Numerical Decanomial with Paper

Age: 7.75

Date: June 2, 2015

Presentation: Numerical Decanomial with Paper Rectangles and Squares

After the Decanomial layout we did last week (or was it the week before?  The days blur…), Thumper is doing the numerical layout this week.

You can watch a video of how it’s done on youtube:

You’re basically doing the decanomial layout using paper.  Now, when I was making my Cultivating Dharma album, I got confused by the writeup because it was not very clear.  I had to cross reference with the video and other write-ups to come up with my current version.  In the video, you will see that the papers are all the same size.  But after doing some research I thought using graph paper and having sizes that are equivalent to their actual multiplication size (1×1, …10×10) is better.

You have 10 envelopes, on the outside labeled Decanomial 1, 2, 3, etc and one labeled Squares 1-10.  On the inside, you have two sets of numbers for each decanomial:

  • Decanomial 1: 2, 3, 4, 5, 6, 7, 8, 9, 10
  • Decanomial 2: 6, 8, 10, 12, 14, 16, 18, 20
  • Decanomial 3: 12, 15, 18, 21, 24, 27, 30
  • etc.

Basically it has the numbers for that decanomial, assuming you haven’t used it already in the previous one.  For example, decanomial 1 is 1×1, 1×2, 1×3, etc.  “Decanomial 2” is 2×1, 2×2, 2×3, 2×4, etc.  But since 2×1=2 was already in decanomial 1, and 2×2=4 is in the “Squares 1-10” envelope, you don’t need to include these.

In the write up, you have the child lay out diagonally the squares 1-10 first, then you build Decanomial 1, 2, 3, etc.  You can talk about the multiplication table, to pick up the tickets in random and place them etc.

What We Did

Given how old Thumper is, and my own laziness, I really did not follow the presentation.  I basically showed her my write up and said, we’re making this table, which is a paper representation of the bead layout you did last week.  I showed her how you would mark off squares and rectangles on the graph paper and cut them out; reminded he she needed to write the value of the rectangle on the paper; that she does NOT have to mark and cut in order.  She could very well do 1×1, 1×2, 2×1, 1×3, 3×1, etc.  I know she knows half of her multiplication table so there is no need for order for us as part of learning process.  I also told her she would glue these rectangles on our Ikea roll.


Work in progress

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十項式排列 Decanomial Layout

Age: 4.75 & 7.75


  • Tables Layout
  • Decanomial Layout: Finding Squares
  • Adjusted Decanomial: Commutative Law
  • Adjusted Decanomial: Tower of Jewels
  • Stacking the cubes

Even though I’ve shown Thumper presentations from squaring & cubing, and squares and cubes, we actually never finished some of the introductory exercises in squaring and cubing.  After reading a blog post about it from What Did We Do All Day, I finally found time to do it with the children on Sunday. What Did We Do All Day has a really great post about the exercises and also the variations within the different albums.  I don’t have the NAMC or Montessori R&D albums, though I do have another really detailed AMS album I used as reference when creating my math album this semester.  Ultimately, since my goal was just to show the children these exercises without accompanying written work, I went with  my album.  Partly because I feel I don’t have a strong grasp of how to implement followup work, but also partly because I just wanted Thumper to have the physical experience without all the written work, which often discourages her from working. IMG_6144

These presentations for me are just exercises that are fun and arouses the children’s interest in the squaring and cubing material, all the while giving them a sensorial experience of what it means to square and to cube.  I took all the related exercises together and we just flowed with it.  The whole thing took about 2-4 hours, with lots of breaks in between.  In the classroom, you would do this with at least two children.  I think 2 is a good number.  Anymore and the layouts get messed up way too easily.

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Introduction to Squares and Square Root – 平方與平方根

Age: 7.5

Presentation: Transformation of a Square, Passing from One Square to Another, Concept and Notation of Square Roots, Extracting a square root for numbers less than 225.

After our introduction presentation about a month ago to squaring and cubing, I gave some more to Thumper on Monday, as part of my studying for my oral exam.  Maybe a terrible reason to give presentations, but, I’m discovering, we’re actually at exactly the right age for learning how to make squares.  She really loved the material and learning.  There were many, “aha!  I know why” moments.  I wish everyone could learn their squares this early!


