As usual, new school year brings a spate of new learning, until we all get tired and drop everything except reading and math memorization.
Last week, I introduced decimal fractions. This is something that should be introduced in second grade, ending in third. Though I’ve seen it introduced later in other AMS style albums.
Last year, I was very anxious and worried that I didn’t start on decimals or fractions. But I’m finding that it was good to wait. Since Thumper learned her multiplication/division operations last year, it is much easier to introduce decimals and fractions additions this year.
To a certain extent, on paper, decimals are very easy. All you’re doing is manipulating numbers and learning where to place that little decimal point. You don’t really have to understand why it works if you don’t really want to.
Montessori wants you to. So there is a lot of time spent on concrete manipulation of decimal cubes. To understand exactly what 100th or 1000th or 0.00003 means. She really wants the child to know that 0.3 means 3/10 and that 0.03 is 3/100. That 10 of 0.01 is = 0.1.
For specifics on how the presentations go, you can check out Eric Johnson’s presentations on YouTube. It’s presentations 105 and 107. The three presentations basically first shows the children the concrete material, link the material to abstract numeral presentation, then only use the abstract numerals.
Presentation #1
Last week, we had a quick Presentation of Quantity. The presentation starts with mapping the circle fraction metal insets to the decimal numbers. So placing one dark blue decimal square onto a 1/10 fraction metal inset.
Then we talked nomenclature: what do you call 0.1, 0.01, 0.001, etc? It’s one tenth, one hundredth, one thousandths. The important part was discussing their relationship to each other, that they’re multiples of 10 to each other.
Concept-wise, this is the important part. Because it is the opposite of what it sounds like for the large numbers and can be confusing for children. 10 1-hundredths makes 1 1-tenths; but 10 tens make 1-hundred.
We jumped through those quickly because Thumper already has been exposed to decimal fraction in her daily life and she understands the concept. I ended the lesson by asking her to take note on her journal on what she learned today.
I realized from her note, for the first time, how Montessori’s representation of numbers is the exact opposite of how number lines are taught to children. In the primary materials, she’s teaching place values, so you always lay your materials out from left to right, 1000, 100, 10, 1. But in a number line, the numbers increase from left to right, 1, 10, 100, 1000.
Something to watch out for.
Presentation #2-#3
The next week, we continued our introductory presentations Symbol Linked to Quantity and Formation and Reading of Quantity. For example, 0.332 can be labeled “three tenths, 3 hundredths, 2 thousandths” or “332 thousandths”. Now, we don’t say this as adults. But I think it’s there to reinforce for the children what those numbers actually mean.
Yes, for those sharp-eyed, that’s a blue 10 on the 1 unit place. Too lazy tired to go dig out my unit bead from storage.
Chinese Vocabulary
Now, here’s the funky part about Chinese vs English decimals. Chinese actually has names for each decimals place value.
- Decimals – 小數 xiao3 shu4
- decimal point – 又
- 0.1 – 分 fen
- 0.01 － 釐 li2
- 0.001 – 毫 hao2
- 0.0001 － 絲 si
So 1.153 is “一又一分五釐三毫”. In English you can say, “one and 153 thousandths” or “one and 1 tenths, 5 hundredths, 3 thousandths.” We had a lot of fun trying to pronounce the strange ways you say decimals in Chinese.
And then I learned, there’s also a mapping to English counters like milli-, centi-, mega-, nano, etc.
- 0.1 – deci 分 fen
- 0.01 – centi 厘 li2
- 0.001 – milli 毫 hao2
- 0.000 001 micro 微 wei2
- 0.000 000 000 1 nano 奈 nai4
On a side note, while I’m so busy reading up on the different ways they’re named and find it so interesting, I think that is the point evoking interest in the children through presentations. For this presentation for example, if a child were to go and look up different place value names, then it goes into the realm of studying language. Everything is inter-connected! I feel like I’m not quite equipping Thumper with the tools to go all out and follow her interest like this. Either that or she’s not old enough yet.
We spent a long time on this, which is a change from how I used to teach these nomenclature. I’ve learned from the last two years that I cannot quite believe a child when they say they understand. Because there is understanding what I tells her, vs being able to produce or teach someone else. It’s only when she can regurgitate and put in words herself a concept that she really gets it.
Oh, lastly, one thing I learned from fellow public school teachers is that schools teach “and” to denote that decimal point. So you need to be very careful not to use “and” when you describe other numbers. For example saying “one thousand and one hundred eleven” for 1,111, which even I find myself doing colloquially.
In the Montessori album, there is a fun presentation used to teach kids to read decimal numbers without the material. You tell the child the board is the kingdom of the unit (1). You put a little crown on the unit circle and then place a drawing of a candelabra, stating the unit supports the whole structure.
Other than making an impression, the point of all this is to show that 10 is related to the 0.1, in terms of appearance and also multiples. 10 is 10 times 1. 0.1 is 1/10 of 1. 100 is 100 times of 1 and 0.01 is 1/100th of 1.
In appearance, the 0.1 is like a 10 turned upside down with a little dot drawn in. It is to help the children read the decimal points. Because when you get to 0.001, and you turn it upside down to read it like a 1000, you can now easily see that 0.001 is 1-thousandth.
One reason why I never presented decimal fractions last year was because I was just way too lazy busy to find the construction paper to draw that candelabra. Thankfully, I’ve relaxed around Montessori presentations in a year’s time. I was all set to skip this presentation till I spied a piece of paper and an old toothbrush laying around the table. I grabbed them to represent my crown and candelabra.
If I may say so myself, I think the toothbrush made just as much an impression to Thumper and Astroboy. I applaud my own genius.
In fact, Astroboy, who was supposed to be doing his own work, had to come over once we put that toothbrush on the unit circle, and insisted on giving the toothbrush his eraser as crown.
I can’t stop myself from laughing when I see this picture. A toothbrush in a decimal fraction presentation!
