primary Math - new question post #19

[Rose-Roget]

I feel like I was a lot better versed in this last year and have somehow managed to forget what I knew. DS is in his 2nd year of Montessori, but possibly in the last year of the primary program (we were latecomers to M.). He also changed schools this year.

He never gives me a play by play of the works that he does, so I have a hard time knowing for sure what he's been introduced to/working on. Today, he said he did the 100's board, which is a step up from the 50's board he's been talking about. He explained that this is where you line up the numbers in order, which I remember someone saying last year was debatably Montessori. I asked if he ever did Math with beads or spindles and he said no.

When ds was 2-1/2, he was developing the concept of addition and subtraction, not by rote but by figuring out the concepts with small numbers. Today, he still can't count to 100 reliably, even though this seems to be what he has been working most on, by his report, and even though he is fully capable. When he is counting with me, we talk about the pattern of the 10's, but I don't drill it.

The other day, he was practicing writing his numbers and "quizzing" us - what's 1+1? 11, what's 5+3? 53. And another time, he took some our numbers and made 479+625 and added all the numbers together (4+7+9+6+2+5) using little tokens we had. I didn't jump in and give a lesson on decimals and hundreds because 1) I'm not sure where he is working in Math and don't want to jump ahead, 2) I don't know how to teach it appropriately, and 3) I didn't have beads/hundred squares available.

So my questions are: What is the progression of Math works in the Primary cycle? I understand that developing the underlying concepts is most important, but what point should a bright enough child be working at at this age? Are those types of math errors due to a developing curiosity/emerging understanding, or is he somehow stagnating/regressing?

I don't know if I've expressed myself clearly. I'm not really complaining; I just want to know that things are on the right track and to know what questions I could ask to make sure he is.

[ADirrim]

I will do my best to take a stab at your questions:

Originally Posted by Rose-Roget

Today, he said he did the 100's board, which is a step up from the 50's board he's been talking about. He explained that this is where you line up the numbers in order, which I remember someone saying last year was debatably Montessori. I asked if he ever did Math with beads or spindles and he said no.

There are different ways of presenting the hundreds board, but the standard presentation involves children sorting tiles representing the quantities 1-100 and placing them in order on a board. It is certainly a challenging work. In order to sort the tiles, the child generally needs to have an understanding that the numerals have different values of tens (eg. all of the teen numerals have a one in the tens place); additionally, it will teach the child to be able to recognize (and read) the numerals 1-100. Generally, the child would have had lessons on the teens and the tens board prior to doing this lesson (which would involve using the bead material to associate the quantity and the symbol).

The spindles on the other hand are designed to teach the child to associate the quantities 0-9 with their corresponding symbols (mainly to give the child their first introduction to zero and the idea of a set), so these materials would have been used much earlier in the progression (he is more advanced from your description). Also, he should have used the short bead stair to count from 1-10 already. So, I am not sure how to take his statement that he has not used beads or spindles (you should ask the guide).

Also, as a side note, I'm not sure about the debatably Montessori part- it is certainly considered to be a regular Montessori material (although a 50 board is some type of extension- not standard). I'm sure someone will clue me in.

Originally Posted by Rose-Roget

When ds was 2-1/2, he was developing the concept of addition and subtraction, not by rote but by figuring out the concepts with small numbers. Today, he still can't count to 100 reliably, even though this seems to be what he has been working most on, by his report, and even though he is fully capable. When he is counting with me, we talk about the pattern of the 10's, but I don't drill it.

Honestly, counting to 100 can be challenging and it is a big, time consuming, work (how old is your son?). The hundreds board is even more challenging, because in addition to being able to linearly count from 1-100, the child also has to recognize the numerals (and understand the concept well enough to sort them). What would come next in the Montessori progression would be one of two things (there are kind of two different tracks the guide could take)- one would be using the decimal materials; the other would be linear counting the short or long chains. No matter which track the guide pursues, mastery of the hundreds board (or the ability to read the numerals) would be important (whether it would be to understand performing mathematical operations with the golden bead material or to correctly place the arrows alongside the chains).

Based upon your description, the hundred board sounds like a great match for his abilities (it will assist him with counting to 100, although that may take a while). If your child also wanted to do some addition, it would not be inappropriate for the guide to have him do it with the number rods, but this is considered to be earlier in the sequence. Another thought is that it might be beneficial for him to go back one lesson to the tens board (this is where children really practice the counting from 1-99), if he has really not done so (he indicated that he had not used the bead material- if that is the case it would explain his difficulty). There is no reason he could not work on these lessons simultaneously (they reinforce each other), but the hundreds board probably won't help him to rote count from 1-100 as effectively.

