Addition Table with the Bead Bars

Here is a super-simple presentation of an addition table from one to five.  It is mainly to work on memorization of basic addition, but it also shows the pattern of sums.

I printed a 6x6 grid (3x3 grids on four pieces of cardstock, cut and pasted together), though you could easily draw it out on bristol board.

We left the upper left square blank, representing zero.  Then we put in our addends: one through five across the top row and one through five down the first column. I got her started by going across the second row verbally: "One plus one is two."  (Then placing the green two bar in the square where the red ones meet.)  "One plus two is three."  (Place the pink three bar.)  "One plus three is four." (Place the yellow four bar.)  And so on.

Ella took over with the bead bars and when I came back I discovered that she had decided that because by six the bead bars were too long to go straight across the squares that she could bend them so they would fit in nicely!  She doesn't do that anymore.  I showed her (as you can see with the sixes and sevens) that they would fit if put diagonally.

This lesson can easily be expanded to 10 + 10 by expanding the grid.  And let me emphasis the importance of saying aloud each equation.  Ella cannot understand why I insist on this, but it does help with memorization and (very important for busy mums) you can hear what your child is doing while doing housework/writing papers/changing diapers/folding laundry or whatever else needs doing!

Square of Pythagoras - Some Extentions

Yet again we used the square of Pythagoras, although this time we worked together at arranging it, Ella dividing up the pieces between us, although she always kept the squares.  "After all, I'm the student!" she explained.

Ella and I have made the square of Pythagoras.  Then I directed her to build the tower with the pink cubes.

Without any prompting on my part she took the smallest square and put it on the smallest cube of her tower and said, "Look Mummy!  They go together!"

We then "related" the pink cubes with the square of Pythagoras.  I asked how she knew where each would go and she explained that each side of a cube was a square and that therefore the pink cubes would go on the actual squares in the square of Pythagoras.

Ella then proceeded to relate the two materials in the same way as she had done with the red square and the smallest pink cube.

Then she did it horizontally.

After we had put away the pink tower and reorganized our square of Pythagoras I got out the bead square for her to relate them together.  She told me that it was pretty easy since they all matched in colour.

Here she is almost done.  I then extended the lesson a little by giving her an early glimpse into multiplication; though we are not yet "naming" it as such as I would like her to absorb the ideas sensorially first.  I would touch each bead square, while it was superimposed on the square of Pythagoras and say, "One one is one.  Two twos are four.  Three threes are nine," and so on.  After each statement I would pause for her to touch the square and repeat what I just said.  I expect this activity will continue in a variety of ways until we are actually doing "real" multiplication.

Our "fancy" storage container for the square of Pythagoras - a plain wood tray from the dollar store.

Square of Pythagoras - First Presentation

Thanks to all who inquired.  No, I'm not dead!  It's been almost a year and a half since my last post, and while my life has been moving quickly onward I've been facing it mostly with glazed eyes and an air of confusion.  Such is life with twin boys.

My sons are now 2 and are healthy, mischievous little fellows.  I've begun to do some early Montessori work with them, which you will hear about in an upcoming post.

Ella is now "officially" grade 1, in that she is 6 and that is the typical grade 1 age.  And there ends all that is "typical" about her.

Evidently she's doing some things on a grade 4 level, and she was teaching our 17 year old babysitter about geography.  Her reading is about average, or perhaps a bit above average for her age, and her handwriting in abominable!  Though much of what she writes looks like the hen-scratchings of a preschooler, what she writes is very well done.  I guess we just didn't do enough of the sandpaper letters or the metal insets.

I have a vast backlog of photos and lessons to share and I hope to do so over time.  For today, here was our afternoon lesson: the first presentation of the square of Pythagoras (also called the deconomial square).

The square of Pythagoras is basically the multiplication times table in visual form.  The smallest square is 1x1 cm and is red in colour, which coordinates with the colour of the bead bars.