Finding the 2 squares in the big square

Transformation of a Square is the first presentation.  Here is where you hold up a 10 square, and using rubber hands, divide it up so that it has 2 perfect squares and 2 rectangles.  For our example, we divided it into (4+6)2.  Then you talk about what you just did.  This 10 got split up into a 4 and a 6 and 2(4×6) rectangles.  All this gives you: 10 = 4 + 2(4×6) + 6.


Had to give a bored Astroboy a task to do.

Thumper tried it herself with other squares.  She has not quite figure out that the equation uses only 2 numbers.  So you can predict what the equation is even without looking at the concrete square.  But I was so good, I did not tell her.  Of course, being an elementary child, she wanted to try this “division on a 1-square and 2-square, which doesn’t play as nicely.  After playing with the squares, we wrote down the equation on our whiteboard and made some mental math calculations to verify that it all adds up.  The next exercise in the presentation had her drawing her own squares.  She drew a 17×17 square.  But was unable to write out the equation correctly.


Passing from 5-square to 6-square

We then moved on to Passing from One Square to Another, which she immediately wanted to do when she was playing with the squares during the first presentation.  Because the first thing she did when trying to see what squares fit into the big 10 was to try to use the other bead squares to reconstruct.  In this presentation, you first try to go from a 4 to a 5, write the equation out.  Then you go from 4 to a 9.  Because she had learned how to transform a square, she knew how to write the equation out.  9 = 4 + 2(4+5) + 52.


Passing from 5 square to 9 square

Technically I really shouldn’t have continued on with square root.  But I wanted to practice.  So I just gave her a short intro with the Concept and Notation of Square Roots and Extracting a square root for numbers less than 225, since those just use her knowledge of what a square is.

In these two presentations, you basically learn what a square root is, and then you try to figure out the square root of a number by using pegs.  See how many perfect squares you can build before you run out of pegs.  For example, we started with trying to find the square root of 14.  You do it by first building 1 square, then building it into a 2 square, then 3 square.  You have to stop after 3 square, with 5 as a remainder.

I stopped after that because she needs more practice, especially with her not being too familiar with multiplication yet.  And the next steps starts getting into algebraic expressions, using and instead of numbers.  I think that’s a bit too early for her.

But isn’t the material and concept fun and easy to learn?  I never thought about squares and square roots this way, but it makes so much sense when you are playing with concrete materials.

I find the children get math in Chinese because of the way we count.  BUT, they get confused by the place values because it’s not the same as in English.  So we will continue to teach math in Chinese, with English thrown in when I introduce nomenclature.

Chinese Nomenclature

Here are some words you will need to know to teach squaring and cubing in Chinese.

  • square – 平方
  • cube – 立方
  • square root –  平方根
  • cube root – 立方根
  • radical – 根號
  • radicand – 開方數

Montessori Fraction Charts

After the confusion that was the fraction charts last semester, I sat down and re-did them this week since my fraction section of the album was due.  It helped that I’ve actually seen and illustrated the whole fraction section of the album.  It made organizing and understanding why the charts were there.

I do want to note that my trainer said it’s better if the kids are making these charts.  So I don’t know whether or not I’m going to be printing these out for Thumper.  A few weeks ago, she made a multiplication table when she needed them.  I kind of want the same thing to happen with the fraction charts.  It’s no use as a reference if she doesn’t need them.

I have 22 fraction charts.  It’s based on the album and also copies of my trainer’s fraction charts (which were not complete and also were missing charts or had extra charts).  I also used these three sites as reference:

It’s two more than the first two websites, because of chart 8 and chart 11.  Their charts do not have these “statements”.  One is imbedded in another chart.  But since my album says so, I made mine that way.  There’s also a mysterious chart 22, which isn’t listed in the album itself.  I’m guessing it probably should belong within a multiplication chart, but the trainer didn’t have space for it.  I also made a change from the original charts so it said like and unlike denominators.

