Math and Young Children

    Have you ever wondered what it's like for young children who are just beginning to learn math? I have, especially since I started working on my Early Childhood Education Associate degree.

    While some people try to show this concept to college students training to become teachers by forcing them to use a different number system (such as base 8 instead of base 10), I'm not that mean--I'm much, much meaner.

    See, while using a base 8 number system for the first time will mess with your head after a lifetime of using base 10, kids aren't transferring from one number system to another--they're learning a number system for the very first time, plus a whole set of new-to-them terminology that they have to memorize. So, in addition to giving you a base 8 numbering system to work with, I am also going to provide you with new terms to use. However, I'm too lazy to invent new symbols to represent my numbers, so you're just going to have to work with the whole words.

    Before I get to that, though, let me explain a bit about how number systems work.

    In our base ten system, we have ten total symbols for our numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we count, we count from 0 to 9, assuming there's an invisible zero on the front of each number (01, 02, 03, 04, etc.). When we reach ten, we run out of unique numerals to use, so what do we do now? To make this work, we add a place value by turning our invisible zero into a visible 1 and resetting the 9 back to zero (from 09 to 10). This is why we do math while accounting for a 1's place, a 10's place, a 100's place, and so on.

    Now that we're at 10 and have "reset", we continue counting from the 0: 10, 11, 12, 13, and so on. When we hit 19, we have to stop again and reset again. We don't want to add another place value just yet because we haven't maxed out our first number--only the second. So we turn the 1 into a 2, reset the zero again, and have 20.

    If we were to choose a different number system (such as base 8), the rules would work exactly the same way: count from 0 to 7, then turn the invisible front zero into a 1 and reset the 7 back to 0. This is where things start to mess with our brains, because, used to the base ten system as we are, we're used to calling "10" "ten"--and in this case, it's only eight. 

    Now, because the numerals of 8 and 9 don't exist in this system, we're going to count straight from "eight" (10) to "eleven" (11), and from there count up to "seventeen" (17). At this point, we have to reset again, because we have no numeral for 8 or 9. So from 17 we go to 20, from 27 to 30, and so on.

    Numerically speaking from a base-10 standpoint, we're essentially skipping two numbers with every set we count. From a base-8 standpoint, we're not missing anything, because 8 is 10, and 9 is 11, and 20 is 16, and 30 is 24, and so on.

    When adding in base-10, we tend to base things around the numbers' relationship to the number 10--such as the fact that 9 is 1 less than 10, so you can just add 10 to a number instead of 9 and take away the extra 1 after. (4 + 9? How about 4 + 10 - 1 = 13?)

    To add in base-8, we use the same principle. (4 + 7? How about 4 + 10 - 1 = 13?) The tricky part here is that the values are different, so what looks like "13" in base-8 is actually "11" in base-10, and vice versa.

    But what if we're working with bigger numbers?

    Same principle. Remember, base-10 has two symbols that base-8 doesn't have, which throws off the values we assign to different groups of symbols. That's it. So, in base-10, 13 + 6 = 19. In base-8, 13 + 6 = 21 (and to be fair, I did have to count on my fingers to work that out, just like preschoolers and kindergarteners do when they're first learning base-10 addition). However, the values are different: the base-10 "19" is 19 as we think of 19 in our base-10 training (exactly as you'd expect). The base-8 "21" has a value of 17 as we think of 17 in our base-10 training. If you want to get a value of 19 as we think of it in base-10 using base-8, you would want to add 15 + 6 = 23 (As I counted it on my fingers: 15 + 2 = 17; 17 + 1 = 20; 20 + 3 = 23).

    Do you have a better understanding of why math is so hard for kids now?

    What's more, if you are an adult reading this post, you actually have an advantage for learning a new math system compared to kids, because your brain is more fully developed and has learned to understand abstract concepts like this. In kids, the ability to think abstractly doesn't start to develop until around age 8, and they aren't fully capable of totally abstract thought until sometime around or after age 12 (at least according to Piaget).

    In other words, if you're confused, then no wonder your preschooler/kindergartener just learning basic math is confused. Furthermore, because they think so concretely, it is also no wonder that using concrete objects (commonly referred to as manipulatives in my field) is so helpful for them, because they are legitimately, developmentally unable to have abstract thoughts that aren't tied to physical objects. This is also why kids will often rely on math tricks to help them calculate numbers, because it's easier to repeat a rule about how numbers add up then it is for them to do the math in their head (for example, "When adding a number to 9, add the number to 10 instead and take away 1."). Later in life, these tricks become so ingrained that we don't even think about them anymore and can do the calculation automatically.

    However, if I switch over to base-8, I can only math at the level of a kindergartener--I no longer know how to do algebra, and I have to count on my fingers to pull off the simplest of addition and subtraction problems. I haven't even thought about multiplication and division yet; I'd need to group dots on a piece of paper to start putting my multiplication table together.

