.

Continue this sequence of numbers!

Okay, teachers and homeschoolers and other readers, I want you to think:
How does this continue:

0, 1, 1, 2, 3, 5, 8, 13, ...

Answer will be here next week.

If you already know, stay tuned anyway; I have a point to make besides the math.

Continue this sequence of numbers!

Okay, teachers and homeschoolers and other readers, I want you to think:
How does this continue:

0, 1, 1, 2, 3, 5, 8, 13, ...

Answer will be here next week.

If you already know, stay tuned anyway; I have a point to make besides the math.

Fun math and math appreciation

Some of us are math enthusiastics: we love math, think it is full of beauty, logic, and patterns; we think it can be fun, exciting, amusing even. (I include myself in this category.)

And then some of us don't care for math, didn't like it in school, didn't do well in it, don't want anything to do with it anymore. And I'm sure there are people in between these extremes, too.

As a teacher, you would like your students to belong to the first category, of course.

Now, I sincerely think that the biggest factor in whether student ends up liking or hating math is what kind of math teacher and teaching he/she gets. Parents' attitudes play a role too, but teacher's attitude to math and the way math is taught are the weightiest factors, I feel.

But books such as The Adventures of Penrose The Mathematical Cat by Theoni Pappas can help too. It is intended for fun reading, exploring exciting mathematical topics in an easy-to-understand way.

The storyline includes Penrose the mathematical cat who seems pretty lazy but often gets interested in her mistress' math papers and thus ends up learning about this and that in math.

For example, you get to encounter 0s and 1s (binary numbers), fractals, infinity, Mobious strip, Pascal triangle, golden rectangles, paperfolding, tessellations, abacus, magic squares, tangrams, nanoworld... For math geeks, these are familiar topics. They show some of the beautiful or fun or intriguing aspects of math.

You could almost say this book is about "Math appreciation". We are told to teach our kids art appreciation, to teach them to appreciate fine arts, paintings, sculptures. In my opinion, learning about these fascinating math topics can go a long way towards teaching children to appreciate math.

Read the complete review here.

Fun math and math appreciation

Some of us are math enthusiastics: we love math, think it is full of beauty, logic, and patterns; we think it can be fun, exciting, amusing even. (I include myself in this category.)

And then some of us don't care for math, didn't like it in school, didn't do well in it, don't want anything to do with it anymore. And I'm sure there are people in between these extremes, too.

As a teacher, you would like your students to belong to the first category, of course.

Now, I sincerely think that the biggest factor in whether student ends up liking or hating math is what kind of math teacher and teaching he/she gets. Parents' attitudes play a role too, but teacher's attitude to math and the way math is taught are the weightiest factors, I feel.

But books such as The Adventures of Penrose The Mathematical Cat by Theoni Pappas can help too. It is intended for fun reading, exploring exciting mathematical topics in an easy-to-understand way.

The storyline includes Penrose the mathematical cat who seems pretty lazy but often gets interested in her mistress' math papers and thus ends up learning about this and that in math.

For example, you get to encounter 0s and 1s (binary numbers), fractals, infinity, Mobious strip, Pascal triangle, golden rectangles, paperfolding, tessellations, abacus, magic squares, tangrams, nanoworld... For math geeks, these are familiar topics. They show some of the beautiful or fun or intriguing aspects of math.

You could almost say this book is about "Math appreciation". We are told to teach our kids art appreciation, to teach them to appreciate fine arts, paintings, sculptures. In my opinion, learning about these fascinating math topics can go a long way towards teaching children to appreciate math.

Read the complete review here.

Adding in the kitchen

I like to try make it a habit asking my daughter math questions here and there. Today, while working in the kitchen, I said we're doing "plus one" questions. I asked, what is one plus one, which was easy for her. Then she told me a question. I did another one, six plus one. After that we dealt with one plus six and how to make it easy.

She suddenly said eight and six makes two. Well I should have dug deeper into why she thought that way but this time we just did it with tiny rocks on the carpet. I put the rocks out and asked could that make two. She agreed it couldn't and counted the answer.

After that she said 1 + 2 = 4. Well I asked her to make it with rocks again and then she came up with the right answer.

Then she wondered herself what is 3 + 1, and found the answer.

Then, one more time I got to hear the wrong math: while eating her soup, he announced 2 + 2 = 5. This time I asked her to use her fingers to get it straight.

We've done similar things with plastic bears and blocks too.

I do love the fact that she's interested, and is doing it on her own even.