  1. Introduction to Quantity, Symbol, and Language
  • Chart 1 – Fraction as circular units divided into pieces from 1 to 5.
  • Chart 2 – Fraction as circular units divided into pieces from 6 to 10.
  • Children may make square charts.
  1. Equivalence
  • Chart 3 – Equivalence for 1/2 fractions
  • Chart 4 – Equivalence for 1/3 fractions
  • Chart 5 – Equivalence for 1/4 and 1/5 fractions
  1. Addition
  • Chart 6 – Addition of fractions with like denominator, ending with whole number
  • Chart 7 – Addition of fractions with like denominator, ending with fractions
  • Chart 8 – Statement: How to add and subtract a fraction with like denominators
  • Chart 9 – Addition of two numbers with unlike denominators
  • Chart 10 – Addition of three numbers with unlike denominators
  • Chart 11 – Statement: How to add and subtraction a fraction with unlike denominators
  1. Subtraction
  • Chart 12 – Subtraction with like denominators
  • Chart 13 – Subtraction with unlike denominators
  1. Multiplication
  • Chart 14 – Fraction times whole number
  • Chart 15 – Fraction times whole number plus rule
  • Chart 16 – Fraction times whole number and fraction times fraction
  • Chart 17 – Fraction times a fraction plus rule
  1. Division
  • Chart 18 – Fraction divided by whole number
  • Chart 19 – Fraction divided by whole number plus rule
  • Chart 20 – Group division
  • Chart 21 – Fraction divided by fraction

Chart 22 – Extra chart on multiplication

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More thoughts & notes on Montessori elementary math

My math class is chugging along.  Once again 1/3 of the album is due and I’m madly catching up after taking 2 weeks off to making Chinese materials.

Now that I’ve seen a lot of the presentations, I feel less anxious and less confused about the math scope and sequence.  Even though if you were to ask me how I ought to teach math, I still could not give you an answer, just like 3 months ago.  But, the only difference is, this time I can tell you why I can’t tell you.

I feel that AMI math training (or at least AMI flavored AMS training) takes a very big view.  Whereas AMS math training (at least the albums I’ve seen) takes the small, detailed view.  What I hear often from my teacher is, “it depends on the child’s interest,” “follow the child’s interest”, “the most important thing is to invoke the interest of the child.”   Yes, there is a general….path, because come on, you cannot really learn the more advanced stuff like squaring and cubing without knowing your addition and multiplication facts.  But, within that, there seems to be some leeway.

Because I’m just giving Thumper a bunch of different presentations whilst we work on our addition/multiplication memorization and multiplication operation, I see this at home.  When a presentation is at that perfect N+1 level, the level that engages her, makes her think, we can spend 1+ hour on it.   Case in point our Multiplication Checkerboard presentation, or our commutative property presentation.  Versus the stuff where she drags her feet and doesn’t want to do, like our attempts at fraction.

As my trainer have told us several times, she would start with a lesson plan for the year, and typically she ends up not following the plan.  Because the children will show her what they’re interested in.

Anyways, obviously saying this helps noone if they’re having the same problems I had, which is, “What do I present for the first year?!”  So here’s the super high overview I now have in my head.

Supposing your kid is in a super traditional program (not many are), then by age 6, they would have, after learning how to count, worked on memorization of +/-/*/÷ , big number operations (+/-/*/÷ 4 digits), been introduced to fractions, skipped counted, etc.  The first elementary year, you introduce them to digits up to 1,000,000.  You do a lot of multiplication of various kinds (commutative property, Least common multiple, greatest common multiples, multiples, etc), factors, continue with +/- probably of big numbers, fractions, intro to squaring and cubing.  And depending on how far you go, you could also introduce long division, divisability, and decimal fractions.

On a side note, squaring and cubing or learning the quadratic equation can start in earnest in third grade!  “Why do kids learn this?” my trainer asked today.  We gave her a bunch of answers like, “It’s good to learn it deeply rather than abstractly, starting from young age,” or “Kids learn better by manipulating materials,” or “it’s good to use multiple senses to learn a concept,” etc.  But the reason is rather simple: “Because kids find it interesting.”  It tickles me to no end hearing this answer.

In any case, look at how many things I could do!  Thumper ought to be interested in at least ONE of these things. And they all work on those basic skills (multiplication, addition etc) somewhere.  I’m still going to figure out my scope and sequence when I get a chance.  But I think it’s more for me.  Just so I have an idea in my head of how the various presentations are linked together.