The next three shapes are green, like the "2" beads and are 2x1 cm, 2x1 cm, and 2x2 cm square.

The next five shapes are pink, representing "3" and are (2) 3x1 cm, (2) 3x2 cm, and a 3x3 cm square.

In the above photograph, Ella is laying out the 4s.

I find that she really enjoys making the square of Pythagoras as she thinks that the pattern is beautiful.  I think so too.

The completed square (as done by Ella).  As you can see it goes from 1x1 cm to 10x10 cm.  It is meant to coordinate with the pink cubes, the brown/broad stair/rectangular prisms, and ultimately can be used as a guide to laying out the bead bars in the deconomial.  Hopefully I will be showing these extensions in the near future.

As for the Square of Pythagoras material itself.  It is really hard to find to purchase. Ideally it is supposed to be made out of wood, but I found it much easier to make it out of foam sheets using a large paper-cutter.  (An office-sized paper cutter can be invaluable when making materials.)  I've also seen beautiful square of Pythagoras lay-outs made of felt or other fabrics, plastic duo-tang covers (though I don't know how easy or difficult it would be to find the colours you want), and painted cardboard or bristol board (though they can tend to warp).

Initially I planned on making it using inches as a base, rather than centimetres.  Bad idea!  It was HUGE and very unweieldy.  I'm very glad I downsized, though what I'm going to do with all that extra foam is beyond me!

The first presentation of the square of Pythagoras is not meant to teach the multiplication table, but like the binomial and trinomial cubes it is a sensorial introduction to a more complicated mathematical concept.  Right now we call all the green rectangles and square the "twos,"  all the pink the "threes" and all the yellow the "fours."  It will be some time before we begin to name individual rectangles 2x1 or 4x3.

To present it I simply named the material, "This is the square of Pythagoras." and I began making the square, beginning with the red, then the green, then pink, and so on.  It didn't take long before Ella got the idea and just took over.

She was rather frustrated by the fact that her square of Pythagoras wasn't as "perfect" as mine, but with a bit of reassurance that it took practice she became satisfied with her results.  I'm looking forward to her reaction to the extentions.

Montessori Mathematics: The Complete Bead Material - Part 1

When those who are not much familiar with the Montessori Method hear it referred to they often have interesting preconceived idea of what it entails. Some thoughts that I've heard are as follows:

--"Oh, the schools where they have no tests and they play with educational toys all day!"
--"You mean those really expensive private schools?"
--"Montessori schools are where the smart kids go."
--"Don't they mix up grades there?"
--"Children from those schools don't adjust well to 'regular' schooling!"
--"The teachers just leave the children to learn on their own."

If you are really knowledgeable about Montessori, then you can see how most of these statements have a grain of truth enhanced by a dose of ignorance. My usual response to the above includes such explanations as:

--"No, they don't have traditional type tests in Montessori schools, at least not in the early years, but the children are constantly being evaluated on what they have learned. And all of the educational materials used are graded as to difficulty, so there is a natural progression as the child learns to master each level."
--"Yes, many Montessori schools are very expensive, but in some places the Montessori schools have been adopted into the public school system. And if you cannot afford to send your child to an actual school, they can still have a Montessori education if you put your mind to it!"
--"Montessori schools are not only for children who are intellectually gifted. In fact the method began when Dr. Montessori was working in a mental institution with the severely disabled. A Montessori classroom is an excellent environment for learning for children of any age and of any intellectual ability. Whatever the level of intelligence, any child brought into such a place and taught accordingly will end up smarter, though, and at his/her own pace!"
--"In a Montessori school each classroom is composed of kids of various ages. The idea is that those who are more advanced can teach and help those who are just learning. It is a cooperative environment."
--"Children who learn through the Montessori method become used to active learning. This is the best way for children to learn because they learn best by doing, not by explanation and reading. It also engenders a love of learning so that as they grow they become independent learners, wanting to find out for themselves about subjects that interest them. Unfortunately, 'regular' schools don't often offer this to their students."
--"Teachers often work as facilitators, teaching a lesson, then re-teaching the lesson occasionally, each time becoming a little less involved in the 'teaching' as the child takes over. This continues until the child understands the lesson enough to 'teach' the teacher, or the other children!"