And I'm sure most kids do the same - we educators just should try not to kill the little child's enthusiasm as school years go on.

Adding in the kitchen

I like to try make it a habit asking my daughter math questions here and there. Today, while working in the kitchen, I said we're doing "plus one" questions. I asked, what is one plus one, which was easy for her. Then she told me a question. I did another one, six plus one. After that we dealt with one plus six and how to make it easy.

She suddenly said eight and six makes two. Well I should have dug deeper into why she thought that way but this time we just did it with tiny rocks on the carpet. I put the rocks out and asked could that make two. She agreed it couldn't and counted the answer.

After that she said 1 + 2 = 4. Well I asked her to make it with rocks again and then she came up with the right answer.

Then she wondered herself what is 3 + 1, and found the answer.

Then, one more time I got to hear the wrong math: while eating her soup, he announced 2 + 2 = 5. This time I asked her to use her fingers to get it straight.

We've done similar things with plastic bears and blocks too.

I do love the fact that she's interested, and is doing it on her own even.

And I'm sure most kids do the same - we educators just should try not to kill the little child's enthusiasm as school years go on.

Number sense

I recently found a
nice article at Apples4theTeacher.com that summarizes 10 research findings as to what teaching practices and methods are the most effective. For example,

8) Teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important.

What is number sense? It's something where the child is comfortable with numbers, knows what numbers mean, can do mental calculations, can connect numbers and real world situations, and can make comparisons.

Kids with number sense have built a 'mental number line'. They can tell that 11 is more distance away from 15 than what 14 is.

How can you help develop it?

Think how can children get to be FAMILIAR with numbers. It's when they have lots of opportunities to use them and to play with them.

Games are good. Observing how parents use numbers in daily life is good. Doing lots of calculations ('drill') helps too. And why not have the kids draw number lines. I'm sure you can think of other ideas (welcome to comment!).

Number sense

I recently found a
nice article at Apples4theTeacher.com that summarizes 10 research findings as to what teaching practices and methods are the most effective. For example,

8) Teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important.

What is number sense? It's something where the child is comfortable with numbers, knows what numbers mean, can do mental calculations, can connect numbers and real world situations, and can make comparisons.

Kids with number sense have built a 'mental number line'. They can tell that 11 is more distance away from 15 than what 14 is.

How can you help develop it?

Think how can children get to be FAMILIAR with numbers. It's when they have lots of opportunities to use them and to play with them.

Games are good. Observing how parents use numbers in daily life is good. Doing lots of calculations ('drill') helps too. And why not have the kids draw number lines. I'm sure you can think of other ideas (welcome to comment!).

Math wars: is it "drill and kill" or "discovery learning"?

Today, Barry Garelick sent me an article of his which explores the topic of MATH WARS.

You may have heard of them. Math wars refer to the debate or controversy about how math should be taught in school. There have been two main camps: those who emphasize discovery learning and calculators, de-emphasizing algorithms and memorization - and those who claim kids need to learn and memorize basic facts and algorithms such as long division.

Mathematicians have been fiercely opposing the math education reforms that are based on the discovery learning postulate.

In my opinion, I think kids surely need to memorize their basic facts, and learn the basic algorithms. Learning by discovering is a fine teaching method, but it shouldn't replace teacher telling or explaining how things work.

I guess what I feel is that learning by discovery (or explorations) can be great but you need to use it right and not let it 'rule'. It is an instructional method or a tool - not a goal in itself.

Anyway, if you are lacking in this topic, Barry's article summarizes nicely how math wars and fuzzy math began, what has been going on, and how the wars are now (unfortunately) turning into politics instead of being purely an academic debate.

Math wars: is it "drill and kill" or "discovery learning"?

Today, Barry Garelick sent me an article of his which explores the topic of MATH WARS.

You may have heard of them. Math wars refer to the debate or controversy about how math should be taught in school. There have been two main camps: those who emphasize discovery learning and calculators, de-emphasizing algorithms and memorization - and those who claim kids need to learn and memorize basic facts and algorithms such as long division.

Mathematicians have been fiercely opposing the math education reforms that are based on the discovery learning postulate.

In my opinion, I think kids surely need to memorize their basic facts, and learn the basic algorithms. Learning by discovering is a fine teaching method, but it shouldn't replace teacher telling or explaining how things work.

I guess what I feel is that learning by discovery (or explorations) can be great but you need to use it right and not let it 'rule'. It is an instructional method or a tool - not a goal in itself.