Back to the class itself….

What I love about my math class is seeing how easy a lot of math concepts are when you manipulate materials rather than numbers.  You start seeing patterns (a state standard that is not explicitly taught in Montessori but is everywhere) in all the materials you’re laying out.  I remember a concept so much better after manipulating the materials with my hands.  There’s also something about seeing a concept illustrated visually.   It really drove home for me something I hear over and over, even in non-Montessori circles, that movement of the hands really helps with learning.

I’ll stop here.  Here are some more “aha!” notes I took from my class.

  • One of the bigger curriculum in elementary math is actually squaring and cubing.  The kids spend a lot of time on it.
  • My trainer didn’t like the higher fraction pieces they sell now a days.  By the time they get to these materials, they’re often too old for them.  Also, 1/11…1/20 is seldom used in equations due to lack of equivalence in many of the numbers.  The goal is to move away from concrete materials, so there’s no need to keep adding concrete materials if you don’t need it.
  • There is no need to do followup work if a child is not really interested. (What a relief to know this!)
  • I finally realized that lesson plans are just like regular schools in that you could plan for multiple concepts to be learned in one year in some sort of sequence.  But this does not mean you’re done.  You will come back again late, maybe the next year with same concept, but at a higher level abstraction.  It isn’t that I work on all fractions till I’m done.  Maybe I do fraction addition for a few weeks, then I move on to another concept.  I don’t have to concurrently run 5 different threads at once.  Whew!
  • The large bead frame and checkerboard are the same concept.  But kids don’t like repetition so this is a way for them to work on the same skill using different materials.
  • Presentations are kept really concise and without a lot of talking or extra info.  The point of these lessons aren’t necessarily to “teach”.  I’m the guide provoking a child’s interest in math.  The real learning comes when they’re manipulating the materials anyway.  So, unlike a teacher, I don’t need to tell them the secret, the trick, the how.  I need to make the presentation interesting so they will want to work with it after I’m done.  If I give all the secrets away, that may be one reason they don’t want/need to work with it after.
  • The stamp game has no 2 digit division.

Everything piling up, especially math!

My first 1/3 of the math album was due tonight.  I didn’t finish.  But we all got an due date extension.  But rather than working on that, I’m starting yet another post (I have 47 draft posts!).  I feel like I have so many balls I’m juggling right now in the air and they’re all in my head.  If I don’t write them down I’m going to stress out.


For our math class, we’ve finished learning about numerations (Least Common Multiples, factors, etc), multiplication of various kinds taking you to 4 digit x 4 digit in abstraction.  We started last week in Division and this week finished it and started on fractions and decimal fractions.  Here’s the scary part.  ALL OF THESE you’re supposed to have covered somehow between first and second grade, mostly first except the advanced stuff.  You’re supposed to at least have given lessons on it.  Thumper is 7.5, technically she is 2.5 years into elementary.  We just started covering this at home.

I told my trainer tonight that I’m getting confused by all the different threads of topics you can seemingly cover all at once because many of the instructions are: “You should cover this at least by age 7.5” or “You start this thread in first grade and study it for 2-3 years”.  She said, just follow the child’s interest.  That made me feel better because I know she’s right.  I already hear in class about children who still need review or study what we’re covering for lower elementary in upper elementary.  So I know you just go at the pace you need to go.  But still it’s freaking me out because there is too much information.  Too much info overwhelms introverts.

That said, I’ve learned a few more thing about the math presentations.  It’s really great I discovered two bloggers who really document what they do and follow the same album I use sometimes.  I can compare their pics with what I’m learning and ask questions.

Anyways, here’s what I learned about multiplication curriculum over the last few weeks.  It’s basically a series of different materials that gradually moves toward abstraction.  My biggest takeaway was that it’s the PROCESS that’s important, not the product.  This is why there are no specified worksheets or specified problems.  Because the goal is for the child to understand the process of multiplication.  They can maybe do 3 problems associated with each presentation.  And they will repeat the process again with another material that is slightly more abstract.  So at the end they would have done quite a few multiplication problems.