As for the materials themselves, if someone has heard of them or seen them the impression that they come away with is often, "that's a lot of beads!" The reason being that the Montessori method wants to start children off by learning things not in the abstract, but in the concrete. Teachers don't explain the difference between 1, 10, 100, and 1000; we show the difference! The way we do that is with beads - lots and lots of beads!

I got lucky on eBay and was able to buy the Decimal Golden Bead Material at an excellent price. Thing is, getting the complete math bead material is a massive investment. Its uses are almost endless and reach into upper elementary. It makes traditional math lessons look positively boring, and it makes it easy for children to grasp what are usually considered to be complex concepts. The decimal golden bead material is an excellent example of this.

Above, Ella and I are doing a lesson on the decimal system. The initial idea is for the child to visually and tactilely note the difference between a unit (1), a set of ten units (10), a set of one hundred units (100), and a set of a thousand units (1000). The unit bead is simply a single bead. The ten bead bar is ten beads identical to the unit bead connected in a line. The hundred bead square is essentially 10 ten bead bars connected into a square of beads. The thousand bead cube has 10 hundred bead squares connected on top of one another making (obviously) a cube. Both in appearance and in size there is a vast difference between the unit and the thousand cube.

The next stage of the same basic lesson is to bring together 10 unit beads and set them together by the ten bead bar to demonstrate the relationship between 10 ones and 10, making sure that the child counts them out. It continues by laying 10 ten bars together, counting them out, and then showing how the result is identical to the hundred square. And finally you put 10 hundred squares together so that they become visually the same as the thousand cube. (Note, though, that the hundred squares tend to interlock when placed on top of one another - so for the purposes of the demonstration, I held them up on their edges making the cube vertically, rather than horizontally, which worked very well.)

Thing is, the golden bead decimal material is all I have. And the time has come that Ella and I can't go much further in math, at least not using Montessori lessons, without getting more bead material. If you've done much searching online for Montessori materials, you know that they can be very expensive. And when it comes to bead material, there is a lot of different items you'll need. Deciding where to get it, when, how much, and in what order can be extremely overwhelming.

I've searched all over for the least expensive mathematical bead material and the least expensive buy-it-all-completed place I've found (though if you've found one less expensive, please do share!) is IFit Wooden Toy Supplier in Vancouver, BC (And at the time of this writing, they've got a good sale on - take a peek!).

If you want to get every component needed for everything - i.e. every lesson, game, and bead activity, you don't have to buy everything. You need only buy the parts that have enough components for every lesson, game, and activity. For example: the decanomial beads have 55 of each of the bead bars for the numbers 1 through 10. If you use material from the decanomial beads, you won't have to buy any other bead stairs, nor will you have to buy additional golden 10 bead bars for the golden bead material. Albeit, you won't have all the tidy boxes and cabinets for storage, but if you can contrive to make those you'll cut down on your costs considerably.

For enough components to do all bead activities, you'd need to buy:

• the Decanomial Bead Bar Box
• the Elementary Negative Snake Game
• the Complete Bead Material
• Forty-five Golden Bead Units
• Forty-five Golden Hundred Squares

Today (November 6th, 2008) the price for the above materials from IFit Wooden Toys is $523.28 CAN or $422.00 USD - which is partly because of the fantastic sale they have on. But don't forget shipping costs when you make up your budget! For example, if I were to order it, I'd have to add 5% GST (Goods & Services Tax - Canada's national sales tax) and shipping costs ($62.86), so my total would be $612.30.

