What the colors mean (part 1), plus an exploration
about 4 years ago – Thu, Apr 30, 2020 at 08:46:46 AM

There are 162 cards in the multiplication by heart deck. Of these, 100 are array cards. And if you lay out those 100 array cards just so, you get a 10 by 10 multiplication table that's quite beautiful!

Whenever you're designing something like this, you need to decide what colors to use and how to use them. In this case, we wanted the colors to help with the way the deck is used. And they do that. But they also allow us to see some interesting and counterintuitive structure in the multiplication table. 

Let's zoom in on the top left corner of the table. Is there anything we might find with those L-shape arrays of the same color? 

I'll tell you that there are exactly 64 of the little blue squares, which you can check if you add up all the blue arrays. They're all of the form 4 x __ or __ x 4, with a number less than or equal to 4 in the blank. But there's a quicker way than just painstakingly adding. 

Try putting the pairs together as matched up by arrows. We end up with four 4 by 4 squares: one square for each pair joined by an arrow, plus the one that's circled. That's 4 cubed = 4 x 4 x 4 = 64 squares in all! Pretty cool. 

So how many orange squares are there? And can you find the sum of the numbers on any of these L-shapes? What do you notice?

In mathematics, these trains of thought can lead to formulas that are at once simple and intimidating. Like this one:

But when you play with the visuals, it's a reminder that every line of complex math symbology is a reminder of an observation or argument that made sense. Math is incredibly powerful, and sometimes that power hinges on a simplicity and conciseness that hides its tracks. But good visuals keep us alert to the meaning of things, and allow us to do more than just remember. When we understand, we can extend ideas to new situations, and even create new mathematics. 

It's easier to see why cubes add up to squares when you've connected multiplication to arrays in the first place. You might even find yourself with ideas of how to extend the idea.

I want to end with a shout out to another Kickstarter project I think you might enjoy. It's a coding toy for young kids that's entirely screenfree, with a Montessori aesthetic. I've got a 1-year-old now, and I'm aware of the way screens permeate the lives of kids. This project seems pretty awesome in how it allows for creative play with the concepts of coding, without having to put a young kid on a tablet. 

CODY BLOCK: introducing coding one block at a time. Ages 3-9

Cody Block is a Montessori inspired, screen-free wooden toy for  children 3+, introducing the basics of computational thinking through  tangible programming. With Cody Block, children can play for hours  without the need of any smartphone, laptop or screen. We have developed a  specific RFID technology (patented!) that is embedded in every wooden block.

GET IT HERE

Spaced Repetition & Visual Flash Cards
about 4 years ago – Mon, Apr 27, 2020 at 06:08:33 PM

I’ve been promising an explanation of spaced repetition, and how the science of memory and learning fits into our visual flash cards. Here’s where I deliver on that promise.

This explanation draws extensively from Nicky Case’s How to Remember Anything Forever-ish.  If you’re interested in the long version, I recommend you check out that webcomic. All the comic panels below are from there.

How to Memorize Better

Memories decay, and repetition is one of the of central ways we stave off that decay. The problem is, you don’t get much bang for your buck if you repeat something ten times in a row. But space the repetitions out, and you see real gains. The greatest gains come if you recall that fact just before it vanishes from your mind. 

Space the repetitions right so you’re hitting the “sweet spot” of maximum gain, and it not just refreshes the memory, but slows the future rate of memory decay. 

 In the chart above, the “sweet spot” is the yellow band. Notice how you need more recall opportunities at first. Then they get spaced out further and further. Thus, spaced repetition.  

 We’re now tinkering with the design of the Spaced Repetition Guide from the first Stretch Goal. The idea is that you practice cards more at the earlier stages. 

Get a card right and it levels up to the next pocket! You’ll see it again, but there are longer gap before you do. Miss it at any stage and it goes back to the beginning. No problem! It just means you need more practice. Get it right at the top stage and the card graduates out of your practice set, and into your deck of “mastered” cards. You’ll still use these for games and explorations. And you’ll be seeing some of those same problems in new forms later! 

Instead of doing all of your multiplication facts every day, you follow a schedule and just work on a small subset. Since the cards automatically sort themselves as you get them right and wrong, you end up practicing the ones that are harder more often, and the easier ones graduate out earlier. You spend less time practicing, and that time is spent more efficiently. 

That’s the basic idea. But there’s more. 

The Power of Visuals

Want another fun dive into the workings of memory? Check out Radiolab’s great piece on Memory Palaces. 

A key idea in that piece is that memorizing words, facts, numbers, and so on without context is hard. Our brains are built to remember relationships and places. Having strong visuals and an understanding of how things connect is better than remembering facts in isolation. 

The analogy I often return to is remembering lines for a play. It’s a lot of words, and if they’re just words, it’s nearly impossible. But if you understand the meaning of those words, how they connect to the emotions and motives of the characters, they become natural, even necessary. Memorization serves art, but the meaning and emotion in the art serves memorization too.

 And the same happens with multiplication. We introduce the right visuals, the right structures, and multiplication is no longer a set of disconnected facts. It’s a set of relationships and connections. That makes it easier to memorize and more natural.

