One thing that struck me about the math talks given at this weekend's New England Conference on the Gifted and Talented was the emphasis on manipulatives and the concerns about whether children understand place value. Are these the most appropriate things to be focusing on when it comes to students who are gifted in math? The mathematically gifted kids I know grasp place value and other aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade. But the education establishment seems bent on convincing itself that children--however gifted--don't understand place value.

Why would you want to convince yourself of this? Because it gives you an excuse not to teach the standard algorithms of arithmetic. If children don't understand place value, then they can't understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though

mathematicians disagree), using them will permanently

harm their mathematical development.

So, given how nice it would be not to feel any pressure to teach the standard algorithms (because, let's admit it, they are rather a pain to teach), wouldn't it nice to convince ourselves that our elementary school students, however gifted in math, don't understand place value?

But how do you convince yourself of this? As that ground-breaking math education theorist

Constance Kamii has shown, it's child's play. All you have to do is ask a child the right sort of ill-formed question. Here's how it works:

1. Show the child a number like this:

2. Place your finger on the left-most digit and ask the child what number it is.

3. When the child answers "two" rather than "twenty," immediately conclude that he or she doesn't understand place value.

4. Banish from your mind any suspicion that a child who can read "27" as "twenty seven" might simultaneously (a) know that the "2" in "27" is what contributes to twenty seven the value of twenty and (b) be assuming that you were asking about "2" as a number rather than about "2" as a digit.

## 11 comments:

Hmmm. You're making me feel less bad about not attending. At least, it sounds like I wouldn't have gained much guidance for how to deal with the math issues in this household.

I was surprised to find my son's fifth grade math class starting out the year with place value. I didn't expect to see an emphasis on it at that grade level, don't recall it from when my older child was in that grade. The only rationale I could think of was that maybe last year's class had done poorly on that part of the state tests.

At least they didn't use manipulatives! (I think...)

Oh man! This is exactly what I've been noticing about the way reading gets taught these days. Reading teachers are convinced that there's tons of kids out there who read perfectly fluently but don't "comprehend".

This is because they ask the kid a question like "what do you think will happen next?" and the kid says "I don't know." They think this proves the kid "doesn't comprehend".

ARGH! It's like they're just inventing problems out of thin air, as if nature doesn't provide us with enough.

On the other hand, when you complain that your bright child can no longer do simple calculations that she once understood, they say, "but she understands place value!"

I've never understood all this concern about teaching place value. It's not a particularly difficult concept and it is a crucial foundational concept that shouldn't be put off. I introduced my daughter to tens and ones at the age of 4. She's not gifted but she understands it. She is now doing online school with the California Virtual Academy and they cover place value in a K-Grade 1 online math class.

If students really are not able to grasp place value, it can only be because schools aren't teaching it properly. If I'm not mistaken, most countries introduce place value in 1st grade. If their students get it and ours don't, we have to seriously look at how math is being taught.

That is most curious. There are, of course, children who at 4th and 5th grade are still remarkably shaky on numbers in a way that place value work (yes, even with manipulatives to some extent) is useful (I tutor some of those kids). But a gifted and talented conference? What? Those kids (almost) all have a great grasp on numbers (I'm a parent to some of those kids), and going back to manipulatives would be a serious waste of their time. Talk about not knowing your audience!

The idea that you shouldn't teach an algorithm before children can developmentally understand a concept is really dangerous. Often teaching the algorithm first will actually make it far easier to understand the concept later on. There is no harm in telling a child that these are the steps you need to solve a problem. Later on the why will make sense.

We've got this problem now that kids don't understand the concepts because we don't teach them the algorithms and we're not teaching the algorithms because they don't understand the concept. It's like being stuck in an infinite loop.

I think an important idea touched on here is the connection between understanding math concepts and being able to verbalize those concepts. The connection is not necessarily very close. Some people would argue that "If you can't say it, you don't know it". I don't agree with that. And the contrapositive, "If you know it, you can say it" is obviously just as untrue. So what is true about knowing and saying? My perspective would be that saying is very valuable to knowing, and knowing is very valuable to saying. We should try to keep them together as much as we can. But we always need to remember that they can be separate, and often are.