Anyway, if you are lacking in this topic, Barry's article summarizes nicely how math wars and fuzzy math began, what has been going on, and how the wars are now (unfortunately) turning into politics instead of being purely an academic debate.

Study order of the ebooks

Posted a short article about in which order to study the ebooks to answer questions from customers.

Study order of the ebooks

Posted a short article about in which order to study the ebooks to answer questions from customers.

What's going on at Homeschoolmath.net

Today I got done moving the preview lessons from Division 1 ebook to the 'teaching tips' section (in the menu at the bottom of page, under multiplication/division lessons), and then extracting a few example pages from the ebook itself to be shown on the Division 1 ebook contents page.

I still have two more ebooks to go, and do similar changes.

I've also received a few books in the mail that I will be writing about later, after I get some time to read them.

One was like a 'bible' of school math - reference book for teacher. One is math history book titled "Mathematics The Science of Patterns" (which is an excellent title), and two are kind of fun math books where a cat is exploring things. Stay tuned.

What's going on at Homeschoolmath.net

Today I got done moving the preview lessons from Division 1 ebook to the 'teaching tips' section (in the menu at the bottom of page, under multiplication/division lessons), and then extracting a few example pages from the ebook itself to be shown on the Division 1 ebook contents page.

I still have two more ebooks to go, and do similar changes.

I've also received a few books in the mail that I will be writing about later, after I get some time to read them.

One was like a 'bible' of school math - reference book for teacher. One is math history book titled "Mathematics The Science of Patterns" (which is an excellent title), and two are kind of fun math books where a cat is exploring things. Stay tuned.

Calculator in elementary grades?

There is controversy over how much elementary school kids should be allowed to use the calculator. Some say it's a no-no altogether, some think it should be used almost all the time - and some think it should be balanced.

In my opinion, one should severely restrict the calculator use in early grades - until the student has mastered the basic math facts and can quickly do basic mental calculations. Until then, calculator should be used only for some special projects or illustrations.

Why? Because not knowing those facts by heart is a big hindrance when learning fractions, factoring, etc. And because it helps children develop NUMBER SENSE - to be familiar with our number system.

Also kids need to learn to do multiplication and long division by hand - it is needed later when those same procedures are used in algebra.

Read more at this link: Using calculator in elementary math teaching.

Calculator in elementary grades?

There is controversy over how much elementary school kids should be allowed to use the calculator. Some say it's a no-no altogether, some think it should be used almost all the time - and some think it should be balanced.

In my opinion, one should severely restrict the calculator use in early grades - until the student has mastered the basic math facts and can quickly do basic mental calculations. Until then, calculator should be used only for some special projects or illustrations.

Why? Because not knowing those facts by heart is a big hindrance when learning fractions, factoring, etc. And because it helps children develop NUMBER SENSE - to be familiar with our number system.

Also kids need to learn to do multiplication and long division by hand - it is needed later when those same procedures are used in algebra.

Read more at this link: Using calculator in elementary math teaching.

More on focus

I wanted to clarify something about my earlier post on focus and math teaching.

Focusing doesn't mean that you would focus on, say, addition only for a year, and then subtraction for a year, then multiplication for a year etc. No. Mathematical concepts connect together and so you need to study closely connected concepts fairly close to each other.

The idea is to focus on a few topics or themes.

For example, for 1st grade, the focus topics could be concept of addition, concept of subtraction, place value, and memorizing basic addition/subtraction facts.

Several of them connect together too: for example, subtraction/addition have a close connection. So do understanding tens/ones and concept of addition.

Measuring and geometry are needed, to some extent. The less important topics could be fractions and data analysis (if any).

When you are focusing on some topics, you should have a goal: you want the student to MASTER something. After that, you can move on to something else.

Maybe you have a goal for the first grade (or first part of first grade) that the student understands certain things about addition and subtraction, understands tens and ones, and knows by heart certain basic addition facts. You want mastery in those topics, so you concentrate on them.

Then, the next goals could be hundreds/tens/ones, and adding and subtracting two and three-digit numbers.

Then could come concept of multiplication, and memorizing the times tables.

You may have more or deeper goals -- you should check your state standards what they say also; these are just examples. I am NOT saying you should teach exactly this way or in this order. It depends on the child, on various standards, on the curriculum. For example I know some people introduce multiplication concept in first grade along with the other operations.

But focusing on a few closely connected topics, with the goal of mastery, is the idea.
Hope this helps.

More on focus

I wanted to clarify something about my earlier post on focus and math teaching.

