Showing posts from January, 2006

Carnival of Homeschooling

Fifth Carnival of Homeschooling is up and going. Enjoy!

What is proof? Two-column proof versus paragraph proof

I asked you in my previous post if what I wrote was a proof (Click here to read my 'proof').

Well, yes and no.

It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the picture.

But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle".

Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" or AB in text.

Then the other is you need to CONVINCE - to be logical in your reasoning. Also it's not enough to convince a fellow student but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.

But, the form of the proof is not the most important thing.

Numbering your arguments is not the most important thing.

In my opinion, students don't need to write proofs in 2-column format if they want to write them as plain text (prose).

I want to sho…

What is proof?

Another obstacle in high school math are the proofs in the geometry course.

What is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof (well their proofs usually take many many paragraphs worth of writing).

Keith Devlin says in his book, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..."

Proof is about COMMUNICATING in a CONVINCING way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.

Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.

"Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, si…

Can I teach my child algebra?

Homeschoolers often wonder if they are capable of teaching algebra or high school geometry to their child. You know, maybe your own knowledge on those areas is a little shaky, so you're not so sure what to do when the time comes.

I'd like to ease your worries:

First of all, maybe you don't have to do all the teaching, after all.
Quite recently, it seems, companies have popped up with products where you see an experienced teacher solving algebra or trig or geometry problems, from start to finish.

You're watching little video clips on your computer but it's like sitting in a classroom, almost. There's and These would work as a supplement to your algebra book.

And there's Math U See of course, a whole curriculum with videos.

And, there exist online animated lessons of great quality, for example our current advertiser

Or, software that solves algebra problems for you.

Then there are lots of online tutoring services, and …

Scope and sequence

I hope everyone's teaching is going well; I guess this is a slower week for blogging for me. Maybe I'll share some of the questions asked on the site.

Someone recently asked me about MY suggested scope and sequence for teaching math, based on the way I have organized my online resources list.

Well I have never made any personal or suggested scope and sequence... Those pages are just categorized as elementary, middle school, and high school.

The thing is, you can find as many different scopes and sequences as there are textbooks. It varies from country to country, from state to state (in standards), from book to book.

People seem to have vary different ideas when it comes to when to teach which math topics.

But in my mind, you could try get your child to start algebra on 8th grade or thereabouts.

If you set that as a goal, then one should study pre-algebra topics such as integers, percent, ratio, proportion, square root, exponents on 7th grade.

One could try set a goal of going thr…

In which careers do you need math? And what kind?

Have you ever had your youngster ask, "Why do I need to study math? Where do I need this stuff?"

Well if not yet, you can expect the question when they get into algebra. But even before your students ask, take them into certain websites and show them!

First, Math Careers Database at
- it's not actually about careers in math, but instead a list of occupations/jobs and what kind of math topics are needed in that particular job.

I've also written an article (earlier) touching on this topic and collected a few similar links. See Where do you need square roots or algebra? Why study math? - scroll down to the bottom to see the links.



You know that Blogger doesn't support categories. But I found a way to add them by using bookmarking service. So I'm trying them out now on my blog.

How to teach equation solving

I got this question in my mailbox recently:

I am having a problem with show my son how to work out the proportion problem and solving equation by division, like 7.y=105 64=y.8 and find the missing term for the proportion of 3/4=x/16.

Can you help me out?

Always remember this problem solving strategy: when a problem is too difficult, make another, similar, but in some way easier problem, and observe for a strategy to solve that one.

If 7 ⋅ y = 105 and 64 = y ⋅ 8 are difficult, use easier examples first. Have him solve these ones:

2 ⋅ __ = 4
5 ⋅ x = 10
2 ⋅ x = 6


8 = 2 ⋅ __
12 = 3 ⋅ x
9 = 3 ⋅ x

If 'x' intimidates him, you can use an empty line.

Then ask him, HOW did he solve these ones? Well, chances are, of course, that he just 'sees' the answer, or remembers his multiplication tables and gets the answer from those.

But then SHOW him how division, in each case, gives us the answer too:

5 ⋅ x = 10. (We already know the answer is 2)
10 ÷ 5 gives the answer.

After going thru this, th…

Visual Math Learning - math tutorials

Here's a site that was proposed to me recently, Visual Math Learning.

It has tutorials about natural numbers, basic operations, factorization, and all fraction operations, using visual models.

They include nice interactive elements where students can experiment on their own and see the visual models change.

The site has also some nice games. I spent quite some time doing the factor sliders game...

It's all free. Go ahead and check the site out!

Is your math curriculum coherent, focused, and logical?

Recently I wrote a blogpost about the differences between US math curricula and those of the best performing countries in international student math comparisons.

I have put those thoughts into an article form on the site. I also got permission to post two charts that show which math topics are usually studied on which grade in the US, versus in the best performing countries in the TIMSS. The differences are quite striking!

Is your math curriculum coherent?

Homeschooling Carnival

Carnival of Homeschooling, the 3rd week, is up and going- including an entry from my blog. Thanks!

Is right answer important in math?

I want to discuss this topic because of a possible confusion. Recently in a blogpost I linked to the article Math anxiety, which mentions this math-related myth:


Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right.

So is it important to get the right answer when doing a mathematics problem?

Well, yes but it depends. This is not a simple cut-and-dried question.

It IS important if you're drilling multiplication facts.

But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given.

And obviously all childre…

Comments on blogposts

I would like for all of you readers to comment on my posts with your ideas on these math topics! Seriously. We can help each other.

Many of you are subscribing to my blog so it comes to your email address, and some subscribe to the RSS feed (117 people total are doing that! Thanks!)