Unfortunately, that's a lot of money to spend all at once, and most homeschool families just don't have it! After all homeschool families rarely have two salaries! So even though this price is phenomenal considering what you get, it's still beyond many, myself included. Therefore you might want to consider going the route I've taken: making the bead material yourself.

My first piece of advices is: don't go tripping off to Michael's or a Wal-Mart and think that you can buy the beads there. I did and once I began to calculate I realised that it would be much cheaper, and much more efficient to buy the bead material outright! That's provided they even have enough beads in enough colours.

You'd do much better to order the beads, possibly from a wholesaler. Some of the least expensive places I've found online for buying ridiculous numbers of beads are as follows:

Lure Making - Yes, it's a fish-oriented place. But some fishing lures need beads, and they have a decent price!

John Bead - A Canadian company, although, as a wholesaler, they have some restrictions as to whom they will sell. But if you have a friend who works in a craft store...???

Consumer Crafts - Which seemed my best option until I discovered they don't ship to Canada! So, for non-Americans, you may as well just skip this link!

Bolek's Craft Supplies - The company I decided to go with for three main reasons: they will ship to Canada, and they were the least expensive, and they have a big selection of colours.

Now, before you say, "But pony beads are much cheaper! And they're available everywhere in bulk!" let me warn you: pony beads are marvelous for children's crafts, but because they are not round they cannot be used for bead material. Here's why. Pony beads work well for the unit beads, and even for the bead bars, but when you begin to make bead squares and cubes you'll notice right away that the squares are not squares but rectangles, and the cubes are not cubes but rectangular prisms! To make Montessori bead material you need beads that are as long as they are wide, as they are high, which limits you to either cube beads (and you'll never find enough of these in enough colours to use) or round beads. Faceted round beads will work as well.

The traditional colours for the Montessori bead materials are: red, green, pink, yellow, light blue/aqua, purple/lavender, white, brown, dark blue, and golden. And for some special-use beads you'd have to add: light grey, dark grey, black, and a second shade of pink. Obviously, not all companies have all these specific colours, so you might find yourself having to improvise, but if you do, you certainly won't be the first one to do so!

So, how many beads, exactly, does it take to make the bead material? And how many do you need of each colour? Well, you've struck a goldmine now, because I've already done all the figuring out for you! Hopefully all my figures are right!

• 55 RED BEADS
• 142 GREEN BEADS
• 273 PINK BEADS
• 35 PINK BEADS (for the pink/white bead stair in the snake game, you may wish to use a different shade of pink)
• 476 YELLOW BEADS
• 775 LIGHT BLUE or AQUA BEADS
• 1194 PURPLE or VIOLET
• 1777 WHITE BEADS
• 2488 BROWN BEADS
• 3411 DARK BLUE BEADS
• 10 LIGHT GREY BEADS
• 265 DARK GREY BEADS
• 8095 GOLDEN BEADS

Well, that's about all I can write for tonight! There will be other posts regarding the making of bead material in the future, as this is a topic that takes a lot of time, for the Montessori material maker, the Montessori teacher, and eventually the Montessori student!

Math Breakthrough!

Something clicked for me this weekend, and it clicked for Ella too.

All this time, I was frustrated with her counting - HAH! What was I thinking? Ella's problem had nothing to do with difficulty counting, or understanding the relationship between the number of objects with the numeral or the name of the numeral.

She had shown interest, the last time we did math, in trying to match two number rods to make the length of a third. So, using the first five rods we did a little bit of introductory addition. There was no frustration, no "I'm to tired's", no I don't want too's."

Up at my Mum's later while visiting with my visiting siblings, "Uncle" asked Ella, "If you had three kitty cats, and I gave you two more kitty cats, how many kitty cats would you have?"

"Five," said Ella, in a tone of voice which added the feeling that she was thinking, "Isn't that obvious? What a silly question!"

Uncle posed a few more addition problems, all which she answered promptly and correctly, until he asked two plus four, to which she answered "Five!" He cocked his head to one side and look at her, to which she responded, "No, no! Six!"

