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What research has been done on the effects of requiring students to learn to count and do some easy arithmetic in an alternative number base, for example binary, base four, base six, base eight, base twelve, base sixteen, or base twenty?

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    $\begingroup$ What level of "students" do you mean? I think it's very important for pre-service teachers to compute in a variety of bases to truly solidify the base-ten they'll be teaching. I'm more ambivalent about, say, actual middle-schoolers computing in different bases. Perhaps just for "advanced" kids. IMHO, High-schoolers should probably see base-2 to inform their understanding of computer science. $\endgroup$
    – Aeryk
    Apr 16 at 15:49
  • $\begingroup$ @Aeryk I am asking about all levels including kindergarten. Good point about solidifying base ten knowledge. Why the ambivalence? Agree with you about the high-schoolers. $\endgroup$ Apr 16 at 16:17
  • $\begingroup$ See math.stackexchange.com/q/382774/18398 $\endgroup$ Apr 16 at 16:49
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    $\begingroup$ This seems overly broad. $\endgroup$
    – user507
    Apr 17 at 1:53
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    $\begingroup$ There's too many questions here. $\endgroup$ Apr 17 at 6:43
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I can't answer the OP's questions, but I'll just mention that a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5$. It is mentioned in the recent article below. Sometimes he called it "star-fish math."

  • James Henle. "Math for Grades 1 to 5 Should Be Art." Mathematical Intelligencer. 42, pages 64–69, Dec. 2020. Springer link.

"And then there is numeration. Kids can invent a method of naming numbers.

[This teacher] used to tell his sixth-graders every year the tale of the mysterious mathematician Professor Étoile, who invented a new way of notating numbers using the letters $A$, $B$, $C$, $D$, and $O$. His method was lost, [the teacher] explains, when he fell off a boat and was eaten by an octopus.

[The teacher] then divided the class into groups. Each group was tasked with creating its own system of numeration using $A$, $B$, $C$, $D$, and $O$. After a week, they presented their methods in a mini-math conference. [...]

[He] did this every year with great success. His students came up with many creative systems."

Of course étoile is French for: star!

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    $\begingroup$ The article is behind a paywall, so a summary would be nice. $\endgroup$
    – BKE
    Apr 16 at 20:18
  • $\begingroup$ I second that... $\endgroup$ Apr 16 at 20:53
  • $\begingroup$ @BKE & Matthew: I have now quoted the relevant sentences from the article. $\endgroup$ Apr 17 at 0:14
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I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true.

Ideally I'd like math and CS students to learn base 2, 8, and 16, and then learn how base 10 notation is related. From there students can do whatever they want; I sometimes teach student base 36 and use 0-9 plus A-Z. After they learn the general concept of base-$n$ representation, it's all the same.

Using some familiar everyday objects to represent bits can be fun for teaching binary. For example, asking students how they might build a binary clock like the one below can generate some good ideas:

Binary clock

Or, ask students how they would do this if all they had was a single flag that they could either hold up or put down. Or my personal favorite, counting to 31 using only the fingers of one hand (literally, "digits").

For hexadecimals, this scene from The Martian is really good.

Finally, should language departments teach this? Heck no! Why should they get all the fun?

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  • $\begingroup$ I thought perhaps it might be a way to make room in the curriculum for a thorough mastery (fluency) in counting and addition and subtraction, say, in one or more alternative bases by allowing a student to learn them instead of French say, while studying the same amount of multiplication, division, geometry and so on, in base ten. In other words, categorize it as foreign languages. The language lab and foreign language teaching expertise is in the language department, too. Just as getting fluent in French does not mean you can write or even appreciate great French prose, mastering an alter- /1 $\endgroup$ Apr 16 at 20:49
  • $\begingroup$ native base does not mean you can do or even appreciate advanced math in that base. But you will be able to read, write, speak, and listen, in the alternative bases, and thus be able to compare them with each other and with base ten. That's why they should have some fun. /2 $\endgroup$ Apr 16 at 20:52

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