# More thoughts & notes on Montessori elementary math

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

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

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

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

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

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

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

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

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

Back to the class itself….

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

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

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