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I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.

When explaining this in the past, I have never not confused anyone with notation. For example trying to explain the difference between '11' (binary) and '11' (decimal) is hopelessly confusing because it looks like the exact same thing, and saying 'one one' almost always leads someone to think 'well one plus one is two, what is this dude going on about it being 3'.

I think the problem is that we use the same symbol for decimal and binary notation here, while there is not strictly a reason for it (it's convenient if you already know how it all works, but not before that).

So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off', so that I would write e.g.

3 = ●●

and

5 = ●⊗●

Any ideas on this? Would this help? Are there already other symbols being used for this purpose, symbols that don't rely on the glyphs we use for 0-9?

Just to calibrate expectations: at the lower end of that age range, the kids will have only just started learning about the place-value system in decimal. I wouldn't try to leverage knowledge of things like "carrying" in decimal over to binary.

My own kids are at boundaries of your age range, and in first and second grade they use things like "tens frames" and "hundreds frames". You might consider making twos/fours/eights frames or perhaps better: blocks, $1times 2$ joined block pairs, $2times 2$ squares of joined blocks, and $2times 2times 2$ cubes of joined blocks. Then show them (or let them explore) how you only ever need at most one of each block type to make a number. Binary then just consists of a table of the number of each block type that you used.

Use money. Tell your students to image a country has bills that are in denominations of powers of 2: 1, 2, 4, 8, 16.... Now tell them this: For a given number, count out that money with as few bills as possible. You use a "greedy algorithm:"

1. Use the largest bill possible.


2. Repeat this until you have counted out the amount.

Questions to ask:

• Did you ever use more than one bill from any denomination?

They should observe this is never necessary.

Now write a 1 for the largest bill. If the next denomination down is not present, put a 0; otherwise, put a 1. Repeat until you descend through the denominations. You have the binary expansion of the integer you specified.

This is concrete and very easy for students to grasp. This "greedy algorithm" works for any base: Use the largest bill you can can count out as many as you can at each stage.

We made bracelets in APCS Principles (9th - 12th grade). I bought a bunch of blue and white (school colors) beads. Blue was on, white was off. They then made bracelets of their names or initials using 5 bits. A was 1 (w-w-w-w-b), B was 2 (w-w-w-b-w), etc.

Kids seemed to enjoy it, and I caught a few wearing their bracelets walking around the halls days later.

My own kids, 6 and 9, came in after school and saw the beads and wanted to make their own bracelets. By the time they were finished the 9 year old had made 6 or 8 bracelets for different friends and was going to explain how they worked the next day at school.

Only thing I'm planning on changing for next year is adding a different bead color to represent the start / end of the string.

I was going to start by asking what was wrong with $11_2$ and $11_10$, and then I noticed the ages you were referring to. Six year olds!

First, a bit of frame-challenging advice: I'm not sure that counting in binary is what I would focus on at that age at all. You might have better luck with finite state machines, which are great CS concepts that you can play lots of physical games with. You can put states on the floor and have kids jump around, or you can have kids pass picture-notes to one another based on state and observe the results. You can even have them try to draw out simple machines by watching behaviors of other kids acting out your "script".

However, if the goal really is binary counting at 6-8 years old, I would start with symbols like "•" and "◦", or even start with symbols like

□□ □ 
□□ □ □ L |

... using "count the lines" as a property, and then extend that to dots, so

□□ □ 
□□ □ □ L |
• ◦ • ◦ •

... becomes ...

□□ 
□□ □ |
• • •

... becomes 21. And only when they're familiar with the dots, at the very end of the lesson, I'd get into something very silly that people do with this system! They use 1s and 0s for •s and ◦s! If the kids laugh, then your joke landed, and the kids will be beginning to see.

Then see if they can connect 10101 with

□□ 
□□ □ |

Good luck!

Please see the end for an important caveat.

Before you get too deep into this, you should look at the work of Piaget. For example: http://alumni.media.mit.edu/~stefanm/society/som_final.html. Note that children of that age have almost no ability to do abstraction and won't for several years. Whatever you do has to be very concrete.

For a wider discussion, see: https://www.verywellmind.com/piagets-stages-of-cognitive-development-2795457.

In particular, while symbols can be meaningful to them, they are still likely pre-operational.

You might also think about contacting a local Montessori school to see what goes on there with students of that age.

Note, of course, that the reason that they can't grok using old symbols for new ideas is, precisely, the lack of abstract thinking ability. Catch them again around age 12 or so.

So, thinking a bit more on this topic, let me suggest that introducing new symbols might be counterproductive. They may get the idea firmly embedded in their minds that you are teaching them things that are important and should be retained. Then, in junior high or so, they will have to unlearn what you so carefully taught them. In my view, it is a serious error to teach people things they later need to unlearn/forget/ignore/reject.

I'm starting to think this may be a minefield best left unexplored.