I was feeling rather good about how he was doing in math even though I knew that we weren't going through the sequence as fast as some children do. He is just about to start the stamp game. For those of you who are not as familiar with the sequence, this means he has done:
- number rods
- spindle boxes
- cards and counters
- all of the golden beads work
- teens boards
- tens boards
- 100 chain
- 1000 chain
- 100 board
For those of you who that don't know what those materials mean in terms of skills:
- He can count up to 1000 by either ones or can skip count by tens.
- He has not just learned this by rote so that the numbers are "words you say in a certain order", but he understands teens as "ten and another number" and a number like "80" as "8 tens."
- He knows the difference between even and odd up to 100.
- He understands a number both as a single entity and as a "set."
- He can read any number correctly up to 9999.
- He can do addition, subtraction, multiplication, and division with numbers up to 9999 using beads as counters. In the stamp game he will learn to do it without beads. He has not memorized his facts yet.
- He knows his addition and subtraction facts up to 10.
If Kal-El were in public school, this week he would beginning the second half of kindergarten (K5). I know that I couldn't do most of those things when I was in kindergarten and I went to very good public schools. I know that they are not doing most of those things in the kindergarten down the street that he isn't enrolled in.
One of the main reasons I decided to homeschool was because I knew that the boys could learn what the kids were learning in an six-hour day at school in a fraction of that time at home. This extra free time would allow them more time to play and chose Montessori because it allowed them to learn in the same way that they play. The extra free time also allows more time to be involved in other things we wanted to do like sports, violin, and bible study. Despite the fact that many parents who know nothing about Montessori choose it anyway because they have heard about "advanced academics at young ages," I had no ambitions for them to be "way ahead of everyone else." I had a pretty good hunch that they would get more rigorous academics at home than at school whether we intended them to or not, but I have not kept them home for rigorous academics. I kept them home so that all of our lives could have the breathing room needed to be more enjoyable. The only area in which I am intentionally setting out to be more rigorous is in "critical thinking."
The "perplexed" in the title of this blog post has to do with the pacing of the Montessori math sequence. All of my primary math albums (Montessori by Hand, Karen Tyler, Gettman, Montessori for Everyone: Comprehensive List, Primary) go through the large bead frame. The large bead frame is a long way off from where Kal-El is in the math sequence, but I wasn't worried about it because I thought I understood that there was a large degree of overlap between the end of primary and the beginning of elementary. The age ranges for the Montessori classrooms are 3-6 and then 6-9. Ideally whether a six year-old child would be in primary or elementary would be based on their plane of development, but I suspect that in many areas it has more to do with the child's birthdate. I called a local Montessori school when Kal-El was 2.5 to check on tuition and they said that the child had to be 3 by a certain September 1st to be placed in the 3-6 class or wait until the next year. So, in my mind, Kal-El would be in his third year of a 3-6 class and still has a semester and a summer before he starts "elementary." However, as I have been deepening my study of Montessori elementary, I have encountered the idea that there isn't really overlap in math from primary to elementary. If that's the case, we have a lot of work to do before Kal-El becomes an "elementary" child in the fall.
Then, I read the following in my Elementary course materials in a section called "Memorization and How Society Works Against it":
Another big factor in our schools is the age at which the children enter the primary class. Ideally, the children are meant to enter the casa between the ages of 2 1⁄2 and 3 – this way, they have time to get through all of primary level materials, exercises and work. Some schools have problems entering children under age 3 due to licensing policies; many public Montessori programs require age 3 by a certain date (or wait a whole extra year) or even 4. These factors are negatively impacting how much the children can achieve in a given period of time. Children who start primary at ages 4 or 5 are just not going to get through all that they should have gotten in three to three and a half years. We need to look at our goals, and what we can do to get closer to where we should be.
So...at some schools Kal-El would have started primary at 2.5 and would be halfway through his first year of elementary by now. So did we just become an extra year behind? That same section also talks about the importance of the child getting through entire primary math album before elementary:
We hope that the children going through the primary class have the opportunity to work with all of the mathematics materials developed for the first plane of development so that their foundation and factual knowledge in mathematics is as highly developed as it can be. That means that we hope the children are at least in the process of learning the math facts, because that process belongs to the first plane of development when they have the absorbent mind...One thing that is likely to be seen is the struggle that children go through to learn their math facts, if they don’t already know them at age 6. At the second plane of development, memorizing these facts involves an act of the will. They are not just soaking them in anymore, they have to choose. We have to find that motivation that will encourage them to make this choice, if we have children in that situation. We must remember that because the characteristics change, a second plane child cannot be expected to work through all of the primary materials that he might have missed. The materials are designed for children with different characteristics. If we have gaps that must be filled in at the elementary, and usually there are some, the adult will take certain selective pieces of primary material and use them in a different way to help the children learn what they need to know.
Fortunately, the next section is called "How Do We Choose What to Use and How?". The course also provides a large section that we are apparently going to need titled "Remedial Mathematics." Boy, do I feel like I missed the boat. Now Kal-El needs "remedial mathematics"? How did this happen? What happened to my overlap? Gettman divided the primary experience into seven periods and Kal-El sits pretty much in the sixth period of these. I couldn't find the specific spot where it says, but I thought there was some overlap of the sixth and seventh period with elementary.
