UPDATED past post: Fraction Materials

I recently updated this post on superfluous Montessori materials. Mostly just tweaks.

But I added the information below about the fractions materials and thought it deserved its own post because of the strong emotional response that has been sent my way when I calmly say, "This material is unnecessary." So yes, here I am a bit more emotional ;) Because I really stand behind the Montessori principles involved.


As I state in my other post, I am not intending to offend anyone, but I know this will sound that way to some people. Please consider the Montessori principles at hand; the century that has passed with consistent observations and the development of album pages appropriate to the various stages of childhood. While not everything may be solidly set in stone, the youngest ages do have a very firm foundation. It is the adolescent age where we are still playing with the details.


Fraction sets for 1/11 through 1/20 - there is a keys-based presentation for helping the children work with fraction above 1/10. If a child is given the material for fractions 1/11 and higher, he is losing out on the opportunity to explore on an abstract level. He will always be hindered in his work because he was given "the fish" when he needed directions on "using the fishing pole".

I have had 3 moms tell me their 5 or 6 year old was SO STRONGLY into fractions and since the child was SO young, this hands-on material would be appropriate for them. NO! Can I say that loud enough!? NO!!! NO! NO! NO!
(before we go on: my son at age 4 1/2 was there too - ALL things fractions! a "fraction genius" even! I really thought I had a guru on my hands - and I was ready to feed that guru-ness wherever it took us, but something happened...)


There is a reason that fractions are in the primary album for children ages 5 and 6 - BECAUSE they have an intense interest! So we are already feeding that interest! But we are feeding that interest by providing the FOUNDATION. A STRONG foundation in the basic facts and uses of fractions.

Then if your child is truly a fractions-genius and his interest hasn't yet waned (most children's intense interest in fractions wanes for a year or two, because they 1) need to consolidate and 2) are moving on to other areas of development and 3) there is just so much else to DO) - there are a series of elementary level presentations that are worked on FIRST (and the little genius can certainly do them all in 1st grade if he wants! rather than spread them across 1st-3rd), before you even get to the need for doing 1/11 and beyond. Once you get to the presentation for 1/11 and beyond - WE DON'T STOP AT 1/20! We go to 1/100 and 1/360 and BEYOND even that! So when a child is ready - he's READY! And we don't hold him back! But we lay the foundation FIRST.

Therefore the fractions materials for 1/11 through 1/20 are superfluous at best; a hindrance and a crutch at worst. Don't bother with them!


SUMMARY:
If a child is a fraction genius and it HOLDS, then he has all the primary presentations AND all the elementary work at his fingertips, before he is presented with how to address fraction smaller than 1/10 --- and at that point, a true fraction genius, will be entirely ready for that materials that are provided, or he will be genius enough to make his own material - thus consolidating and applying the fraction concepts at hand in a much more efficacious manner than just handing him the material.

:)

It all works out in the end :)

Multi Base Material - Upper Elementary Mathematics

(UPDATED! For clarity and to add search results at the end)


We strive to be keys-based around here. I provide the keys and the opportunities, my son uses them to their fullest potential at the right times.

Great.

Until I won a giveaway through Alison's Montessori for a beautiful, non-essential, non-keys Power of 3 Cube. This is an *abstract* presentation in my album - it is not intended to utilize a material because at that point, the material becomes a crutch to the child. If the child needs a material at this point, the child *creates* the material himself.

So. We're using that one for sensorial exploration for now - as a puzzle - in lower elementary; then I will pull it from the available materials for upper elementary. It's a neat little "puzzle" so I'm ok with that.


But then I placed an order for our remaining Montessori materials several weeks ago. Received everything; then realized I still have a couple of small purchases to make. Oops. That company doesn't have those things anyway, so it's fine. I get my list ready and figure out how much money we'll need to save and I have it ready for the right time. Ok :)

But then I went and pulled this out of storage (given to us a while back - I'd tucked it away without thoroughly checking it beforehand). I thought it was the Non-Decimal Base Board, but I never thought to wonder what was in the BOX (non-decimal base board does not require a box).

This is the one sold at Alison's' - ours is identical, but I don't know where we got it.
It's pretty - it looks neat - but it is definitely NOT an essential material.
Why do companies sell the non-essentials and LEAVE OUT so many of the essentials? 