    If you think there's an ideal numeric system that makes more sense than base-10, I'd love to hear about that! I've heard some people say that a base-5 or base-12 system makes the most sense because we have 5 fingers or 12 joints on our four non-thumb fingers, respectively, which makes it easier to finger-count (the only trouble I have with base-12 is that it would require two new numerals for 10 and 11 so that 12 can be the "place value reset number" at 10.). The only reason I went for base-8 in this post is because that's the number system I ended up with whilst creating the giant list of new number terms you'll see below.

    Also, if you don't want to bother reading the giant list of strange number terms, you don't need to--I made the point I wanted to make here. However, if you're interested, I recommend reading the first 16 entries on the list to understand how all the numbers are put together, then skim to the end to read the thousands and such out loud and laugh at how ridiculous and tongue-twister-y it is. It's not practical in the slightest, but it is hilarious!

    All right, with all the sensible explanations and mechanics out of the way, how about I introduce you to a totally crazy system of numbers that will make you glad you didn't have to do math this way?

    What we're working with here is a Roman-Numeral-esque system of numbers, but it's base-8, not base-10, and there are no simple symbols to shorten things up and make everything easier to keep track of; I was too lazy to start assigning symbols to my number values, and I didn't want to have to memorize what names went to what symbols when all the names are similar enough to make my brain freak out, even though I am not at all dyslexic. (If you are dyslexic, I apologize for the travesty my older brother and I have created.)

    For the sake of space in my number system chart, I counted by one's up to 96, skipped up to 100 to help my own brain (since I'm a base-10 gal through and through), counted by 100's to 1000, and then counted by 1000's to get to 5000 (I didn't really want to go any higher at this point, for good reason). With this information in hand, you should be able to deduce any number you need to by taking the terms for your chosen thousand and tacking the terms for your chosen hundred, ten, and one onto the back. Note also that the higher the place values get, the more parentheses I have to add to make it clear which terms are contributing to which place values. Just as in our number system, the term sets that come first are higher place values than the ones that come later, so just because 110 takes about twenty times longer to say (and is even worse to spell out), the hundreds, tens, and ones are all in the right places.

    Without further ado, here it is:

1 – Single (01)

2 – Couple (02)

3 – Triple (03)

4 – Double Couple (04)

5 – Double Couple Single (05)

6 – Double Triple (06)

7 – Double Triple Single (07)

8 – Quouple (10)

9 – Quouple Single (11)

10 –  Quouple Couple (12)

11 – Quouple Triple (13)

12 – Quouple Douple Couple (14)

13 – Quouple Double Couple Single (15)

14 – Quouple Double Triple (16)

15 – Quouple Double Triple Single (17)

16 – Double Quouple (20 -- If I wanted to make this a base 16 system, I could call this number a Douple, but I thought about it too late, and so we're just going to have to deal with base-8.)

17 – Double Quouple Single (21)

18 – Double Quouple Couple (22)

19 – Double Quouple Triple (23)

20 – Double Quouple Double Couple (24)

21 – Double Quouple Double Couple Single (25)

22 – Double Quouple Double Triple (26)

23 – Double Quouple Double Triple Single (27)

24 – Triple Quouple (30, and so on)

25 – Triple Quouple Single

26 – Triple Quouple Couple

27 – Triple Quouple Triple

28 – Triple Quouple Double Couple

29 – Triple Quouple Double Couple Single

30 – Triple Quouple Double Triple

31 – Triple Quouple Double Triple Single

32 – Quadruple Quouple

33 – Quadruple Quouple Single

34 – Quadruple Quouple Couple

35 – Quadruple Quouple Triple

36 – Quadruple Quouple Double Couple

37 – Quadruple Quouple Double Couple Single

38 – Quadruple Quouple Double Triple

39 – Quadruple Quouple Double Triple Single

40 – Pentuple Quouple

41 – Pentuple Quouple Single

42 – Pentuple Quouple Couple

43 – Pentuple Quouple Triple

44 – Pentuple Quouple Double Couple

45 – Pentuple Quouple Double Couple Single

46 – Pentuple Quouple Double Triple

47 – Pentuple Quouple Double Triple Single

48 – Hextuple Quouple

56 – Septuple Quouple

64 – Octuple Quouple

72 – Nontuple Quouple

80 – Decatuple Quouple

88 – Hendecatuple Quouple

96 – Dodecatuple Quouple

100 – Dodecatuple Quouple Double Couple

200 – (Double) (Dodecatuple Quouple Double Couple)

300 – (Triple) (Dodecatuple Quouple Double Couple)

400 – (Double Couple) (Dodecatuple Quouple Double Couple)

500 – (Double Couple Single) (Dodecatuple Quouple Double Couple)

600 – (Double Triple) (Dodecatuple Quouple Double Couple)

700 – (Double Triple Single) (Dodecatuple Quouple Double Couple)

800 – (Quouple) (Dodecatuple Quouple Double Couple)

900 – (Quouple Single) (Dodecatuple Quouple Double Couple)

1000 – (Quouple Couple) (Dodecatuple Quouple Double Couple)

2000 – (Double Quouple Couple) (Dodecatuple Quouple Double Couple)

3000 – (Triple Quouple Couple) (Dodecatuple Quouple Double Couple)

4000 – (Double Couple Quouple Couple) Dodecatuple Quouple Double Couple)

5000 – (Double Couple Single) (Quouple Couple) (Dodecatuple Quouple Double Couple)

    If you want a number that's more than just basic multiples of 10, 100, and 1000, here's 5111: (Double Couple Single) (Quouple Couple) (Dodecatuple Quouple Double Couple) (Dodecatuple Quouple Double Couple) (Quouple Triple).

