Navajo Code Talkers, Plato’s Cave, and Dots

Since the students have been creating their own codes in an attempt to solve our signaling problem, we began class 3 with a lively discussion of the history and cryptology of the Navajo Code Talkers of WWII.  Almost everything in this discussion was new to everyone, but V helped out by explaining what the Marines and anthropologists are.  Then we returned to last week’s question of whether we had really solved the signaling question.  We reviewed both codes that the students invented, noticing that these codes were really codes representing another code.  (Remember, in the original problem, “27” meant “safety,” and the question was how to signal it without enemies deciphering it, since we don’t know whether the enemies had cracked that code.)  As we were congratulating ourselves on devising not just one code for a code, but two, S proposed that we could embed the message more deeply by alternating codes by day.  So we had a code for a code for a code.  Not bad!  I announced that the students were now ready for Level 3 of this question.  In Level 1, the people in the house could move the 5 candles from room to room; in Level 2, they could not.  In Level 3, the challenge is more like that of the Navajo Code Talkers:  there are many messages that must be encoded.  In fact, there are 30.  Could our codes handle that?  The students thought for a second then shook their heads “no.”

We took a break from this problem to finish class with another round of Exploding Dots.  To insure that the students figured out and could apply the rule, I placed the dots in the boxes in evenly spaced rows of five, instead of the random placement we used last week.  I asked them what I was doing differently.  When everyone figured that out, we talked about that often ignored area of mathematics:  counting strategies. Then I asked them to predict explosions (“How many dots must be added to generate an explosion?” – essentially a subtraction exercise), which everyone was able to do.  So I asked them to predict what would happen if I placed 30 dots in the right-most box.  Many ideas sprang forth:

•             “Write down all the dots and move them 10 at a time.”

•             “Move them all at once and make it explode 3 times.” (D)

•             “Just put 3 dots in the middle box.” (A)

•             “Just point to us without writing all 10 down and we do the explosions.” (M)

Most people agreed that all of these strategies would work, and wanted to dramatize the explosions.  I proposed putting all 30 dots in that box.  A expressed doubt that I could do it.  So I did, giving us another chance to play with counting strategies.  We ended on the following notes:

•             a reminder that next week we will enter Level 3 of our signaling problem

•             a suggestion by X that we consider Morse Code

•             and a suggestion/demonstration by V that Pig-Latin might help.

I’ll let my assistant R take over the writing now, as this narrative moves into class 4.

This week’s Math Circle began with a game of Exploding Dots. The children were slowly trickling in, and, as Rodi explained to them, she had a story she wanted to read them. But, since she wanted to wait for everyone to arrive first, a game of Exploding Dots began.

The game had hardly begun when A entered the room, carrying a colorful contraption. All eyes were immediately on her, and everyone started talking at once.

“What’s that?” most of the group chorused.

“It’s a Morse Code generator… Each letter has its own code,” she explained “Like if I wanted to do M, which is something like ++-, I would press the button.”  Everyone was captivated.

A went on to explain that, since her generator used sound, you could change the pitch of the sound by pressing another button. “Why would you want to change the sound?” Rodi asked.

“Maybe a high pitched long long short would be different than low pitched long long short,” S suggested.

“They don’t have voices so it would have to be with light,” X reminded the group.

“They don’t have voices?!” J exclaimed, half kidding, half not. After all, this was a hypothetical situation.

Moving on, A said, “We’d have to make our own Morse Code.”

“Would the enemy know Morse Code?” Rodi asked.

“Probably,” X replied.

“Or we could use real Morse Code,” D suggested.  But the enemy might know it, the group reminded him.

(Now the writing returns to Rodi.)    We set aside the idea of our own version of Morse Code for future consideration.  Since our group has also been very interested in using shadow puppets as a solution to our problem, I told them a story about shadow puppets: Plato’s Allegory of the Cave.  R and I read the dialogue, with her playing Socrates/Plato, and I playing Glaucon.  About halfway through, our Circle member C arrived.  Sadly, V, who still hadn’t arrived, was absent.  This group is starting to develop a collective curiosity, and it felt different without one of our members.  C’s arrival was a welcome chance for the children to recap the story for her.  No one individually had any clue what was going on.  But the group made sense of it for her because of 2 things:

1)            Collaboration:  each child understood different key phrases and words and could contribute the part he or she understood.  For instance, X has been tuned in to Plato’s view of the role of the Guardians.