I am now the keys-based, flummoxed owner of TWO non-keys-based materials that I didn't know I was getting! ;)


I *thought* it was the board for Non-Decimal Bases. But it's not. And I have *nothing* in my albums for this Multi-Base material. Thus it could be a great work to do! But it is not keys-based - it is an extra. I am "ok" with that to an extent, but if I can't figure it out in under 30 seconds, with my experience and training, then is it really going to be necessary for the child?
NO.

But my son MIGHT be interested in it and it might make a FANTASTIC follow-up if that is where his interests go (no, the local educational requirements in our area will not require what this board teaches, so I can't even pull it in for that purpose!).

Wish I'd double-checked first! Because now I don't have the Non-Decimal Bases board - but that should be easy enough to just draw out on posterboard. And anyone else who needs Non-Decimal Bases Board and you've already made a chart for primary use of organizing the golden beads into their categories (that chart being another non-essential material that could be useful and even necessary in some cases - it is 4 columns, with a row at the top to designate that units, bars, squares or cubes go in a particular column)... Just use that! Fine. I'm good with that. Easy and cheap!


But what do I do with this Multi-Base material? I am flummoxed and, frankly, borderline annoyed - both at myself for not checking a long time ago and at companies for making this stuff. I am looking for an album page, a set of album pages - or someone to just tell me which company sells an album that has it - just to save me time figuring it out and making up the album page(s) myself.

Any ideas?


I have contacted 5 companies who sell it, of the 3 who have responded thus far, they have no idea either! So my next question is where did they get the instructions on how to make the material? Many of the discount places get their materials from the same place in China I am sure, but not everyone gets their stuff from there; so where did the original pattern come from?

It is very sad and very disheartening that the gray line between "essential" and "extra" (but could still be good) is so blurred, that there are sometimes "extras" that just aren't even good, but are still being called Montessori - but that is for other posts!


Downer post - sorry! It gets better tomorrow!


(UPDATE 12/29/12: All but two companies have responded back to me; I contacted 3 others who all responded. Not a one of them has any clue how to use this material, or even where the pattern actually came from. Wow.

This pdf document makes ONE reference to it:
Workshop
“How Mathematics forms the brain–Montessori and
Multi-base Material” by Ms Han Kamphuis, pedagogical
consultant at Nienhuis.

Thus it appears to be AMI, and it appears to be relatively new.

Not one AMI elementary training center has responded to my messages, to date. Not even the one I attended.)

Will update further as anything new comes up.

Dot Game Fun!

I love the dot game - I don't know what it is - all the dots maybe? Nah - it's the organization of it - the beauty of it - the essential teaching of it!

And it is the one time that markers are typically used in a Montessori class - if the dot game is kept in a glass frame or laminated and mounted - the children can use a dry erase marker and eraser and use it again and again!


Legoboy did it at school, and wanted it to do it at home (that year I alternated subbing at Montessori schools and working at home - and homeschooling - it was a crazy, but wonderful year!) --- well, I just printed it out and he filled it in. We downloaded this version from Montessori Materials - a great place for some of the printable materials.

Then I had to print more - because he just ate it up!

Here is his first one... It's so pretty! I wrote the first set of numbers and everything else was on him.

The children check off as they do a row, so they can track where they are. They connect up the dots to create a "bead-bar of ten"; then carrying over is noted in red the first several times they do it, to really reinforce the extra work that goes into carrying. But these are all concepts that are actually KNOWN - the child has been doing this work throughout use of the golden bead material - for Legoboy, it had been almost two years with the golden beads, probably a year and a half with carrying, before he actually did the dot game. 

Montessori works in baby steps - so there are only two new things here: 

1. doing the work ALL on paper, in typical addition format (on the right side) --- but connecting it back to their previous work of no more than 9 in any category! 
2. introduction of the comma. Until this time they have not really been using the comma. There are many adults (ahem, me) who try to sneak that comma in there sooner - before the child is ready (too many new things at once if it is introduced sooner!) and most of the time the adult doesn't even realize he or she is doing it! It's so instinctive! But we want the children to focus on place value, so we don't worry about the comma until around this time - this is close to the time they will also learn about millions with the wooden hierarchical material - so the next step is to be introduced to the repetition of the categories (unit, ten, hundred; unit of thousand, ten of thousand, hundred of thousand, etc.). 