I have a few years experience teaching college freshman math. Here is something I have noticed again and again when trying to help a student in my office. I will have the student attempt a problem, and I will offer as much explanation as I can. But time and again I will be struggling to put together a string of words that concisely expresses whatever idea or information the student is missing, when the student will suddenly say, "Oh, I get it!" That's my cue to shut up.

I am not a constructivist in the usual meaning of the term these days, but the constructivists are very much right about the central idea of the learner constructing his or her own knowledge. People who call themselves constructivists, however, don't seem to apply the really important meaning of constructivism. When a student comes to me for help, and after a bit of preliminary groping to understand the situation, I say, "Let's try this problem here . . .". In doing so I am setting up some elements that the student may use to construct knowledge. The student takes the problem and tries to assemble the elements of that problem into some sort of meaningful structure. I offer help as best I can, which typically includes verbalizing the essential math concepts that are involved. But my verbalizations are imperfect. The student's understanding of what I am saying is imperfect. Understanding comes when the student manages to take the elements of the problem and see how they are to be assembled together in a meaningful way as required by the problem.

I think a very common fallacy for teachers is to think, "I explained it. They understand it. Therefore my explaining caused the understanding." That's not totally fallacious, of course, in most cases. A good explanation can be a powerful contributing cause to understanding, often even a necessary cause, but not necessarily a complete and sufficient cause. We don't just explain. We also assign problems. Every problem is a set of elements that the student must assemble into some form as required by whatever mathematical idea is being taught. Students must indeed construct their own learning. It's strange that adherents of constructivism seem to want to do anything but deliver to the student a carefully crafted "learning kit", which is what the combination of a good textbook, a good explanation, and a well chosen assignment is.

All this points to the idea that understanding, or the lack of understanding, is not necessarily easy to diagnose. Suppose I ask a student, "How many thirds make a whole", and the student looks a little puzzled. (And I have had a few occasions to suspect college students may not understand the basic meaning of a fraction.) Whether in third grade, seventh grade, or even college, a blank look may mean the student is wondering just what I mean and where it is leading, or it may indeed mean the student really doesn't understand the very simple primitive meaning of a fraction. Or, I may ask a student that same question and the immediate response is "three" with no hesitation at all. Can we take this to mean that of course that student understands the basic meaning of a fraction? I don’t think so. We should not take this as full and definitive confirmation that the student has the basic understanding of what a fraction means that we want that student to have. It is simple one bit of evidence

Understanding, I would argue, is never easy to assess. It normally must come from observations and analysis over time.

Unless you're an ideologue, of course. Then it's amazingly simple and easy, as examples given attest.

And, if I may mention it, I have elaborated on what I consider real constructivism at http://www.brianrude.com/constv.htm.

Thanks for all the great comments! Look for some of these later in my "favorite comments of 2010" series.

"Discovery" or "constructing ones own knowledge" comes about through careful hints and prods as Brian Rude so accurately points out. In the examples he provides, what he is talking about is "scaffolding". Start with something the student understands and use that as the springboard for the concept/procedure you are trying to get across.

If an assignment is constructed properly, sudents take away a good understanding of the material by the time they are done whether conducted in class or as homework Sometimes this manifests itself in an an "aha" experience brought about by procedural fluency as Brian Rude pointed out with his "Oh, now I get it" example. In tutoring students, I've seen that the procedural fluency resulting from the exercises helps clarify the concept, even if it wasn't fully understood before starting the problem set.

I also agree with Brian's characterization and thoughts on "understanding". (See http://www.educationnews.org/commentaries/opinions_on_education/93277.html

I recall walking along line of black cherry trees during first grade (1956-57) to pick up sticks. [Apparently we were too poor to buy "manipulatives", or our teachers knew when and when not to spend precious resources on such things.] The sticks were brought into the classroom where some were kept as “ones”. Others were bundled into “tens” with rubber bands. Some of the tens bundles were (without removing the tens rubber bands) grouped into ten and bundled with a bigger rubber band into “hundreds” bundles.

These bundles could be grouped and regrouped, (carried to and borrowed from) much more easily than can be done with those yellow cubes and sticks and flats that cost a lot of money. Let’s see what happens when a student tries to “unbundle” (break) a stick or flat.

But I digress. We learned place value both on the blackboard and with physical models that showed what we were doing on the blackboard. We combined rote with logic and visualization. It worked.

Why do we have so much trouble remembering what works?

Mark Bohland

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