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.

The digitization and current state of Chinese classrooms

I went to a talk today hosted by the SFCCC (Culture Center of Taipei Economic office) today about the state of teaching Chinese today, the usage of digital devices in the classroom, and how parents can help their children learn Chinese.

The speaker is a teacher teaching high school Chinese.  It seems that her viewpoint is from one of watching children who are learning Chinese in Saturday Chinese school or high school level.  Basically children who are not necessarily in immersion programs, bilingual programs, or somehow going to school daily to learn Chinese.

Though this is not where I am coming from, I still found the talk to be interesting and had some take away points.

She started by talking about the rise of Chinese language education in the US, in terms of government focus and investment, which arose after September 11.  There was a realization that learning a foreign language is a good thing.  This lead to the establishment of StarTalk program, which trained teachers and also conducted language summer camps.  Then they went down a level and established the AP Chinese test to entice students to learn Chinese.  Supposedly after that they realized learning a language from 9th grade on was still not enough; since Chinese is a hard language to learn.  Thus the establishment of immersion schools and children learning Chinese from an early age.

She talked a bit about common core and AP Chinese tests.  This was a revelation #1 for me.  The emphasize on these tests is the ability to communicate (hence listening and speaking skills very important), and the breadth of knowledge in Chinese (thus not so important to learn all about culture, which we heritage learners like to emphasize) such as global issues, science and technology, etc.  She noted that Chinese schools have been so concerned with the culture part but really they’re not too important or relevant to the children here.  It’s not that we shouldn’t learn them, but there are other things the children should know how to talk about other than the lady on the moon or the origin of the Autumn Festival.  And she also talked about how students who are learning Chinese as a second language (e.g., no native speaker parents) sometimes do better than the students who have native speaking families because of what they’re taught in the classroom.  That parents who send their children to Chinese school have too low of a threshold, often for them attending Saturday Chinese school is enough.  She gave an example of a father writing homework for his son.  But we really, as parents, should ask for more than that from our Chinese school.

Her take away point here was that the Chinese classroom needs to emphasize speaking skills.   She described the AP test and gave an example of a sample question, where the tester is given a scenario and needs to listen to some multiple choices on what a proper response would be.  She noted that in textbooks there are often yes and no answers but native speakers don’t speak that way.  In the slide she showed, a “corn” person says to a “popped corn” person, “Are you hot?”  And given the illustration of the “popped corn” doesn’t look too happy with the question and his answer wouldn’t just be yes or no, but rather maybe something sarcastic like “You think?”

The speaker spent a better part of the talk also talking about how classrooms are taught now a days.  In the traditional Chinese classroom, you’ve got a lecturer.   In the last 20 years, we have moved on to a more group method of teaching.  And she says that in the last few years, we’re starting to move to a “self-learning” method of learning.  Where coursework is individualized with the help of technology.  This was not-quite revelation #2.  I say not quite because really, this is the Montessori way of teaching: auto-education.  This also allows the students to practice their speaking skills because they’re now not waiting for 1/28, 1/30th of a teacher’s time to speak to the teacher.  They can practice with each other instead.

She showed us pictures of her high school classroom.  There is a picture of students all sitting in front of laptops.  But they’re not necessarily all doing the same thing.  For example, maybe she has given the assignment of practicing writing.  You could have some students using the laptop to type, some using it to practice hand writing digitally, and others not wanting to use the laptop at all.  The student chooses how they want to utilize the tools, and the teacher is the guide.  She joked that she spends most of her time taking pictures while the students work.   There were lots and lots of photos of students engaged in fun learning activities (games), that really helped them do group work to practice their listening and speaking skills.

Another thing she mentioned was how technology is changing what we need to learn.  Revelation #3 was the non-importance of learning to write, proper stroke order, etc.  Of course these are all good things to learn if you have time.  But remembering where she’s coming from (AP students learning Chinese), does it really matter in the long run when the computer can finish your typing for you.  For all we know in 5 years, we can just dictate and the computer will type for us.

Though I work in technology and I know this to be true, this was really hard for me to accept when I imagine what the classroom curriculum should look like.  I obviously am still most comfortable with the way I was taught Chinese, which involved lots of repetitive writing, memorization, etc.  This relates to revelation #4, which is that what we teach should be relevant to the students.  Again, all familiar ideas to the Montessori teacher.  It is important to teach things relevant to students because they can learn better.  She gave an example of textbooks always having sample conversations about “What sport do you like to do?”  And she questions why we need to do that.  A good teacher will know the goal isn’t to learn all 20 vocab words related to sports at the end of the class.  But rather, the goal is for the student to be answer the question (again the emphasis on language used as a communication tool).  So every student could have a different answer and learn different vocab, but it’s all good.