This material is only used for addition although multiplication could be done on it. There is more other work for the child to do and the point here is the two points listed above. So further operations are not necessary. 

By the 3rd sheet of this work, the Dot Game turned into a balloon popping game - pop 1, pop 2, pop 3, etc. until number 10 was the biggest POP of them all! 

oh the days when it wasn't about battles and weapons.... it just balloons popping! How I miss those days ;) 

Mathematics Materials: A Photo Journey

I started two separate posts, but there is so much overlap here! The materials can be pricey or tricky to make at first, but, as noted, most of it goes for several years!


A photo journey of our primary mathematics materials:

Numbers 1-10 (the candle is for decoration, polishing,
and it makes a neat representative of the number 1 ;)
Basket underneath for golden bead collections.
Most of this material is JUST for primary.
(except the small bead frame!)

There is also a small bead frame- but if you know you are going to do elementary as well, get the large bead frame and use for both (that is what we did). The dot game is downloadable. Teens and tens boards can be made of paper and cardstock (that was ours!). The memorization charts can be downloaded.



A photo journey of both primary and elementary Montessori materials:

At-home multi-purpose use ;)
This material is used FOREVER in Montessori!
It is WORTH it to have every piece of it - do NOT skimp here. 

Mathematics mats - especially for primary

Circle fractions are for both primary AND elementary

Division with Racks and Tubes
Test Tube Division
Can be used in primary for short division
Elementary for short and long division
and the materials can be used SEVERAL other places as well

Golden bead material - 45 of each, 9 thousands
(only one non-AMI primary presentations needs
more than 9 thousands - and you can make those)

Decanomial bead bar box - we get away with one box
by adding a few bars of pony beads on colored pipe cleaners.
Primary and Elementary
AND you can use this material for SO MUCH - just pull from it what you need!

The stamp game, large bead frame - belong here. Also, get a negative snake game and you'll have all you need for all the snake games.




A photo journey of JUST elementary Montessori materials:

flat bead from for elementary only

Elementary: power of 2 cube
(power of 3 cube is NOT necessary)

Montessori Protractor - this one is 0 - 360
Another one goes 0 - 100
Both are necessary in some form. 

Fraction skittles for division.
Some albums have them in primary for another purpose.
They are intended for dividing fractions by fractions
at the elementary level. 

sample of squaring and cubing in elementary
all material comes from the bead cabinet

too pretty of a picture not to share!
all material comes from the bead cabinet

Decimal Checkerboard for multiplying decimal fractions (decimals)

There is more I have not yet pictured: the squaring and cubing material; the pegboards.

Mathematics is huge! The nice thing is that primary materials DO carry over; and there is a good deal of the elementary material that can be consolidated, or made of paper, cardstock, or other items around the house, along with a few basic purchases. Or is multi-purpose for both primary and elementary with slight modification.

While this area is the most intense, materials-wise, it is also the most efficacious of all the Montessori materials. It is worth every second and every penny! 

:) 

Multiplying Sum by a Sum

Legoboy loved this work intensely - for about 3 weeks. Then he's done. He found the shortcut of adding up the quantities within parentheses and then multiplying, so we had to move on to unknown quantities to keep him practicing the proper procedure!

The idea is to start with lower numbers, with bead bars within one set of parentheses and number cards in a second set of parentheses; then show proper procedure for multiplying each bead bar by each "command card", finding partial answers and later putting them all together into the final product. We turn over the multipliers (commands) not being used, so that we isolate just the one individual being used at the moment.

Talk about good habit-forming right from the beginning and before EVER doing the abstractly!

So. He LOVED the work. And he loves drawing. But for some reason, he always rushed through the few drawings he made of his work onto graph paper. Go figure! Here is a sample - he started off well except for parentheses ;) As Grandma says: Funny boy!
(ordinarily the partial answers are placed underneath their corresponding bead work - not to the right as he shows it here - but he likes to have all of his work of this sort fit onto one piece of paper when possible). He still had the final product on a different piece of paper which he then tucked away somewhere I can't find to show you all! He does that with his favorite work until he is ready to share. I can't complain! At least he wrote the numerical answer at the bottom :)

Sample of 2-sum by 3-sum

1 unit by a sum of 2 addends, then switch it around. 

sum by a sum - basic presentation :) 

Long Bead Chains - Photo Journey

Just browsing through photos looking for something else, and thought, "I should share these!"

