Hacker News Reader

I don't know what the answer is, but we're currently going through this with our kids, but the opposite. There is now a push to not memorize the multiplication table, but treat multiplication as repeated addition. So to get 6 * 7 you do 6+6+6+6+6+6+6 or 7+7+7+7+7+7 (I forget which is 'correct' as it's been a few years). That's good because it gives understanding, but when you're trying to multiply 2 4 digit numbers, it's really useful to have your tables memorized! We're now going into a new school and their biggest advice to new students is to memorize their multiplication tables, as they'll have trouble otherwise. Maybe they should do understanding followed by memorization?*

I picked up this trick intuitively as a very young kid, perhaps 7-8

I break multiplication and division up into chunks of 1's, 5's, and 10's so that it's mostly just adding a zero to the end, or doing that and then cutting it in half.

I never learned multiplication tables, long division, lattice multiplication or anything else. just did this in my head.

So say you want to find 86 * 32

86 * 10 = 860

860 * 3 = (2400 + 180 = 2580)

2580 + (86 * 2) = 2752

It starts to break down for bigger numbers depending on how "uneven" they are but it's been pretty useful in general

Multiplication tables memorized in school usually stop at 12x12. They're super useful for doing the exact thing you demonstrated, in your head, very fast, using some tricks to turn the problem into smaller problems (with memorized answers, so they're solved almost instantly).

Memorizing that multiplication table was a more valuable use of time than the vast majority of time I spent learning mathematics. It's useful daily. Can't say that for much else.

How do you know that 800 × 3 = 2400 if you haven't learned multiplication tables?

(8+8+8)*100

Addition

8 x 3 = 24

8 x 300 = 2400

Add two zeros

I meant, how do you know 8 × 3 without a multiplication table?

Ah I see what you mean In my head, I do this:

8 + 8 = 16

16 + 8 = 24

> I forget which is 'correct' as it's been a few years

God I hope they're not teaching one of them is correct and one of them is incorrect.

Unfortunately it is according to some of the common core grading rubrics (in the US).

5 x 3 can be understood as five groups of three items, or three groups of five items, and students at a particular grade level are "encouraged" to use one interpretation over the other to the point of taking off points when they naturally use the commutative property (and or choose to chunk differently). I forgot what it was exactly but I read about a parent (rightfully) complaining about this.

I don't think this is anywhere in the common core standards, but probably the teacher/school was using some pre-packaged rubric and the teacher didn't bother to override using better judgment.

> Instead I took the approach of trying to understand what the mathematical meaning of multiplication is.

There's not much to think about tho, is there? You take 3 boxes of 4 apples each, how many apples is there?

Or were you thinking about groups, rings and fields of integers? I highly doubt it at that age ;)

Ultimately you need to know multiplication tables instinctively to be able to do algebra, so they make you learn them. I don't think there's any way to skip it, just like you can't be good at football without doing a lot of cardio.

For the record - I also hated memorizing multiplication tables and thought I will be a writer because clearly math isn't for me. Fortunately in the next semester we had word problems and I loved that part of math.

I think there is. For example, multiplication is not visually intuitive on a number line, but it is very intuitive shown as a grid of row x columns. Doing the math out helps build an intuition for things like commutative properties and what not.

Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.

> Doing the math out helps build an intuition for things like commutative properties and what not.

Well yes, but that's the things in boxes example. Doesn't matter if there's 3 boxes of 4 apples each or 4 boxes of 3 apples each. That takes like 1 lesson and most kids understand it intuitively anyway.

I remember we had 1 lesson of introduction and then a whole semester to learn multiplication tables and use them to solve simple word problems with multiplication. I'm not sure how that time could be better spend thinking about multiplication in abstract.

> Multiplication on a number line? I guess you can take X and repeatedly jump by it Y times.

Of course you can, but it's not intuitive to most. Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.

Kids absolutely do not grasp commutativity immediately. That they can solve 3x4 is 12 and 4x3 is 12 is not the same thing as understanding with confidence that nothing changes between these swaps.

> Most notably it requires understanding that in 3+4 you start at 3 and move 4 spaces to the right. In 3x4 you start at 0 and move to the right 3 spaces 4 times.

That's a weird way to describe it. Doesn't make it more understandeable and isn't useful for solving problems. So why bother? We were taught the definition of multiplication (it's just repeating addition). So if you have 3+4 it's starting at 0 then jumping by 3 and then jumping by 4. If you have 3*4 it's the same as 4+4+4 which is starting at 0 and jumping by 4 three times. Or vice versa.

I don't think we had that jumping on number line on that lesson, probably not since it's kinda obvious and provides little value when you know the definition. We had a lot of word problems and whoever was the fastest would explain how to solve it to others. So basically somebody was the first to realize you don't need to do 2+2+2+2+2+2 when you can calculate the solution as 6+6. Kid got reputation boost for being smart in front of others and others stole the technique to be the fastest the next time. This got me into math.

That is entirely the point. The number line is awkward for multiplication.

But mapping the relationship between multiplication and the area of rectangles is very visually intuitive. Geometric reasoning is powerful. Being able to use your understanding of multiplication rules to derive the formula for the volume of a cube, or a cylinder, or whatever is probably more insightful than realizing some arithmetic tricks.

> That is entirely the point. The number line is awkward for multiplication.

Strong disagree. The number line is one of the best tools for understanding multiplication of numbers (not just whole numbers, but any decimal). The problem is that it is no longer taught. Example: American rulers have both metric (cm) and Imperial (inch) units. You can use it to visually multiply and divide by 2.54.

To generalize multiplying positive numbers x and y, (1) Mark x on a number line, (2) create another number line with the number 1 placed where x would be on the original line, (3) find y on the new number line, and (4) the corresponding point on the original line is the product x*y. The point: Multiplication is scaling (stretching, shrinking, whatever you want to call it). The Common Core tries to address this.

As a kid, the row x column thing never made sense to me. I was horrible at understanding visual metaphors, I don't think I "got" the row x column thing until well into high school, and even then it was "I have to think about it".

I always had better luck with just manipulating the numbers mentally.

Metaphor seems like a poor term. Mapping concepts to geometric concepts is pretty important. Multiplication is simple enough that you can just do it.

But having that capability is pretty important for understanding other concepts. For example, many concepts in calculus.

Visual explanations of geometry, trig, and calculus made sense.

But as a kid I never got the multiplication thing. Even now as an adult I understand what they are trying to do, but it isn't "obvious" to me.

Then again I am one of those people who doesn't read comic books because all the pictures get in the way of the words. Heck as a kid I read illustrated books and didn't realize until I had finished the book that it even had pictures.

I'm not sure that's a great example.

The principles of multiplication are pretty straightforward. However, especially before calculators were commonplace (one can probably debate if there's sufficient value in memorizing multiplication tables today), there was a lot of benefit in doing multiplication and other simple operations quickly and mentally. There are a ton of basic things that we memorize without necessarily knowing how to get there from first principles off the top of our heads.

This is true with programming as well. Sure you may look up APIs or specific syntax but if you had to look up every detail every time you put fingers to keyboard, you'd be really slow.

>I felt like a total failure. It's much faster to memorize vast multiplication tables and recall them from memory than trying to understand first principles.

and this is why kids are taught to memorize these lookup tables. When doing algorithms such as long division or multiplication these lookup tables can make the process much faster.

>This is actually a really good point, but unfortunately this is actually how public education through high school tried to teach math.

Also university-level math for everyone who doesn't get up to taking proof-based analysis.