"Researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.And why would I, the Cuisenaire Rod enthusiast, find anything to agree with in this?
"Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods."
Charlotte Mason wasn't always in favour of commercial manipulatives and models, either. (You can see a set of math manipulatives (by Adolf Sonnenschein) that she described here.) [2012 update: that link has changed, but there is a similar photo here.] She didn't want pre-made models getting in the way of students doing their own thinking. She didn't want them getting too dependent on rods. (She also didn't like "drawing in chequers.")
However, she did make use of both beans and dominoes as teaching aids. (Dominoes were used as a sort of addition flashcard: young students were to learn all the combinations in the set.) Longtime CM user Lynn Hocraffer wrote an interesting article, "Seashell Math," about using a bag of shells to help her son who couldn't "see" what the numbers in arithmetic were about. She says, "We did use all the sections and activities [of the math program] with my son, but I was dense and didn't do the manipulatives until the end. Dumb me! I knew my son was a kinesthetic learner, but because he recited so well I took a while to realize he didn't know what he was saying! He could "Reason", that is he could follow in order, but it had no meaning, no application."
I've talked to a lot of homeschoolers who have gone one direction or another in choosing math materials, sometimes after trying several approaches. I know people who have had enough with the "math toys" and are now back to using the Victorian-era Ray's Arithmetic. I know other homeschoolers who never really felt they "got" math themselves until their kids started using Math-U-See with its colourful blocks. The division might be right there, between those teaching parents who are already comfortable with math and who do better without all the "blocks and whistles," and those who can communicate the same concepts without gimmicks. I believe that there are also children who learn just fine without having to see or handle manipulatives; and there are others who need that visual or kinesthetic boost to make sense of it. But is there a place for those of us who aren't math majors but still feel like we have a pretty good grasp of what needs to be taught and just prefer to teach it with some kind of manipulatives? (And for how long?--are manipulatives to be encouraged in the primary grades but not in the upper years?--or are they to be eschewed all the way along?)
I remember being in the first grade (in a rows-of-desks classroom) and not being able to figure out why 1 - 0 = 1. I was supposed to be one of the smart kids in the class; I went to the second grade room to do reading every morning. But those arithmetic drill sheets with their 0's really threw me. Nobody ever gave me a clear illustration or explanation of why 1 - 0 didn't equal 0.
The rest of my elementary math education (at a push-the-desks-into-groups school) was so forgettable that I've pretty much forgotten what we did do. I remember our spelling series perfectly (waste of time--I already knew how to spell), but I don't even think we used math textbooks. This was during the experimental '70's, in what was supposed to be the most up-to-date local school (the one with all the learning centres and extended classrooms). But we were still being taught by teachers who had learned more traditionally themselves; so what I do remember is a rather schizophrenic mishmash of times-table drills, problems on the blackboard, reams of purple "ditto" pages, and occasional forays into manipulatives. "Here are the attribute blocks. Follow the directions on the Learning Cards." We looked forward to those manipulative occasions not because we were learning anything but because--obviously--they were a chance to play in class.
And this--I think--is where Mr Person and I are getting onto common ground. According to the article he quotes, teachers like manipulatives for various reasons, one of which is that giving kids something to "play with" may keep them out of trouble longer. Educational suppliers like manipulatives that are required to use a particular curriculum--they really like them! Kindergarten suppliers had that one figured out over a hundred years ago. I don't think that homeschoolers are quite as bombarded by silliness as classroom teachers are--most of us couldn't afford all that stuff even if we wanted it. However, we too can be bewitched by all the neat stuff on the conference tables. It's colourful and it's fun. But the big question is, always--does doing or using whatever it is help you learn the subject better? (I go back to that question constantly; and it's expertly expounded in Mary Pride's book Schoolproof.)
In our own homeschool, with our children, using rods the way we use them, the answer is yes: I've posted about that here, here and here. I've written elsewhere about our oldest, who never quite saw the sense in Cuisenaire rods. She was more of a "just show me how to do it" math learner; however, I used them with her during the first years of school anyway, and had her complete the book Spatial Problem Solving with Cuisenaire Rods later on. Our middle one ("Ponytails"), on the other hand, relates to relationships, and the point of Cuisenaire rods is relationships. When I asked Ponytails whether she thought that the rods had actually helped her learn math better, she enthusiastically agreed and started talking about how the "one rod" could be a "ten" and vice versa. Of course she did use a variety of manipulatives and methods throughout the primary years, including a hundred chart and an abacus--so I suppose that was one way we avoided having her depend too much on one particular aid.
Now that she is in the fourth grade and done with Miquon Math, I notice that we hardly use the rods in day-to-day work; obviously they're not helpful in the nitty-gritty work of learning to multiply multi-digit numbers on paper! And although some of this heavier-on-the-pencil math isn't easy for her, I've never noticed my rod-user having any particular problem transferring her knowledge onto paper, either.
Is there a difference, then, between what we're doing and what Mr. Person is debunking, and what Charlotte Mason didn't like? What our teachers were doing when they occasionally hauled out the rods or the attribute blocks, or brought in the new-fangled Betamax to show us a video with a catchy song about finding area? I think there is, because in my homeschool classroom manipulatives aren't just busywork; I have no reason to want to make my lesson take longer than it has to! (The weather's good and the kids want to go outside...) If the rods or other manipulatives can illustrate a point or a relationship clearly (and more conveniently than having to count out multiple piles of beans), then we will use them. And although our preschoolers (Crayons especially) made an art form out of Cuisenaire floor constructions, that's not what we do during math time.
While I can see exactly what Mr. Person is getting at (and why one of his commenters referred to "Happy Meals" and "McMath"), I'm still going to hold on to our rods.