Counting up the 1000 chain!

Drawing it afterward - elementary children

sample of elementary work with the long chains

the lighting was bad ;) 

1000 chain from my student teaching days
I still have that skirt!
And I still remember the happiness of the boy
who worked on this presentations
- the TRUE self-esteem -
the pride he took in his work
the complete turn-around in his behavior
having been given some REAL work to do. 

5 1/2 year old SO PROUD of his BIG WORK
(this boy did this work BY HIMSELF!)
It was probably one of the most fulfilling things
he had done in his life to that date!)

Your Business Math - Pet Store

As with all elementary Montessori work, the idea is to get the child exploring and learning of his own accord - provide him the keys and let him do the unlocking ;)

Along that vein, when Legoboy came across some information about Your Business Math, he was really excited! And started saving his money for it.

While business math is ideal for the adolescent age, there is nothing wrong with some sidework in the business world at the elementary age too!  They have three options: pet store, book store, or sports store. We have the e-book version of the pet store.

There is a bit of printing with the e-book, but it saved shelfspace and allowed us to just print what we need so I'm good ;) You can print straight from the pdf they send you or they have links on their website to download just the needed pages.

He got started on it the week before Thanksgiving and while he would be happy to fly right through it, I am intentionally slowing him down so that we truly focus in on each step of the process.

You work in "months" which take about an hour each to do if you fly straight through and a bit longer if you are more mindful.

As I said, he started to fly straight through, because
he was SO excited. He mis-spelled a word -
but he's corrected that since I took the photo.
I would have added more color, but
he preferred keeping it simple. 

We spent one evening just setting up. I sat at the computer and worked; he came to me for the next step to read the instructions, print as needed, then he was off. When he needed guidance he brought his binder to me and we worked through it. This way, I was "occupied" but available and would not fall to temptation to micro-manage. ;)

He chose a binder and some dividers; drew out a logo and selected a name for his store:

The first divider - all inventory sheets, order forms,
customer orders,, ledgers, etc. for each month
are behind their own tab. 

We like it. I wish it was a bit more intense, so we could more easily beef it up and make it a younger adolescent experience - but this is solidly 8-12 as their website specifies - and I could see most Montessori 11 and 12 years old wishing for more intensity.

The nice thing with families - is that each child can have their own store - the instructions are the same, the order details are different - and they can place orders with each other. There are other ways to provide variations, such as other family members placing pretend orders, choosing not to accept a program-provided order on moral grounds (ie one customer wants to feed his animal purchase to his other animal, so would you as a *pet* store owner want to allow that, etc). The instructions note when you should develop certain policies such as these. And there are certainly your own variations you can make as you like.

There are chance cards and cards for "additional in-store sales" (AISS cards). These also add some variations and allow for varying situations.

At the end of each month, profit and loss are calculated; as well as at the end of the year. Evaluate how things went, what worked, what didn't - and how would this all work in the real world.


Down-sides: 

• The ledger system is a single-line - good for some purposes, but some accounting records need double-entry - this might be an item to add to an adolescent experience, following using this program in elementary - build up complexity in increments. 
• The labels of debit and credit are incorrect - reflecting what your personal bank statement looks like rather than what a real business ledger account would look like. This is easy to fix by simply crossing them out and re-writing them - or leaving those words out altogether. It is also a good discussion point for the children - when is a debit "money in" and when it is "money out". 
• the orders have no actual dates, so we make them up. The program has you calculating all ledger entries at the end of each "month" instead of as it happens. This has benefits and drawbacks in the real world. Another discussion point for the children. 
• the chance and AISS cards can be a tad confusing - how to record them for example: if you had $450 in additional in-store sales, what inventory are you replacing? It doesn't say. So we chose to mark it down as services rendered - perhaps pet-boarding for the day or the weekend; or delivery service. We just have to ignore the other expenses (extra food used, gas, mileage) - this is where this is program is solidly elementary, but would not be appropriate for adolescence who want real world experience. So we make the adjustments, gain the experience and move on. 
• If a chance card says increase earnings by 10% - 10% of WHAT!? It doesn't say, so I created another set of chance cards, that when this sort of card is picked, he draws from the new stack to tell us the percentage of what - AISS, monthly profit/loss, total income for the month, etc. 