So I am experimenting if it will work to have the "Comments" link in the main body of the post so it would show up in the RSS feed and in the emails. We will see.

Can girls do math? A paper folding story.

This story is cute, and even though it happened several years ago, it was new to me, and I really want to share it with you.

Maybe you've heard that it's impossible to fold paper more than 7 or 8 times, no matter how hard you try?

Well, that turned out to be a myth. Then high-schooler, Britney Gallivan tackled the challenge of folding paper 12 times and succeeded (in 2002). She experimented and thought about it, and found out what exactly limits the folding process. She discovered an equation linking the minimum length of paper needed, thickness of paper, and number of foldings.

Then, she calculated she needed paper about 4000 feet long to do 12 foldings! Britney found one special toilet paper roll that would do, went to a mall with her parents and started folding... It took 7 hours for her and her parents to do it, mostly on hands and knees.

Don't you ever think or leave that impression that girls can't do math - even if you're a female yourself and weren't good i…

Math as a career - why not?

Would your youngster enjoy a career in math?

The article at Businessweek Math will rock you world tells us how mathematics is used more and more in advertisement world, in internet technologies etc.

Use of mathematics is on the rise because it is used in more and more places to model various things, including people's behaviour.

New math graduates can easily land a job - often with a 6-figure salary.

Says Tom Leighton, an entrepreneur and applied math professor at Massachusetts Institute of Technology: "All of my students have standing offers at Yahoo! and Google."

But math programs in universities and colleges aren't filling with enthusiastic American young people: estimated half of the 20,000 math grad students now in the U.S. are foreign-born.

So it looks like math graduates are in need. And you don't have to be a boy to study math, remember.

US math curricula are incoherent

You probably know that in international comparisons, US students don't do real well in math.

Research into curricula in the best performing countries versus US is giving us one clue as to why this is:

US curricula tend to be
incoherent and a collection of arbitrary topics instead of focused and logicalAverage duration of a topic in US is almost 6 years (!) versus about 3 years in the best-performing countries. Lots of spiraling and reviewing is doneEach year, US textbooks cover way many more topics than the books in the best-performing countries What this means is that typically, any particular math topic is NEVER studied very deeply in any given school year, and that a vast amount of time each year is spent reviewing.

Is this really the best and most efficient way?

Just check your own math books (if you have them for several grades): do you find for example the topic of fractions on each of the books from 1st till 8th grade? Or, does your chosen math curricula teach the concept of per…

Solving a system of equations vs. doing dishes

I was getting ready to face the 'challenge' of a kitchen counter filled with dirty dishes, when I started thinking about the task ahead and how it compares to problem solving in math.

Then I found a funny little comparison between washing dishes and solving a system of linear equations.

You see, when you tackle that counter, you first need to organize things and move dishes around so you can see your kitchen sink again.

Just like if you have, say for example, these equations:

2x - 4y + 3x = 2z - 4/5x + 67y
4y - 0.3x - 0.98x + 34z = 5z + 90x - 0.2y
z - 2/3z + 3y -0.3x = 7y - 5/6z + 90x - 45x
Just looks messy and almost discouraging! x's and y's and z's all over the place!

But then you start moving things around (moving x's, y's, and z's all to the left side), and making piles of plates, putting pots in one corner, containers in another pile (combining x's, y's, and z's), making the other side of the counter empty (zero) - and voila - it looks much s…

Square root of 2 and Pythagoreans' shock

Last week I asked you two questions pertaining to the image below:

1) How does this connect with irrational numbers?

Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).

If both sides are 1 and diagonal is d, then Pythagorean theorem says:

d2 = 12 + 12

Solving that, you get d = √2

And, √2 is an irrational number.

So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).

2) How does this connect with history of mathematics?

Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or f…

Things not to learn in school

Sorry this is kind of off topic. Just too good to pass... It's comparing real LIFE and school tests:

Things not to learn in school

Square and its diagonal

Two questions for you to think about:

1) How does the above image of a square with one diagonal relate to irrational numbers?

2) How does this image connect with history of mathematics?

Answers will be here next week... : )

Learn to like math as the teacher

Good teacher loves his/her subject matter. That's old wisdom.

Obviously not all people who teach math actually like or love it - I've had homeschooling moms admit that, for example.

The sad thing about that is that you're probably transferring your attitudes to your students: your students will end up not liking math either.

What to do?

Well, try change your attitude:
--> Think back and try find the reason as to WHY you don't like math. Was it some bad experience in school? See if you can overcome or ignore that factor, whatever it is.

Maybe you had not-so-enthusiastic math teachers yourself. Maybe you don't like math because you don't understand it. Or, you feel "I'm just not a math person - I'm no good at math."

The last one is probably not true... it's largely a myth. Any normal person with normal intelligence can learn basic math.

Maybe you as the homeschooling parent suffered from math anxiety yourself? If so, I encourage you to read t…

Growing to love or hate math?

Have you ever met anyone who feels, "I hate math", or "I don't understand math at all", or "I'm not a math person" ?

Wonder when and where those feelings got started?

I recently read an article called Formula to bring back thrill to math.
The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math.

He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math.

I just wonder, is there a connection between the lack of good math teache…

Geometry and Euclid

You know, some math history can help spark interest in whatever you're teaching, and enliven the math (make it 'live math' so to speak). So today I want to educate you just a little bit about Euclid and geometry.

Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!

So the theorems your student is learning date back 2300 years!

Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.

The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.

So Euclid gives his name to Euclidean geometry - also called plane geometry.

In the beginning of his book, Euclid stated some defi…