Earlier today, as we were still at my Mum's, and as a result of visiting aunties and uncles, we hadn't had "school" in a while, I noticed her playing with a group of her "Little People" animals. I laid down in front of her and told her we were going to play a little game. Since "little zebra" is her favourite, he and she were going to be on one team and they were going to figure out the mysteries I made.

I started with four little people in one spot, and one separate from them. I made up a little ditty and sang something along the lines of "How many animals do you see?" while pointing to the four animals together. She and zebra sang back at me, "Four, four, we see four!" Then I sang the same ditty while pointing to the solo animal. Zebra and Ella sang back "One, one, we see one!" Then I slid them all together while singing, "So four plus one makes ____." Zebra and Ella sang right out on cue, "Five!"

We did this for quite a while, when she began asking for harder "mysteries." We went through 2+3, 5+1, 5+2, 6+1, 3+4, 7+1, 3+5, and 4+4, when she (and zebra) asked for a really tough one. So I put out 6+6, which at first she said was 11, but when I said, "You'd better check again," she figured out it was 12.

So, now I'm going to have to buy/make the next stage of math materials, such as the mathematics bead material (now there's a huge project!), the teens and tens boards, and a second set of number rods. I'm really glad now that I received the Montessori Research & Development Mathematics Manual. (In fact, I got quite a few Montessori R&D Teacher's Manuals, and I'll be writing about them at length in a future post.)

I doubt my math woes are completely over, but at least now I know it wasn't that the material was too hard, but that it was too boring. I have a funny feeling that when it comes to making/buying Montessori material, I'm really going to have to work to keep up with Ella!

Homeschooling Frustrations

Some days go better than others. Some days Ella just drives me nuts!

I've mentioned before that for some reason (could be genetic - neither I nor my husband is partial to mathematics) Ella would choose almost any other activity over one pertaining to math.

Today, my Mum called and informed me that Ella can now count to 30. THIRTY??? In school I can barely get her to count to 10!

Here she is using the spindle box, as it is meant to be used. She always starts off very well, counting out the spindles to put the correct number in its proper section. But around 6 or 7 she still just mentally wanders off. From her behaviour, I'm guessing that she can quickly and easily visualize what number of objects makes the numbers up to six, even without actually counting, but beyond that, she just doesn't want to bother with the effort of counting them out, rather she just makes a guess and throws in about how many she thinks might make 7, 8, or 9.

Here she is still doing the activity correctly, dropping them in individually until the correct number of spindles are in the appropriate section.

And here she is grabbing a handful of spindles to toss in the 8 section! If I thought she was just having difficulty grasping the concept, then I don't think I'd be bothered by this, but she doesn't even seem to try. It's the same story with the number rods and the numerals and counters.

On the bright side, I was sitting, typing up a letter to my brother the other day while Ella watched Winnie-the-Pooh (the original) when Ella turned to me and said, "Bear! Bear starts with 'buh'!"

We haven't yet done "buh" in school. I was impressed and decided to see what else she could figure out. "And what letter makes that sound? What letter says 'buh'?" I asked.

She thought for a few moments and then exclaimed, "BEE! Bee says 'buh' and 'buh' is at the beginning of 'Bear'!"

Until now, she had been struggling with the concept that the sound that begins a word can be separated from the rest of that word to discover what letter begins the word. And today Mum also told me that they were playing a little "game" where Mum would say a word and Ella would tell her what sound began that word. Evidently Mum was quite impressed.

As frustrating as some days are, I have to say that when something like that happens, it's a real encouragement. I call them "breakthrough" moments, and they are as exciting for the teacher as the student. The moment when you see the dawning light of understanding overcome your student, the moment when everything "clicks" together and the student finally "gets" it.

Often these moments are followed by a sudden leap in the accumulation of knowledge, a period when the child, who until now has only been doing what he is expected to do, is able to absorb the information quickly and deeply, embedding it in himself to use for the rest of his life. Those breakthrough moments, and the rapid, exciting learning done afterwards are what makes homeschooling a joy, despite frustrations.