Focusing doesn't mean that you would focus on, say, addition only for a year, and then subtraction for a year, then multiplication for a year etc. No. Mathematical concepts connect together and so you need to study closely connected concepts fairly close to each other.

The idea is to focus on a few topics or themes.

For example, for 1st grade, the focus topics could be concept of addition, concept of subtraction, place value, and memorizing basic addition/subtraction facts.

Several of them connect together too: for example, subtraction/addition have a close connection. So do understanding tens/ones and concept of addition.

Measuring and geometry are needed, to some extent. The less important topics could be fractions and data analysis (if any).

When you are focusing on some topics, you should have a goal: you want the student to MASTER something. After that, you can move on to something else.

Maybe you have a goal for the first grade (or first part of first grade) that the student understands certain things about addition and subtraction, understands tens and ones, and knows by heart certain basic addition facts. You want mastery in those topics, so you concentrate on them.

Then, the next goals could be hundreds/tens/ones, and adding and subtracting two and three-digit numbers.

Then could come concept of multiplication, and memorizing the times tables.

You may have more or deeper goals -- you should check your state standards what they say also; these are just examples. I am NOT saying you should teach exactly this way or in this order. It depends on the child, on various standards, on the curriculum. For example I know some people introduce multiplication concept in first grade along with the other operations.

But focusing on a few closely connected topics, with the goal of mastery, is the idea.
Hope this helps.

Mathematics is all about... ?

In elementary school, I was under the impression that math was all about NUMBERS. Before 7th grade, I was thinking that there couldn't be much more math since we seemed to be doing the same things over again (spiraling curriculum).

After that, the math I was learning seemed to be all LETTERS: x's, y's, a's, b's, f's etc. And soon it was Greek letters too!

But math is actually neither. Math is all about PATTERNS.

The whole structure of our base-10 number system is a certain pattern. Multiplication, division, etc. are full of patterns to notice. Functions and relations are all about making general rules for patterns.

When first learning about sine in right triangle, it may seem to students like a disjointed piece of information - but if they could get past the initial stage and see the sine curve! Patterns!


But I especially wanted to mention trees, ferns, coastlines, snowflakes, and such. Because they have a pattern too, and mathematicians have unveiled the mathematical laws governing them. Those nature objects are fractal-like.

Check this introduction to fractals or fractal lessons for middle school to learn a little bit. Then go check some nature fractals.

I love it when natural beauty and mathematics connect.

Mathematics is all about... ?

In elementary school, I was under the impression that math was all about NUMBERS. Before 7th grade, I was thinking that there couldn't be much more math since we seemed to be doing the same things over again (spiraling curriculum).

After that, the math I was learning seemed to be all LETTERS: x's, y's, a's, b's, f's etc. And soon it was Greek letters too!

But math is actually neither. Math is all about PATTERNS.

The whole structure of our base-10 number system is a certain pattern. Multiplication, division, etc. are full of patterns to notice. Functions and relations are all about making general rules for patterns.

When first learning about sine in right triangle, it may seem to students like a disjointed piece of information - but if they could get past the initial stage and see the sine curve! Patterns!


But I especially wanted to mention trees, ferns, coastlines, snowflakes, and such. Because they have a pattern too, and mathematicians have unveiled the mathematical laws governing them. Those nature objects are fractal-like.

Check this introduction to fractals or fractal lessons for middle school to learn a little bit. Then go check some nature fractals.

I love it when natural beauty and mathematics connect.

Education and entertainment - children don't differentiate

Like I said, people send me links. This time it was www.Time4Learning.com. They say they're "children’s educational and entertainment online subscription service for ages 3 to 9".

I went and looked at the site and some of their free examples - and it seems like a quality service. It has learning activities and games and worksheets for K-5 language arts and math.

Like I read in a news article recently, children don't differentiate between entertainment and education - they learn and absorb whether they are watching a violent cartoon movie or an educational clip. I would definitely opt for the latter.

So I posted the link at my site's educational links page.

Education and entertainment - children don't differentiate

Like I said, people send me links. This time it was www.Time4Learning.com. They say they're "children’s educational and entertainment online subscription service for ages 3 to 9".

I went and looked at the site and some of their free examples - and it seems like a quality service. It has learning activities and games and worksheets for K-5 language arts and math.

Like I read in a news article recently, children don't differentiate between entertainment and education - they learn and absorb whether they are watching a violent cartoon movie or an educational clip. I would definitely opt for the latter.

So I posted the link at my site's educational links page.