Originally Posted by Rose-Roget

The other day, he was practicing writing his numbers and "quizzing" us - what's 1+1? 11, what's 5+3? 53. And another time, he took some our numbers and made 479+625 and added all the numbers together (4+7+9+6+2+5) using little tokens we had. I didn't jump in and give a lesson on decimals and hundreds because 1) I'm not sure where he is working in Math and don't want to jump ahead, 2) I don't know how to teach it appropriately, and 3) I didn't have beads/hundred squares available.

This really makes me think that he probably skipped the teens and/ or tens board (he probably has not used the bead material- perhaps when he changsed schools they misjudged where he was in the progression). Either way (whether he was not given a presentation or whether he did not completely understand it), I would share this example with the guide and encourage her to give/re-present the tens board (very nice catch by you, by the way!). The teens and tens boards are neither a decimal work nor an addition work per se, but upon mastering them the child should not be conflating units and tens in the manner you describe. They will help him to grasp that adding together a certain number of tens and a certain number of units, produces a specific quantity and teaches the child to correclty assoicate that quantity with the numeral as well as counting from 1-99.

Originally Posted by Rose-Roget

So my questions are: What is the progression of Math works in the Primary cycle? I understand that developing the underlying concepts is most important, but what point should a bright enough child be working at at this age? Are those types of math errors due to a developing curiosity/emerging understanding, or is he somehow stagnating/regressing?

I don't think he is regressing/stagnating. He sounds very bright (and my guess is that he will advance really quickly), but I think he missed (or did not completely understand) the teens/tens boards presentations. Once he gets those either for the first time or as a review, the hundreds board lesson should really click (my guess is that he will move along really quickly after that) and he can work on the short chains (which will teach him to skip count, by tens for example, as you were describing) or performing operations with the decimal materials.

Hope that helps!
Abigail Miller
www.bloommontessori.com
www.bloommontessori.blogspot.com

[Lillianna]

Hi Rose-Roget, I'm not sure that you mentioned how old your son is now??
Here is a rough outline of stages for montessori math:
number rods, cards, spindle boxes, numbers and counters, memory game of numbers, sandpaper numbers
teen boards and ten boards (if necessary) this prepares for linear and skip counting with the bead chains (starting with 100 chain). The 100 chain is the montessori material for counting 1 to 100 (not the 100 board) . Montessori early math combines a symbol with a concrete representation of a quality until the child is ready for more abstraction.
The gold bead presentations for addition, subtraction, multiplication and division can occur simultaneously with the linear and skip counting exercises. (this is most appealing to children around the age of 4 and 5).
Then Stamp Game, Snake Game and various finger charts lead the child through a path that is continually more abstract and also aids in the memorization of math facts.
There is a also a montessori material for fractions (begins about the age of 5).
Good luck!

[Rose-Roget]

Thanks, ladies, for your insightful replies. Ds just turned 5, but like I mentioned, it's only his 2nd year in M. and we did change schools, so maybe some of the progression got jumbled. Maybe he meant that he hadn't used beads at this school.

[ADirrim]

One last thought...
I wouldn't hesitate to ask the guide what he has been working on and when things were presented. Schools generally give parents a progress report (and Montessori guides are expected to keep records) of when presentations are given and when material is mastered, so it shouldn't be a lot of trouble to produce those (it's too bad that his last school didn't supply them).

Also, if you're feeling like you don't have a good sense of where he's at in the progression, in addition to speaking with the guide, you might want to get the booklet A Parent's Guide to the Montessori Classroom by Aline Wolf (published by Parent Child Press- you can order it from their website) is a great resource. It's inexpensive (~$6), explains the materials and the progression really clearly, and has nice photos of the material (it might be worth asking, your school might have copies on hand to sell too- we give them out at the child's first conference).

[Rose-Roget]

I really appreciate all this information! I'm glad to hear that he seems to be progressing, just maybe have missed something along the way. I'll speak more to his guide for more specifics.

[MattBronsil]

Hi Rose. Let me start off with a bit of honesty. I had a long day. Just finished dinner. Ready to head home. I quickly glanced the thread. I am exhausted, but will try to answer as best I can.

The overall math development involves:
1). Learning 0-9 and 1-10.
2). Learning about the decimal system.
3). Learning counting those higher numbers (which is where he is with the 100 board).
4). The operations (Addition, multiplication, subtraction, and division).
5). Higher math skills (Fractions, etc.)

I taught a year in a school that had several rooms. The math materials were in a room, the language in another, etc. I was in charge of the math for the whole time I was there. I noticed how students moved differently through the materials. Some students worked a little with the decimal material and beads, but they were not there fully in their concentration and they moved quickly into the counting exercises.