2)            Modern references:  somehow the song “Mother Knows Best” from the film Tangled came up in relation to this story, and between that song and the idea of shadow puppets, kids got the idea.

One idea that most kids did not get is that this story is allegorical.  Initially, people thought it was a historical account of an actual event (“Why were these 5 babies kidnapped?”  “Did Plato ever let the rest of them out?”).  Once I explained the general concept of allegories, we were able to go on and discuss the symbolism of the sun.  The person who got out of the cave “was enlightened!” announced X excitedly.  And speaking of the one who got out, M was thinking very deeply about this story.  Several times during the reading, I kept kids focused and thinking by asking for predictions about what might happen next and why.  She predicted how the people in the cave would react when the one who got out returned.  Finally, I asked the group how this story relates to what we already know about Plato’s philosophy, and to mathematics.

(The narrative now returns to R.)  After the interesting Plato discussion, Rodi moved the group back to the table.  She gave them two numbers to add, noting that they should “just try”. The numbers were 279 and 568.

She explained that they could use whatever method they wanted to add the two numbers and that she was going to use the “exploding dots method.”

“So, wait, we’re trying to add with exploding dots?” someone questioned.

“Any method you want,” Rodi answered. 

There was silence while everyone worked. Some grabbed a pencil and started writing, or drawing, while others just sat there, deep in thought.

After a few minutes of intense concentration, Rodi asked if anyone had any numbers. C and a few other children hesitantly raised their hands. The numbers were: 847, 837, 747, 843, and 847.  D rechecked his math and got 847 the second time.

Then Rodi explained her method. She had lined up the numbers, drawn a line underneath them, and added a plus sign. Underneath the line, she had drawn three boxes.  In each, she had drawn dots. In the box underneath the 9 from 279 and the 8 from 568, there were 17 dots, because 9+8=17.  She went on to show them the sums in the two other boxes.

In the end her number was 7|13|17.The children looked dubiously at it.  Someone pointed out that their numbers all had three digits, while hers had 5.

“It’s still right though, isn’t it?” She responded.

“That would be too many numbers,” explained A.  My dad told me, he teaches me math, that you have to carry the one.”

“What’s carrying?” Rodi asked.  No one answered right away.

“It’s just something that you do,” someone said. Since it seemed as if no one was exactly sure what carrying was, Rodi went on explaining her Exploding Dots, hoping for the group to discover carrying on their own.

Since we had done Exploding Dots before in this Math Circle, the kids already knew what to expect. Before, there had been three boxes as well, each with only a certain number of dots allowed in them before they exploded. Going from right to left, each box represented units, tens, and hundreds.  The same rule was applied today.

“So, if only 9 dots are allowed in the right box, and there are 17 in there, do we need to explode?” Rodi asked.

Yes! BOOM!

That left 7 dots. Moving over one box, which before had 13 dots, but now had 14. BOOM!

Now there were 4 dots in the middle box, and 8 in the last box: 847!

(Rodi reporting now.)  After this problem, we added 525+546, using Exploding Dots, but with only our usual 3 boxes on the board.  The group debated what to do with the extra digit in the answer, and finally agreed to add another box for the thousands place value, getting 1,071 as the answer.  Before agreeing on this, we had the answer 10|6|11 on the board.  I asked them whether it was okay to write the number like that.

“It’s not proper,” said the kids.

“What do you mean by proper?” I countered.

“It’s not correct,” they clarified.

“What do you mean by correct?” I asked.

“It’s not right,” they insisted.

“What do you mean by right?” I demanded.

“The opposite of left…” X trailed off.

“…and the opposite of wrong!” the rest of the group chorused.

Also, “It’s not good,” they explained.

Finally, I let them off the hook.  I told them that questioning everything and defining your terms are important skills in mathematical thinking.  Then I explained that 1,071 is a convention we use so that everyone can communicate mathematically.  I needed to give the definition of “convention,” since one student mentioned having heard of beer conventions.  Then I stated that the way I showed the answer on the board, with the vertical lines and seemingly extra digits, was technically valid in mathematics – it’s just not the convention.

“What do you mean by ‘technically?’” demanded J.

Score one for the emerging skills of questioning and defining!


Rodi and Rachel Steinig


NOTE:  For background information, see these helpful websites:

Navajo Code Talkers:,,

Plato’s Cave:,  We also used a site that explains the allegory with outstanding clarity, but very bad language.  Email me directly for this reference (


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