For what it is and what it purports to do, it is fun, lays a good foundation and introduces the practical purpose of various math concepts. Can't beat it! It would be great to see something for middle schoolers with more of the "trickiness" included - perhaps that is something to look forward to! 

Passing from One Square to Another

After our work with the cubing part 1 and part 2, I figured it was probably time to officially present "Passing From One Square to Another" since he'd already done the allegedly harder work of "Passing From One Cube to Another" (just subsequent passings at this point - not jumping two sizes).

rebuilding the square of pythagoras from primary
YES, this comes in a LOT in elementary
in particular using the bead bars! 

First he wanted to stack the squares to make a flat pyramid.
Then he turned them a bit to make them "fancy"
(think of all the art ideas a child could do with this work alone!)

Backwards order - he passed from square of 6 to 7;
then he passed from square of 5 to 8.
Hm. Looks like the binomial cube he said.
Then quickly added: but DIFFERENT colors!
(see previous posts on cubing to get the joke)

Displaying the originals with the finals. 

When we were done, we verbalized what we had done. We also noted that pattern on each side - for example, the square of 7 IS the square of 6 plus the square of 1, but with two rectangles add of 1 taken 6 times. The square of 8 IS the square of 5 plus the square of 3 plus 2 rectangles of 5 taken 3 times.

Now let's look at the values: the square of 7 (49) is the square of 6 (36) plus the square of 1 (1) plus those two 6x1 rectangles (12). It adds up!

Repeat with the second square. It still works!



Yep. We did the work out of order - some things are entirely ok to change up! This was all strong interest-driven and the main pre-requisites were in place - that's ALL that matters! We have plenty of time to review the concepts again and again to be sure of thorough understanding. (and see my previous posts this past week on that aha! moment when a child verbalizes something his actions have shown he's known for YEARS) ;)

Yes, scope and sequences are great - they get us organized - but feel free to diverge from it! Cover pre-reqs (could be informally) - that's all that matters!

Cubing - Part 3 - Just exploring

After all was said and done, he really, really, really wanted to just build with the squares and cubes. Ok, I said - but you have to graph out your creations. He agreed and off he went:

The camera batteries died after that - and my own batteries were slowing down so I didn't replace the camera batteries until the next day. When he graphed this on paper, he did it two ways - straightening up the inside pieces as he worked:

1. straight down - so just the tops of each piece. He wasn't satisfied with this because it was a 3-d work. 
2. he tilted it and drew it out on the graph paper with diagonals in some of the boxes to show both the tops and the full fronts of each piece. That is better he said, but still tricky. He colored the sides in a darker shade of each color than the tops (outlining the tops of the white walls in gray, and coloring gray for the sides). 

So what did this work entail: 

• geometry
• art
• aesthetics
• architecture
• planning
• viewpoint adjustment

Good skills going on!

Algebraic Trinomial Cube

This cube goes by several names - but the most important fact is that it is NOT the trinomial cube.

As Legoboy so recently and astutely discovered:

Poorly cropped - but
1) someone skipped getting dressed
2) I wanted to include an image
 of the squaring/cubing

"It's just like the trinomial cube!" He proclaimed when he first saw it. "But the colors are different," he side-commented to me to be sure I didn't ask him about the changing colors as I did with the tower of cubes versus pink tower.

But I still had him one-upped ;) "Hmm. Different colors, alright. I'll let you explore this while I go do something else and keep myself from interrupting you!" He just smiled and proceeded to work.

When he was done, he brought it to me, and said, "Oh Mama! The colors were VERY different! They didn't even match the lid on every layer like the trinomial! This is a CRAZY cube!" (keep in mind our trinomial cube is not in our home right now - it's over at the local school building).