The 100 board comes before addition and the bead cabinetN so the answer that 5+2 is 52 seems right from what he is learning now. He is identifying 2 digit numbers so when you ask about putting them together, he thinks about what he is working with. He might even understand the concept of addition, but not the terminology. Try asking him instead, "If I have 2 apples and I buy 1 more, how many will I have?". Putting it in a full sentence might show he understands the idea, but not the symbols + and = or the words that go with them.

[SomedayMom]

I'm glad you asked this. My son also just turned 5 (on Sunday) and this is his first year in Montessori. He has also been working on the hundred board lately. I was kind of wondering where it fit in as well.

Simon sometimes gets confused on things too. He can do addition and subtraction, but there are times when he'll also say 5+3 is 53, etc. He seems to have it in his head that it means both things right now.

I'm going to have to locate my copy of that book.

On this, while he's been doing the 100 board his favorite game to have me play with him is the "which number is bigger" game. I will give him 2 numbers like 86 and 49 and have him tell me which is bigger or smaller. That and addition/subtraction games are big with him.

[Rose-Roget]

Originally Posted by MattBronsil

The 100 board comes before addition and the bead cabinetN so the answer that 5+2 is 52 seems right from what he is learning now. He is identifying 2 digit numbers so when you ask about putting them together, he thinks about what he is working with. He might even understand the concept of addition, but not the terminology. Try asking him instead, "If I have 2 apples and I buy 1 more, how many will I have?". Putting it in a full sentence might show he understands the idea, but not the symbols + and = or the words that go with them.

This is good to hear. So basically, he's processing the information he's learning and figuring out how it all fits together. You're right, if I phrased it in a sentence, he would be able to solve the problem - I just didn't think of that at the time,

I can tell that his interest is really piqued in Math at the moment, because he, too, likes to play games with Math (although, he does the quizzing - until he's confident in a concept, he does the asking). He also does greater-than games, as well as asking me for mental addition of multi-digit numbers (keeps my brain active ) and what number multiple digits strung together make. Since his interest is so high right now, I feel reassured that he seems to be on the right track, so that he progresses and learns. Gotta have faith in the system...

[BCFD]

Not to hijack the thread, but Matt...where would a job such as the 45 lay out come in? Is that after the hundred board? My DD has pretty much mastered the hundred board and I just had a lesson by the teacher on the 45 lay out and had never seen it before. I was impressed!!

[Rose-Roget]

Originally Posted by BCFD

Not to hijack the thread, but Matt...where would a job such as the 45 lay out come in? Is that after the hundred board? My DD has pretty much mastered the hundred board and I just had a lesson by the teacher on the 45 lay out and had never seen it before. I was impressed!!

What's the 45 layout?

[BCFD]

http://www.ehow.com/video_4403337_th...materials.html

[MattBronsil]

Originally Posted by Rose-Roget

What's the 45 layout?

Here is a picture:
http://3.bp.blogspot.com/_rsgetwotaj.../45-layout.jpg

The right side has unit beads, from 1-9 with the numeral cards. Next to that are the tens...then hundreds...then thousands.

It's called the 45 lsyout or decimal layout. (45 is because each decimal place has 1-9. 1+2+3+4+5+6+7+8+9=45).

[Rose-Roget]

Originally Posted by MattBronsil

Here is a picture:
http://3.bp.blogspot.com/_rsgetwotaj.../45-layout.jpg

The right side has unit beads, from 1-9 with the numeral cards. Next to that are the tens...then hundreds...then thousands.

It's called the 45 lsyout or decimal layout. (45 is because each decimal place has 1-9. 1+2+3+4+5+6+7+8+9=45).

Okay, I've seen that before; I just didn't know what it was called. Thanks!

So where does it fall in the progression?

I ordered a copy of the Parent's Guide to Montessori. I'm looking forward to it. I seem to have a lot of questions arising these days, and I'm starting to wonder if M. is really the best for ds. Can't really think of a better alternative for him, though.

[MattBronsil]

Originally Posted by Rose-Roget

Okay, I've seen that before; I just didn't know what it was called. Thanks!

So where does it fall in the progression?

I ordered a copy of the Parent's Guide to Montessori. I'm looking forward to it. I seem to have a lot of questions arising these days, and I'm starting to wonder if M. is really the best for ds. Can't really think of a better alternative for him, though.

::Matt reaches into his bag of Montessori standard answers and pulls out a random one. He reads it quietly, looks at you, smiles, and speaks.::

"It depends on the child." Calmly, he sits back, proud of the answer he gave.

OK, not a great answer, so let me try to add more.

The 45 layout and the 100 board are very different concepts. The 45 layout teaches the concept of place value. The child mightknow that 2157 is "two thousand, one hundred and fifty-seven," but they do not necessarily know what that means. With this activity, they begin to see and feel those numbers.
The 100 board focuses on rote counting. It is counting and identifying those numbers, but they don't necessarily have to have a perfect understanding of place value to do that material.