This is different from learning the mechanical steps of solving multiplication or even division problems I learned in grade school.  Many other books I’ve read about math education kind of talked about this; that the way we do multiplication or division is just an agreed upon notation to help us when we solve a problem.  It’s not the point.  They talk about how we spend a lot of time teaching children how to write their numbers in a column in lower elementary when many cannot do it because they’re not developmentally there.  Instead we should be doing math problems that teach the concept rather than the mechanical process.

What a relief to know this!  I’m always confused and get anxious when I see children working on equations on blogs because I don’t know where to find them or how many to do.


I’m mostly done with researching how to teach Chinese writing.  Apparently many other people have figured it out before me.  Why am I not surprised.  Huayuworld is your friend.  But anyways I know what to do.  I need to find time to do it.  It’s quite complicated.  There is stroke names, stroke order, character components, component placements, component names, radical names.

Learning character components is a new thing.  It’s not really taught completely in school in Taiwan.  It’s taught a bit in China I think.  It’s mentioned in research papers.  People do it kind of automatically but the system for teaching components isn’t agreed upon in Taiwan.  China apparently has made lots of progress on that front.


You would not believe the state my classroom is in.  It is not befitting a Montessori classroom.  Or the state of the house.  Or my sense of loss of routine because I’ve got too many things going on.  On top of that, our water heater is leaking madly!  And now I have to do some research and get a plumber to replace it.

AND everyone got sick the last 2 weeks.

I also have a few posts I want to write but I’ve been postponing because it takes some time.


Wow, writing it down makes it seem not as bad.  I’m off to finish my album.



Learning math facts: Reflex Math app review

Reflex MathLast month, I got really into wanting to find out how children learn/remember.  The impetus was wondering if Anki would help Thumper learn her addition facts.   Anki is a program that many people use to learn a second language.  It’s basically a flashcard system.  But it shows you the flashcard based on spaced repetition.  Cards you deem as easy to recall won’t be shown to you as often as cards whose content you haven’t mastered.  That got me to wondering about spaced learning so I borrowed a huge stack of books.  I managed to read about 2 books out of the 10 I borrowed.  One was How We Learn.  Another was on children and memory.  I shall put in a review about what I learned later.

In researching Anki, I came across a website which reviewed the use of Anki specifically for 4th graders and up in learning their math facts.  The author found some issues with the app.  One reason I remember was because children can’t quite tell you whether or not they know or not know a concept.  Another was that it was hard to scale/implement for a classroom of children.  They recommended Reflex Math instead.  Since the app has a 14 day trial period, I decided to check it out.

The short verdict is:  I LOVE IT.  

Relay Math is an iPad game where children answer math facts until their answer time is fast enough to be considered a “reflex”.  There are 2 modes in the app, addition/subtraction and multiplication/division.  In each mode, the child starts with two games and can unlock additional games after a certain number of days has passed.  One game has the child answer math fact questions so that the ninja can hop on successively higher wooden branches, dodging birds and what not.  The game starts with some assessment of your knowledge and reflex time.  The problems progress with higher and higher numbers.  So for example, maybe for multiplication it tests you on adding 1 to a number or adding 10 to a number first.  After awhile, it also teaches you about the fact families (5+2=7, 7-2=5, 7-5=2).   You’re shown facts to solve based on what you know or don’t know.

There is a teacher mode so you can see which math facts your child is working on, which one is unassessed, which one they’ve mastered.  You can see the progress they make day to day in graphs and charts.  It has a classroom mode so you can add multiple students.  I created two accounts for Thumper, one for addition/subtraction, one multiplication/division.  There is a timer on the app.  After 45 minutes or an hour it will disallow the child from playing more.  Because the thinking is that more frequent practice is better than playing a long time each day.

Thumper loves the app because it’s a game.  I put in a limit of 20-30 minutes max each time she uses it, in addition to the daily limit imposed by the program.  She likes the fact that there are new games to unlock (she gets bored and likes trying new things) and loves earning tokens to accessorize her avatar.  We don’t allow her to use other games on the iPad, which I think is important as it would otherwise distract her from wanting to use the app.

At $35 per year per child I think it’s a worthwhile and cheap investment.  Given her rate of progression I think we only need it for one year. Continue reading