Math Without Borders

04/01/2026

The relationship of "learning the multiplication tables" to "mathematics" is like the way "doing pushups" is related to "playing football," or the way "playing scales" is related to "playing music." They are important, and they will block your progress if you don't master them, but they are an unfortunate first introduction to mathematics.

That said, how do we get past that barrier?

A few years back, I worked with a group of 8th and 9th graders who were failing in math, and figured they just couldn't do it, but their actual problem was they had never mastered the multiplication tables. This meant everything else they ever came to in math seemed impossibly hard.

In response, I used flash cards with them for awhile, but then I wrote a program to do it better. This program (which you can run right on my website for free without downloading anything) keeps track of not only which problems you know or don't know, but also how long it takes you to respond. It uses that information to structure daily practice to take you all the way to mastery. It feels sort of like a computer game. You should do the program for at least 20 minutes every day, but it should only take you a month or two. Once you reach "Mastery", everything has been pushed into long term memory, so you will know them even when you get to be my age!

Even if you "more or less" know "most" of the combinations, you can use this to polish off your skill painlessly. Tell your friends about this program. It's free and the system really works!

Start today:

Free multiplication tables practice online. Bookmark this page and practice right here. Master the multiplication combinations once and for all "Learn 'em Forwards and Backwards" Current Version 1.5 -- May 19, 2016 [START] (Scroll down to read instructions) Learning the multiplication and division c...

09/10/2024

https://www.youtube.com/watch?v=LpunKsVnQNg

FEMA has allocated $20 million to aid the hurricane and flood victims in North Carolina. What does that mean?

17/03/2024

How did Finland reach the top of the world education ranking?
Groundbreaking documentary filmmaker, Michael Moore, is shocked to discover that Finnish students do better by going to school less.

23/01/2023

I just posted a new video on YouTube:

I have a collection of links to data, interviews, presentations, and videos in pdf format here: https://www.lcurve.org/writings/LocalArticles/VaccineConcern...

17/09/2022

A few years ago I posted a suggested activity for people with drones near the beach being able to observe the curvature of the earth. I recently found this video that provides a dizzying array of video evidence. This is spectacular!

Flat earthers love to say "show me the curve", but then point to the horizon and say, "see, it's flat". Yes, of course the horizon looks flat, except from ve...

12/09/2022

There has been some controversy in my post about the ship crossing the International Date Line and Equator at the beginning of the 20th Century. The controversy arises because of historians not being very good mathematicians. (Sorry, historians, it's the truth!) Hopefully this will shed some light on the matter.
(BTW, I'm glad this post has aroused so much interest.)
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The problem in deciding whether 1900 or 1901 is the beginning of a new century is what happens when you let historians do mathematics. We count with the numbers 1, 2, 3, ... but we measure intervals starting with 0. When you lay a ruler down on a board you put the 0 mark at the end of the board. If we want to measure time from the supposed year Jesus was born (a rather vague marker, actually), the birth of Jesus should be labeled 0 AD = 0 BC. The year 1 should start after one year. You don't have your first birthday on the day of your birth, you have it when you have lived one year. The day of your birth is counted as your 0th birthday. Apparently mothers and midwives do math better than historians.

Historians mess it up even more by calling the year before 1 AD 1 BC. 1 - 1 = 0, not -1. We could leave the 1 BC in place if we counted the birth of Jesus as 0 AD. Century numbering has the same flaw. The interval from the birth of Jesus to 1 century later should be considered the 0th century, and the first century should start after that. That's where we get the convention that the 20th century is the 1900's, the 19th century is the 1800's, etc.

The century numbering issue is similar to the problem of numbering of the days of the year. When are you 1.5 days into the year? Is it noon on January 1? No! It is noon on Jan 2. (Draw a picture and verify this for yourself.) That's because calendar makers don't allow for a Jan 0. Astronomers, who use time intervals to calculate orbits of planets, comets, etc. found this situation intolerable, so they took matters into their own hands. They have redefined Jan 0 to be another name for Dec 31 of the previous year.

So if people get the starting points of centuries "wrong" it is not because they are dumb or that they can't count. It is because whoever set up the messed up system we live with messed it up long ago by not recognizing 0 to be the appropriate starting point for measuring time intervals. As far as I am concerned, we should say World War II happened in the 19th Century and we are currently living in the 20th century. Do I talk that way in public? No, because I need to communicate and be understood according to "convention." But when people start "correcting" each other about picky definitions based on what is a messed-up situation to begin with, I just roll my eyes.

08/09/2022

I found this on another timeline but had trouble sharing it, so I am re-posting it.

GREAT SEA STORY
The passenger steamer SS Warrimoo was quietly knifing its way through the waters of the mid-Pacific on its way from Vancouver to Australia. The navigator had just finished working out a star fix and brought Captain John DS. Phillips, the result. The Warrimoo's position was LAT 0º 31' N and LONG 179 30' W. The date was 31 December 1899. "Know what this means?" First Mate Payton broke in, "We're only a few miles from the intersection of the Equator and the International Date Line". Captain Phillips was prankish enough to take full advantage of the opportunity for achieving the navigational freak of a lifetime.
He called his navigators to the bridge to check & double check the ship's position. He changed course slightly so as to bear directly on his mark. Then he adjusted the engine speed.
The calm weather & clear night worked in his favor. At mid-night the SS Warrimoo lay on the Equator at exactly the point where it crossed the International Date Line! The consequences of this bizarre position were many:
The forward part (bow) of the ship was in the Southern Hemisphere & in the middle of summer.
The rear (stern) was in the Northern Hemisphere & in the middle of winter.
The date in the aft part of the ship was 31 December 1899.
In the bow (forward) part it was 1 January 1900.
This ship was therefore not only in:
Two different days,
Two different months,
Two different years,
Two different seasons
But in two different centuries - all at the same time!

01/08/2022

I just purchased a new document camera and wanted to test it out. In particular I wanted to see how it would integrate with my existing microphone providing the audio. The camera has some nice features on it, such as controls to compensate for tilting, rotation, and "keystoning", so I set up my tablet comfortably on my lap and let the camera compensate to "flatten" and rotate the image to make it look normal. I'm pretty impressed. The math content here is a pretty standard derivation of the quadratic formula, which you may find interesting or useful it it happens to meet you "where you're at", but keep in mind the real goal was to test out the technical features of my new camera system. What do you think? I'm not going to give up my graphics tablet any time soon, but this will be useful on the side.

I did this short video to try out my new document camera integrated with my Yeti microphone. I held the writing tablet in my lap at a convenient angle, used...

23/06/2022

I have just uploaded a new video on YouTube (and included it on the Bonus Topics tab of my website). Here is the YouTube version:

The conventional way to represent the Real Number system is to think of the numbers as corresponding to points along an infinite straight line. The problem ...

21/08/2020

In my new HSC Physics course I have a "Science and Society" section at the end of certain units. Here is one related to electric fields and potentials.

This is one of the Science and Society topics in the Math Without Borders Home Study Companion Physics course (a rigorous, algebra-trig-based physics course ...