I'll share more about those connections later, since this update is long already! But a huge motivation for me in creating MULTIPLICATION BY HEART was that memorizing multiplication facts trips up a lot of kids, to the point that it marks the moment, for many people, where they started to decide they weren't "math people." If we can share how math makes sense, is connected, and is worth loving, a lot of kids will have an easier time memorizing their facts. The creative part of math and is supported by a solid foundation and understanding of the basic facts, and vice versa. 

So thank you again for supporting our campaign! We're coming up on the halfway point and with luck we'll be at double our goal when we get there. Keep spreading the word about the campaign! We'll keep thinking on how we can make the deck more awesome.

We reached our first stretch goal!
about 4 years ago – Thu, Apr 23, 2020 at 09:20:46 PM

Thanks to your support, we've hit our first big landmark after funding: our first stretch goal! I'm working on designing it now, so I don't have great pictures to show you. But it's coming together nicely. 

What is it? A spaced repetition guide to help you track which cards you're going to practice today (and which tomorrow). I'll share pictures when I have them!

The next stretch goal is less than $5K away: three mini-games that will be included with the card set, so that in addition to flash cards, you'll also have games and puzzles you can play with the cards.

Keep spreading the word, and we'll keep making Multiplication by Heart more awesome. Onward!

New Stretch Goals added
about 4 years ago – Mon, Apr 20, 2020 at 08:18:20 PM

Because everyone loves to have more cool stuff unlocked as the campaign grows, I've now added stretch goals to the campaign. Keep spreading the word, and we should see the box include a guide for spaced repetition practice, more games, puzzles and explorations, and, if we hit our big target, an app version of the flash cards!

We've just added our 500th backer to the campaign. Excited as we continue forward! 

First STRETCH GOAL, plus a meditation on the distributive property, and a puzzle
about 4 years ago – Mon, Apr 20, 2020 at 08:02:31 PM

I'm still thrilled from the outpouring of support that allowed us to hit our funding goal in less than 24 hours. Now we get to the fun part: STRETCH GOALS! As we hit these marks, MULTIPLICATION BY HEART will get cooler and better. 

STRETCH GOAL NUMBER 1 is at the $25K mark. If we make it, we'll  design and include hardware that makes the spaced repetition recommended with the cards more natural. We still need to design this, but it will be some kind of container that tracks which cards you've reached which stage of memory with, and which ones you need to practice today. Explaining it all can be a little clunky, so having something physical in the box that makes this easy will definitely make these visual flash cards more effective and easier to use. 

In other news, I've been thinking about the distributive property. It's usually summarized algebraically as:

a x (b + c) = (a x b) + (a x c)

A more general formulation is the old "FOIL" rule you learned in algebra class:

(a + b)(c + d) = ac + ad + bc + bd. (First, Outside, Inside, Last. FOIL!)

It's possible to develop an intuition for the algebra, but not everyone does. And students of all ages tend to mess up these rules and formulas. Wouldn't it be nice if there were a visual intuition for all of this? 

Let's look at what happens when we introduce some good visuals. Like arrays, for instance.

It might just be me, but I can actually SEE that the two cards on the left equal the cards on the right. I'm just taking a 5 by 5 array and chopping it into two pieces with a vertical cut, right? 

I can express this almost as clearly in words if I say what I see: 3 groups of 5 plus 2 groups of 5 is 5 groups of 5. Or equivalent, (3 x 5) + (2 x 5) = (3 + 2) x 5 = 5 x 5. 

This is just the distributive property again! In fact, that's all the distributive property really says: that a full array is the sum of the two smaller arrays you get when you cut it with a single cut. 

What about FOILing then? Let's take a look at those pieces, and see what happens if we cut them into even smaller pieces.

 Here's 3 x 5 seen with a horizontal cut, as (3 x 3) + (3 x 2).

And here's a similar cut for 2 x 5.

Now let's put all those small pieces together. 

It's pretty clear visually that the four cards on the right would add up to the card on the right. Right? I just made a horizontal and a vertical cut, and separated the four pieces. If I were to write out an equation for this, it would be (3 + 2) x (3 + 2) = 3 x 3 [First] + 3 x 2 [Outside] + 2 x  3 [ Inside] + 2 x 2 [Last]. It's FOIL! But this time, we don't have to do any of the annoying memorizing. We just make the natural observation that when you cut a whole into pieces, the pieces add back up to the whole. In other words, we could have expressed those algebraic expressions visually this whole time.

This is what the savvy teacher knows. Visual argument tends to be much easier to intuitively grasp. And when students intuitively grasp what they're doing, they do better in math, enjoy it more, and identify as "math people." The way to prep students for understanding math is to use the right visuals all along. 

There ends my meditation on the distributive property. 

Now if you're looking for a puzzle to see how well you (or a kid, perhaps) grasps all this, let me pose one for you. Find four cards (representing arrays) that add up to 5 x 5, but that are not the four cards from this picture. 

If your kid can do that, algebra is doing to be that much easier.