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My nearly-5 daughter loves math and science, so that's pretty much all we do homeschooling-wise at this age. For math, we've started out with Singapore Earlybird, which she enjoyed until we hit the wall during the book that is almost all learning how to write the numbers. Let's call her a reluctant writer, and at this age I'm not worried, so we plug ahead with me transcribing for her. Nonetheless, she's currently a little spooked with Singapore, so I purchased another curriculum to supplement for now.
I got Miquon. I've been reading through the teacher's manual (the "Annotations" book) and it is unapologetically constructivist ... and yet it does not appear to be totally lame. For example, when teaching addition that requires carrying, they make it clear that at first you let the kids try to work out on their own what to do with that extra 10 in the ones column and come up with their own solution. But here is where it seems to diverge from the more modern day constructivist curriculum -- they specifically state that the reason why you do this is so when you then show them the standard algorithm, the kids will then be able to appreciate its elegance. I don't have Everyday Math in front of me, but I suspect that they don't end on that note.
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" ...they make it clear that at first you let the kids try to work out on their own what to do with that extra 10 in the ones column and come up with their own solution."
Some form of this technique has been used by teachers forever. You would like to lead students along a path to the light bulb effect. It doesn't always work. Actually, I don't think it works very often, and I don't agree that the effort would make students appreciate anything. I think it would frustrate and annoy many of them. It also takes a lot of time if it is done on a regular basis.
Homework problems, however, can be given to better lead students toward discovery in a planned, incremental way, while they are achieving mastery. But discovery using homework is not in fashion. Schools have to waste class time having mixed-ability groups talk and talk and talk. You would think that some kids would claim that their learning style is "quietly". I know that when I have to learn something new, I don't go looking for a group.
Even if kids look for what to do with the extra 10 all alone, the time and effort are not worth it. I have had many light bulbs go on while being directly taught. Constructivism is neither necessary or sufficient for full appreciation, understanding, and mastery. In most cases, it wastes a lot of time.
I see a lot of value in a lot of stuff that's associated with constructivism and I know many homeschoolers have success with programs like Miquon so I wonder why it often seems a good fit for homeschooling but a poor one for school.
I think it's what you mentioned but also think it's that the kids aren't ever on their own. There's still an adult right at their shoulder to poke and prod and guide. Can that happen in a classroom full of kids?
One more thing that occurs to me is that so many homeschoolers don't adapt just one method. Those with Miquon often picks something like Saxon to complement it. The Math-U-See users might also have Singapore. When it comes to math in school it often seems polarized and uncompromising.You're either one or the other and there is no middle ground.
I agree with steve.
the best way to appreciate base 10 and why you carry is AFTER MASTERING the carry. NOT before. kids don't appreciate "elegance". Sophisticates do. Sophisticates know enough to say "wow, that's beautiful and elegant" because they can understand the value of the new invention, and that's because they already unerstand what the invention does well enough to MUCH LATER step back and think about it.
e.g. you really want to appreciate the elegance of base 10? do addition of 5 numbers, all in roman numerals.
but to do that, you'd have to teach roman numerals, and how they aren;t a base system. you'll have to spend more than just a minute of 5 to get there. distracting your child from learning how to carry by having them get muddled about the right way to do it is dangerous as well as inefficiemt. they might just misremember, as they have "mastered" their own confusion about the number scheme more than the rules you'll teach after the fact.
so, i'd say while this may be much better than EM, and yes, this is probably as good as it gets for constructivism, it's still a waste of time for the student and worse, likely to cause confusion and undermine any teaching to mastery.
successful teaching tightly controls the mental path down which students walk, so that no wrong or irrelevant inferences can be made. wrong ones get instantly stuffed into belief systems and it's darn hard to undo what gets builr on top of them. irrelevant ones obscure the truth.
imagine your child decided to add 27 and 18 his own way, by writing down
27
+
18
=
15
30
=
45
he didn;t "carry" the 1, but he did acknowledge and write it down. he scheme works, doesn't it? all good right? except that's the format we use for multiplication....and when he learns that he might get mighty confused mid operation about which calculation he was performing. but you've primed his brain for THIS way first and he might spend weeks (or years) saying "why can't I do it my way..."
Are they shooting for a light bulb effect or an appreciation effect or both?
i.e., greater comprehension versus greater appreciation -- ?
My conclusion thus far is that one-on-one teaching gives you an amazing margin for error.
I suspect that's the reason Reading Recovery seems to work as well as it does. The student/teacher ratio is extremely low. (I think it may be 1 to 1 - ?)
The waste of time issue is huge. The past 3 years I've felt as if my school district was scooping up huge heaping armloads of my child's time and simply throwing it in the trash.
That emotion culminated with the dental dam riot.
The guidance counselor at Hogwarts told the kids yesterday that they have 150 days of school as compared to 180 in public schools. "We get more done in 150 days than public schools do in 180," he said.
He knows this because his wife teaches in public schools.
When you have direct instruction in character ed (requiring monthly character ed assemblies that effectively take the entire day in terms of disruption to the schedule), constructivism in the liberal arts, and heterogeneous grouping you've made an institutional commitment to using time as inefficiently as humanly possible.
The character ed situation is particularly egregious, and I plan to get something posted about it. C. is now in a CATHOLIC school and they don't spend a full day every month attending character ed assemblies.
I don't know if it is what constructivists are shooting for, but understanding place value is fundamental to success in elementary. In my experience at our elementary, the students who grok have no trouble with moving to the algorithm on paper and need little skill repetition to acheive mastery, but rarely does a child who is using procedural memory and rote fact memorization succeed in grasping the concept after practicing the skill thousands of times. The latter group tends to hate math and has major trouble with Math A (now Integrated Algebra I), while those that grok are the math club kids.
Our district does no paper math at all in K-1 - it's mental or with concrete objects unless the child wants paper math. Homeschoolers that do Earlybird sometimes use number stamps and an ink pad to get around the writing issues.
Does Miquon actually call it 'carrying' or do they use the term 'regrouping'?
Most constructivists see the process as the goal and not the end result. They think that there is some sort of required magical something that happens during constructivism that doesn't happen during direct instruction.
Unfortunately, many constructivists see it as a group process starting from a blank slate rather than after a detailed introduction by a teacher. As everyone knows, many students are a long way from understanding even after direct instruction. I've spent countless hours on homework discovering (understanding) what a teacher directly taught in class. It's not a good thing to start the discovery process from a blank slate. You can't discover everything, but it's strange that we never hear about exactly what should be discovered.
When Everyday Math talks about different ways to multiply numbers, they surely aren't talking about an algebraic form of understanding. They don't even get to the understanding that each digit of one number has to be multiplied by each digit of the other number. They could then have kids use that understanding to design their own multiplication algorithm.
Constructivists tend to believe that homework is just rote practice of already learned material. They think it adds just speed and not understanding. This is a very simplistic view of discovery that allows them to implement all of their pet ed school ideas of mixed-ability group work.
It also allows them to reduce the importance of mastery. If mastery is speed, not understanding, then they can unlink the two. That's what Everyday Math does. They assume that mastery will come as a by-product of the process. If that is true, then EM would not be packed with tons of Math Boxes in sixth grade.
I said long ago that there are many levels of understanding, each of which is tied directly to mastery of certain skills. Many people (myself included) find that a focus on mastery of skills is the best approach to understanding.
Skills without understanding can be fixed. Understanding without skills is not possible. The goal is not to be in the audience, it's to be on the stage.
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