These two concepts are taught individually and in an isolated way. The two concepts are put together somewhat in the teen and ten boards, which he may have already done, but really come together in the bead cabinet and many other work, such as the snake game (an addition work) or the bank game (which works on building these numbers and exchanging them).

The math curriculum puts a lot of emphasis on learning a certain skill (rote counting), learning another skill (place value), then connecting them together so the student can make the discoveries and connections. If it helps ease your mind any, take a look at the standards for math in the regular public schools for this age. You'll see how far ahead he'll be at the end of it all.

[Rose-Roget]

Thanks for that information, Matt. I wasn't expecting him to be working on the 45 layout at this point, but it's good to know where it fits in. I don't doubt at all that M. teaches skills far ahead and in a much different way than traditional - I'm just trying to find the right fit for ds. I really believe in Montessori and I need to sit back and let things be. I guess I'm just feeling neurotic at the moment , and keep questioning whether he is at the level he is supposed to be at in reading as well as math. Ds isn't exactly an overachiever, and tends to get people to underestimate him. So I feel like I need to stay on top of it, so that he gets the most from his education...and we get the most from our money. But I digress... Thank you!

[spedteacher30]

I guess then, the question to me would be, "What are you trying to get out of his education?"

We're probably the only parents in our son's class who don't quiz the teachers on his academic progress during conferences. We ask about what he enjoys, and his social interactions, because we have faith that he will learn everything he is expected to learn. it might be on his own timetable, but it is obvious to us that he can learn and that he enjoys learning.

We figure that the Montessori approach is giving him a tremendous luxury: the luxury of time, of space and of opportunity. He is only ahead or behind if we compare him to other children. As long as he's at a Montessori school, he won't be compared to other children, so he's developing on his own schedule.

He probably could be pushed to do some things that he is academically ready for (like reading) but he is not yet emotionally ready for them (he tells us, "I can read, but i don't want to because I want you to read the books to me!") I am sure at some point, he will be pushed or led a little more than right now, but it seems to me we owe all of our children at least a few years of school where they aren't compared to others, where they aren't pushed to meet standards, and where they are free to explore and develop their own interests based on their own timetable.

[Rose-Roget]

Originally Posted by spedteacher30

I guess then, the question to me would be, "What are you trying to get out of his education?"

We're probably the only parents in our son's class who don't quiz the teachers on his academic progress during conferences. We ask about what he enjoys, and his social interactions, because we have faith that he will learn everything he is expected to learn. it might be on his own timetable, but it is obvious to us that he can learn and that he enjoys learning.

We figure that the Montessori approach is giving him a tremendous luxury: the luxury of time, of space and of opportunity. He is only ahead or behind if we compare him to other children. As long as he's at a Montessori school, he won't be compared to other children, so he's developing on his own schedule.

He probably could be pushed to do some things that he is academically ready for (like reading) but he is not yet emotionally ready for them (he tells us, "I can read, but i don't want to because I want you to read the books to me!") I am sure at some point, he will be pushed or led a little more than right now, but it seems to me we owe all of our children at least a few years of school where they aren't compared to others, where they aren't pushed to meet standards, and where they are free to explore and develop their own interests based on their own timetable.

Yikes - I've veered way off the topic here! But thank you for your response, Spedteacher, and you pose a good question. I think I'll start another thread to respond.

[Rose-Roget]

Another question: What are the pre-requisite skills/works to play the Snake game or Bank game? Ds told me he's finished the 100's board. When we were talking about works, he said he wanted to learn the Bank game and the Snake game. I didn't want to encourage him to ask for a lesson if they were just going to tell him he wasn't ready. Maybe I'll suggest it and prompt him to find out what he needs to learn first, so that he can set a goal to keep him on track. I wonder if the mystery of the progression is as frustrating to him as it is to me...

[MattBronsil]

Originally Posted by Rose-Roget

Another question: What are the pre-requisite skills/works to play the Snake game or Bank game? Ds told me he's finished the 100's board. When we were talking about works, he said he wanted to learn the Bank game and the Snake game. I didn't want to encourage him to ask for a lesson if they were just going to tell him he wasn't ready. Maybe I'll suggest it and prompt him to find out what he needs to learn first, so that he can set a goal to keep him on track. I wonder if the mystery of the progression is as frustrating to him as it is to me...

If he knows the 100 board, he should be ready for at least the introductory snake game, for the research of ten. As long as he knows 1-10 and knows the colored beads of 1-10, he should be fine.

I also found this great book on amazon:
http://astore.amazon.com/monteblog-20/detail/B001ETYZQQ

I think it's out of print, but have a friend that had a copy. It REALLY does a great job explaining what the math curriculum is and why it works. I can't find that book easily and it's sold through a 3rd party on amazon and they only have one available. Good luck getting it.

Matt