Most of my other sources implied some overlap as well. My comprehensive list from Montessori for Everyone lists all of the primary math materials on the Elementary list. I have a complete set of 6-9 albums from the Mid-America Montessori Training Institute. Their math album is divided into three books. Book one overlaps my entire Karen Tyler and Montessori by Hand math albums. However, other than tying all the work to the elementary great lessons, the MAMT 6-9 albums seem to eerily contain very little that my primary albums do not, so perhaps that is not something to go by.
The elementary math, language, and geometry albums available at Cultivating Dharma are very different than the MAMT albums and similar in scope to the Keys of the Universe albums. Both CD and KotU have NO overlap from the primary math albums. I looked on the Montessori Research & Development website at the tables of contents for their elementary math albums. Their math album also is a set of three. The first of the three albums overlaps my primary albums by 100%. I happen to own their elementary Language Arts Volume 1 album and it overlaps the entire grammar section of my Karen Tyler album. This also led me to believe that there is some overlap. I'm also influenced by the math materials that I purchased from a homeschooling-mom friend of mine. Her daughter attended a good Montessori school from age three up through the first year of elementary. At that time, they pulled her out and started homeschooling due to health reasons. The first material she had to make for her, for her second year of elementary, was...the stamp game. I also bought from her a complete set of grammar boxes, and the small bead frame. I couldn't buy the large bead frame because, at age 10, she was still using it. The NAMC math guides seem to overlap some of the primary. It seems from the table of contents that their first presentations begin with the addition strip board and I still see the snake games, long and short bead chains, and golden beads mentioned here.
My favorite primary math, language, practical life, and sensorial albums are Meg's albums from Montessori by Hand. These are the albums that I use for the "core" areas to absorb sequence an theory. It matches the Gettman, so I tend to use Gettman for the actual presentations because I can carry one book in my hand instead of four big binders. I do read and re-read the Montessori by Hand albums frequently though so they have likely played a large part in how I think about Montessori. I re-read Meg's math album this weekend and found where she said, "We present the mathematic materials at a very particular time, later than the other areas. There is no rush to start mathematics; it is generally introduced around 4 years of age [my emphasis]." Not the very first, but one of the earliest presentations in her album is for the number rods. Each presentation provides a loose "age" for the presentation. Most albums do with the understanding that those who study Montessori know that in "following the child" these ages can only mean so much. For the number rods Meg's album says: 4-4.5 (after all practical life and sensorial). If you have read this blog from the beginning you may remember that Kal-El showed very little interest in the math presentations I tried to give him until that time. He suddenly "woke up" in math and was ready to go right around that age...at the same time that he finished the bulk of the practical life and sensorial works. If you use the Gettman at all, you may have noticed that math doesn't at all begin until period three (with just the number rods) and doesn't really get going to until period four.
So, I'm perplexed...I find it hard to believe that most children who begin the math sequence around age four would get through the entire primary sequence before age six. I also find it hard to believe that it is inappropriate to wait until age four to begin the math sequence. At the same time I see the argument that these materials are designed for primary, not elementary. Jessica mentions that this is sometimes a matter of switching from AMS albums to AMI albums. However, Meg's training (Montessori by Hand) was AMI as was Jessica's. Both my AMI and AMS albums include all the presentations necessary to get to "the end." I just never saw any indication that I needed to "push to the end" before age six. One of the great things about Montessori is that there IS no "ahead" or "behind" but rather just "right where you are." It doesn't seem likely that in a Montessori environment where every child is working at their own pace that the majority of them would magically reach the end of the math sequence just before their sixth birthdays.
I believe that I started Kal-El on the math sequence at the right time. I also believe I have been "following the child" the best I could. Math is a little different than much of the curriculum in that, as Meg says, "In math, the child can't really discover anything on his own that lies ahead. The teacher's plans are what keep the child moving ahead at an appropriate pace." I did not do daily math presentations when I could have. Should I have? Me Too is currently 4.5 and ready to start the golden beads. If I did something "wrong" the first time, I have an opportunity to do it "right" the second time.
I am depressed because it looks like I've set him up for "remedial" work in elementary. I am depressed because I feel like I held him back from the full benefits of the primary math materials. If it is true that that there should not be much overlap from primary to elementary math we are pretty far off of that mark right now. Jessica specifically states in her albums not to feel like "all hope is lost" but I do feel bad. I never thought I would feel bad about my child not being able to do long division halfway through kindergarten though.
I guess I have a question...IS there any overlap between the primary and elementary? If "no," is that "theoretical" or "a reality." In reality, have most children who attended three years of Montessori primary completed the math sequence through the large bead frame? How closely do Montessori guides control the pace of math work in the third year of primary? Should I do things differently the second time around?
Well, if my imagined "comfort" with Montessori primary led any newer readers to believe I "knew it all" I've certainly blown my cover :) I started out this blog with a lot of questions and in that tradition don't want to hide our "bumps in the road."
Disclaimer: (Added a little later in the day) I had misgivings about posting this not because I didn't want to look bad, but because I didn't want to strike fear into anyone who is in the same place as we are or behind us. I do feel that Montessori IS about following the child and that it will all work out. Please don't let my despair make you feel bad about what you are doing, or Montessori in general. Also, realize that while I quoted from Jessica's albums, I am pulling out just a portion out of pages of what she said. I will make sure she has an opportunity to comment here and please don't let any mistakes I make here reflect on her albums, her thoughts, or her course (which I am enjoying). Seeing quotes like that out of context can never reflect on the actual content of a work.
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