Wouldn’t it make more sense to use lines than blocks? A block would be a number with a width value a height value, and a starting coordinate (x and y values minimum) to have four values in all. A line only has two values, being a starting coordinate (only x minimum) and a magnitude. A point itself is a value (or equal to multiplying times 1 or adding by 0). Your random number would be a line also, but one that acts differently than a transformation line in that you can transform to either the magnitude of the random number or the location of the random number.
[…] Rethinking Arithmetic: A Visual Guide | BetterExplained […]
[…] Arithmetic became a way to transform a number: Addition was sliding (+3 means slide 3 units to the right), and multiplication was scaling (times 3 means scale it up 3x). […]
Here’s a fighting aspect to it: throwing a combination of punches like 1-2-3 is really repeating a natural rhythm of punches, which gives the opponent three times the rhythm information and three times the chances of counterattack. Using this as a starting point can help explain why there is a point of diminishing returns for the effectiveness of long combinations, aside from just the muscle deterioration or energy reserve depletion theories. (mostly, because this holds true in video games as well, even video games that ignore fatigue!)
[…] Numbers: number systems, visual arithmetic, different bases, Prime numbers […]
[…] Rethinking Arithmetic: A Visual Guide (tags: mathematics visualization reference education) […]
absolutely love this page.
@eivo: Thanks, glad you liked it.
You’re splitting , my head, I know I should probably put this on that page but you’re view of calculus doesn’t really help me (i only read it once though) I’ll keep reading to see if I can make something click! 
Awesome site, it helped a tiny bit so far 
Hey man nice article!
I never thought that you could teach a concept like “transformation” with arithmetic. I never learned about arithmetic until upper div. math in college (linear algebra)…
Yeah, I’m studying to become a math teacher, and I’ve read plenty of articles on how most of how math is taught in America (at least) doesn’t actually teach much. Students (like myself) aren’t learning and are turned off from math…
I appreciate the work you’re doing with this site
@Seamus: Thanks for the note – sorry the calculus article didn’t help, my view is being refined over time, and there may be even better analogies down the line.
@Frank: I appreciate the comment! Yes, unfortunately a lot of math education is about memorization and moving on, vs. really getting the concepts we’re talking about. It’s really awesome that you’re going into the profession with a desire to change it :).
i love math, I want to learn every lessons in math, someday i want to be a good student for the teacher in college… “success lies beyond sacrifices”
[…] at the number 1, see multiplication as a transformation that changes the number (1 * […]
Addition and multiplication are way to combine two numbers. Also, they’re commutative.
Threrefore, I think the perfect visualization would start with both numbers lying on the line, and them flipping ans spinning and tugging at each other until they settle into the new configuration.
I’m sure you know how to visualize the commutativity of multiplication of positive integers using rectangles and rotation. That’s the kind of stuff that I have in mind.
So I’m not completely satisfied with the visualizations in this post. I’m gooing to have to keep looking for new ways.
[…] is another issue. What do you think of this guy's approach of teaching arithmetic visually? Rethinking Arithmetic: A Visual Guide | BetterExplained I've never tried it with kids, as I don't know any, but some adults found the approach much easier […]
these are really good things…keep them running with jumps to higher level…and please please keep them free…u’ll earn your name only that way…many of us hackers …or real mathematics insight seekers don’t want to buy a book…they prefer to read for free…support open and free source movement from heart…THANKS A LOT FOR THESE…I’LL PASS THESE ON TO SCHOOLS I KNOW…
Thanks for the good program. I sent in a question on the 1st Oct 12 and since then have been looking for real basic stuff on convention and stuff like that as that’s my only stumbling block, to be able to understand the words, and now I have found rethinking arithmatic: a visual guide and that’s what I have been looking for, something along the lines of basic arithmetic. So hopefully I will gradually find such things as basic as realising asterisks have the meaning of multification, etc. Without going any further afield. Thanks.
[…] see the dot product as directional multiplication. But multiplication goes beyond repeated counting: it’s applying the essence of one item to […]
@Iqbal: Agreed.