Now, if I can just get her motivated to count!

New Montessori Activities

I'm constantly adding and adapting the activities Ella does for school. Here's some of the newer stuff...

This is a practical-life activity. Essentially, she's learning to use tongs to transfer an item (in this case rocks!) from place to place. She soon figured out that she could get more if she sort of "scooped" up as many as she could before closing the tongs together.


This is a colour-sorting activity, which could fall in the category of "sensorial" or "finger dexterity."

For a long time Ella's favourite colour was blue (it's now green), and she still always begins with blue if green isn't an option.

I'm not 100% sure yet, but I suspect that my greatest challenge is going to be making mathematics fun and interesting. We've used the small number rods (pictures previously, but not in use), but in order for Ella to do it, I need to be completely with her, helping, showing, and creating games in order to keep her even semi-interested. Since until now that has been our only math-oriented activity, I've added my version of the spindle box. It was very easy to make. I got a few long, thin dowels and had Dad cut them into 5-inch lengths (45 spindles are needed for the activity), and sanded them. Then, using a "snap together" desk organiser that I got at the Dollarama ($1 for three parts, I needed ten, so I bought 4 - anyone need 2???), which didn't snap together very well! So I glued them together in two sets of five (easier to move, and takes less space), and using a thick permanent marker, I made the numbers 0 through 9 at the ends of each section. She's either not "up" to doing it on her own, or is not able to maintain interest in it on her own yet, though. So we're doing it as a game and we take turns putting the correct number of spindles in each section. I try to be sure that I alternate between doing the even and odd numbers.

She does really well until about 5 or 6, and then (quite honestly) I think she gets bored and doesn't bother really trying to do it correctly anymore. Math just doesn't have the fascination for her that other subjects seem to. I can't really blame her; neither of her parents were fond of math either!

WARNING - NUDITY AHEAD!!!

That is to say, our anatomy lesson - which (strangely enough, lol...) is anatomically correct! (To see the "nudity" you may have to look closely).

I got this anatomy layer-puzzle at one of my favourite toy stores, Hot Toads! (you can peek at my links, if you want to see other great stuff). I never have enough money when I go there. They are primarily an online store, but they do have a small store in Oromocto, New Brunswick, and the owner really knows his stuff. It's put out by Beleduc-Hape Toys.

This was an excellent activity. The bottom layer is the skeleton, then organs and systems, musculature, skin/nude, then dressed. As she did the puzzle, we talked about what the different parts were called and what they were for in a fairly informal manner. She did catch a lot, because I over heard her explaining to her grandmother that the skeleton was bones inside you and those are you "hard parts"!

Perhaps our favourite subject of all - geography! We're doing world continents. Ella adores maps and globes, and I've always like geography and cultural studies. I've made some "Antarctica" cards that we play with, and we're still planning an "Antarctic Expedition," which will take place in a snowy, barren location (maybe our back yard!). I had been hoping to do it tomorrow, but the weather might be against us. Freezing rain - yep, okay, they've got THAT in Antarctica, but by noon it's supposed to be just plain rain, and that will be wet and miserable. I need a cold day for it, not a wet one!

Ella had a surprise visitor - Nanie! Mum, being off from school (yea March Break!), came down to see Ella at her school, and to play. Every time Ella takes a piece out of the puzzle and lays it on its corresponding "shadow" above the written name of the continent, she'll say, "This is _____" (fill in the blank with whatever continent it is). She doesn't get them all right yet, but when she's stumped, she looks up at me and says, "I can't bemember this one!" and I'll tell her. She'll then repeat the name and continue. When re-assembling the puzzle, she also says each name.

She was so proud of herself - she got them all out, on the right shadows, and said all their names correctly when showing Nanie! Nanie (being a typical grandmother) was pretty proud too!