Combine the idea of auto education, individualized learning, emphasize on speaking and listening, she talked about how the teacher is the guide who doesn’t necessarily have all the answers but rather enables students to know how to find answers (very Montessori idea), with the help of technology.  Students can be given assignments and know how to google or use an electronic dictionary for words they do not know.  They can use it to practice listening, etc.  She talked about how she doesn’t need to worry when she leaves the classroom to go on trips because the students all know what they need to do, and all of those learning are done without her to begin with.

I’m going on and on but I’m not sure I’ve got the gist of everything she said.  The slides are on the web anyway.  Her ideas were not all new too me, except the part about really emphasize speaking and not worrying about stroke order, writing etc.

Her students are different from my kids so I’m not sure how much I should let it apply to our curriculum.  I did really enjoy looking at learning chinese through the lens of Saturday Chinese school, after school programs, or just starting in high school.  It is not normally the crowd of parents I talk to, who have much higher standards of Chinese for their kids and do require that they speak at home or send their kids to immersion school.

Oh, one thing she said was that AP Chinese level is comparable to 2nd to 3rd grade Chinese level in Taiwan, other than concept.  Because obviously this is high school/college level, and the test will have vocabs and concepts that are older and a real 3rd grader wouldn’t understand it.

Is there such a thing as joyfully learning Chinese?

I went to a talk given by a bilingual Montessori school last Saturday. It was supposed to be about how to joyfully teach Chinese. I spoke with my friend afterward and she said that she didn’t think it helped. I mostly agreed with her. We were really looking for specifics; teach characters this way, introduce zhuyin that way! But they didn’t really go into that kind of detail. It was more of an explanation of how they teach Chinese at the school, which we felt were like “d’oh, of course!”

However, parents from a FB group I’m in are coincidentally discussing how they teach characters over the weekend and I realized that it IS different from how most people teach Chinese. So I’m going to write it down here for my reference before I forget.

I have to say, not having had kids who are grown means that I don’t know if one way is better than the other.

First they talked about how they don’t teach: No writing characters 5000 times like we learned it when we were in elementary school. They use a variety of tools for the children to practice. For example:

Creating a booklet of calligraphy zhuyin
A cabinet of chinese characters for the children to lookup when they want to write, sorted by zhuyin
Matching of English and chinese words. I did not like this activity but I can see why you may want to use it for non native speakers.
iPad game app that lets you practice chinese stroke order
Another iPad app to practice zhuyin
Initial sound boxes

They also talked about the way we learn, that we need to recall (test) with various time lengths in between. Hence they have the variety of activities.

The presenter of course talked about why you want to learn zhuyin and traditional characters. The zhuyin reasons are the same as mine. Except they added the fact that it allows you to introduce both zhuyin and the alphabet between 3-6.

They also talked about introducing 80 component characters by the time you finish kindergarten, which is different from how the local Chinese charter school teaches. The idea is that with these 80 characters you can use them to combine into more complex characters. Along with learning zhuyin it will allow you to begin writing once you start elementary.

Oh and lastly their ambitious plan is to introduce 750 characters by 3rd grade. That’s almost at grade level in Taiwan. To which I say not likely unless the child has been in preschool and knows 150 to 200 by the time they start elementary. It’s just too hard on top of learning English. But that’s just me speculating.

I’ve been mulling over the talk the last few days. What I got most out of the talk is the recall and testing. It jives with what I just realized last week. The Montessori materials provide a variety of ways to practice a concept, sometimes in a spiral learning pattern that spans years. There is no time or restriction which says we will cover this over one month and be done though most schools do have a curriculum/theme they follow. This means the child is free to revisit a concept again and again, which aids retention of what they’re learning.

Really GETTING this now means I’m not stressing so much thinking how is Thumper supposed to cover all the different math topics if she’s only doing 3-5 units of work a week? I know now to provide a variety of presentations and activities for her to work on over weeks and months and this is a better way for her to retain info rather than working on one skill/concept at a time only. Supposedly the “How We Learn” book that just came out covered this too.

I have my own ideas of how I will teach but they are similar in concept if not in implementation. Another post, another day.