    So, number chart in hand, imagine yourself as a small child learning to count with these numbers, with no prior knowledge with numbers (and therefore no handy numeric translations on the left of the table) to work with. Because of the way the terminology has been created (thanks, in part, to my older brother and older sister), this is an extremely clumsy way of counting, but it's actually very similar to Roman Numerals--the only differences being that Roman Numerals are in a base-10 system with symbols, and this is base-8 without them.

    Now, I'm going to give you two math problems: one addition and one subtraction. First, I will show the problems in the crazy-math system. Then I'll show you what they are in our normal base-10 system.

    Please keep in mind that I have intentionally kept the math problems simple: Neither addend will be over 20, which means that the end sum can be no higher than 40. Similarly, the higher number in the subtraction will be no higher than 40.

    Also, as I explain my solutions, I am bolding the parts of my answers as I get them.

Crazy Number System: Addition

Quouple Double Triple + Quouple Double Triple Single
To solve this problem, I'm going to need a few steps:
First, I'm going to add my Quouples to get a Double Quouple.
Second, I'm going to add my Double Triples. This is a two-part process, as a Quouple is a Double Triple plus a Couple, which leaves me with a Double Couple left over. This means that adding my Double Triples will leave me with another Quouple and a Double Couple.
Third, I'm going to tack on my Single.
Fourth, I'm going to combine all my Quouples into a Triple Quouple.
This leaves me with a Triple Quouple Double Couple Single.

How much easier would this math problem have been if I just wrote it as 14 + 15? Our normal method is much simpler; just add the 4 and the 5 (making 9) and leave it in the 1's place, then add the two 10's (making 20) and leave it in the 10's place (making 29). Another way to do this is to recognize that 14 is 1 less than 15, and that 15 + 15 = 30. Therefore, 15 + 15 -1 = 30 - 1 = 29 (Which is how I'm usually more inclined to do my math when the numbers aren't written out in front of me).
However, the mental gymnastics we went through to work out the math problem in the Crazy Math system are extremely similar to the maneuvers children have to make in their minds as they learn our base 10 system. While mine is significantly more complicated (mostly due to the lack of symbols/numerals to represent terms), we use the same grouping techniques in both, and the only reason we don't realize we still use grouping in mental math as adults is either because we've memorized the arithmetic for lower numbers already, or because we've gotten so quick at it that we don't have to think through every step individually anymore.

Crazy Number System: Subtraction
Triple Quouple Double Couple Single - Quouple Double Triple
Even though this problem is the inverse of the last one and technically has fewer steps, I actually found it more complicated than the last one, which I understand to be true for children who are learning subtraction in general. If you're ever teaching a child math and they pick up addition easily, but struggle to no end with subtraction, that is because this process is inherently more complex and more difficult to understand.
Starting with my Triple Quouple Double Couple Single, I'm going to systematically remove elements of my Quouple Double Triple.
First, I'll remove the Quouple, leaving me with a Double Quouple Double Couple Single - Double Triple.
Second, I'm going to take the Double Triple out of another Quouple, leaving me with one full Quouple and a Couple leftover. This leaves me with a Quouple Triple Couple Single.
Third, I'm going to translate my answer into standard form--a Triple Couple is the same amount as a Double Triple, and the Double Triple is more easily recognizable as its own number, so I'm going to flip the terms around, leaving me with a Quouple Double Triple Single.

Once again, how much easier would this problem have been if I'd just written 29 - 14? Once again, we can recognize that 29 is 1 less than 30, and 14 is 1 less than 15. By adding 1 to both, we recognize that 30 - 15 = 15, which is the correct answer. Otherwise, we could just do 9 - 4 (which is 5) and leave it in the 1's place, and 20 - 10 (which is 10) and leave it in the 10's place (creating 15).

    In other words, if you hate math, go through this process and tell me that our regular math process doesn't make sense (and if this version does make sense to you, I applaud you for your insanely wacky brain; please ask me to put together a base-10 version of this list to help format traditional math to fit your brain better). I was a stout and stubborn math-hater as a child, but doing two simple math problems in the system I have here has pretty well convinced me that traditional math is much easier. It's so much nicer to have proper numerals to work with and a system that makes sense, with unique names for each numeral and sensible ways for the numbers to mix together as you go up the place-values.

    At any rate, that's all I have for today. I'd love to know what you think about all this mathing I've done, and if you see any mistakes please let me know, because I think my brain shut down partway through the giant list of strange terms and I ended up just following patterns until I gave up.