I told him as soon as we get our trinomial cube back into our hands, I will show him the story of the Three Kings. Of course, just to be contrary, he says, "I already know about the Three Kings! I can skip that presentation!" (sound like some adults we know? I already know it (based solely on the title or perhaps the title and the main headings, then skip the actual content, which just MIGHT actually be nuanced different, if not entirely different... and then wonder why things aren't working right;) hehe - Yep, that describes me!)

"Oh no, my son, not THOSE three kings! These are the Cubed Kings!" That got his interest up! He's been asking for a week now. Perhaps next week we'll get to it. I like to keep up some anticipation (and I do want to review a couple of other concepts before working with the Kings.).

Yeah, we had fun with this one!

Stylized version - I know how tricky the illustrations can be!
Tricky - but NECESSARY
for each and every step!

Don't know the Kings?

Pick up a Keys of the Universe mathematics album ;)

HONEST review here - from a homeschooling mom - NOT from a Montessori trainee:
There is a free elementary math album available online, with which I usually agree as to the order and general set up of presentations, but I am thoroughly perplexed by its version of the Three Kings - (this part comes from me as a trainee: and I KNOW it! And I know what is trying to get across and what comes next.) Back to being JUST a homeschooling mom: But the instructions are confusing, the story is BLECK-boring until you realize that the *actual* story comes after the presentation in the album and is actually mildly interesting but leaves some strange questions, the whole set up begs to know what pieces to move when and as a mom I want it all together so I can SEE what is going on, and I really-really want those black pieces to be the same height as their king - it just makes sense when you see the other attendants the same height as their own king. But this album has them all laid flat.    It just does not look appealing - while mathematically correct, there is another way that makes the math more clear, leaves out the strange questions and keeps the appropriate parts of the presentation together in order. I'll stick with the Keys of the Universe version ;)
Sorry! I really do LIKE that album - I just don't "love" it and I'm very honest about things these days.

Cubing Material - Part Two - the REAL fun begins!

We had a fun morning exploring all of our "new" materials and it was too much for just one post ;)

After building the pink tower with the cubing material:

He finally asked me, "So what is this material for?" He has seen it at work in a classroom but it has been a while and he was not yet ready for it. He has been working with the early squaring and cubing and playing the decanomial games and calculating the values of pyramids and towers (with the bead cabinet material). So now we are on to something new.

I challenged him (before doing the actual album page) to construct one cube into its next cube.

Little secret - shhhh! Don't tell Montessori "purists" (of which I consider myself one, but with a different definition)... but Legoboy has not officially done the work with building from one square to the next square! Ahah! I have RUINED MY SON! Hahahaha - not really! I have followed an interest, in the moment of pure joy over discovering he already knew something about this material, helping me organize it - now is the perfect moment to introduce the concept!


Thus, the challenge:

Can you turn the cube of 3 into the cube of 4,
using ONLY the materials here????? 

Well, he tried all sorts of ways - some with relative success - some were just tedious and annoying (a wall of 1s) and he started figuring out where he could exchange squares of 1s for squares of larger numbers. He was presenting a series of problems (issues) to himself - knew I had an answer, so he continued to keep working, to 1) see if he could figure it out himself and 2) see if he could formulate the right question to get the answer he wanted (he knows I don't easily answer questions - that I am going to guide him in his thinking anyway, and he finds it "easier" to do as much himself first ;) ).

eh!? it was a first attempt to see what he could find
in patterning and sizes.... 

Replaced the cube with a series of squares
just to see what could be done.
he recognized patterns from the decanomial work! 

Not bad - and getting there. But what about all those squares?
isn't there something easier?
I modified the challenge to state, "build this cube of 3
into the cube of 4" (I'd reviewed my album page
in the meantime - yep, even I have to review ;) )

Back to the original cube.

A wall of ones - this got tedious - FAST. 

  

But he managed to get it. Then it all fell off. 

He thought, "what if I use the cube AND add squares?"

I didn't tell him - but this is VERY close to the actual
presentation - all the right components -
just in the wrong spots.
He built the squares at odd angles around the cube. 

It's kind of pretty from this angle ;)
But doesn't express the mathematical formula.
So I told him, "Ah, you are so close!
Is there one more easier way to rearrange just
THESE pieces???"

SUCCESS! And now the mathematics is displayed!

 Yes, I then presented him with the actual album page presentation.