Saturday 9/25 - late afternoon:

My notes from Wu discussing the teaching of division with remainder:

Actually, this was a big "Duh!" moment.

Prior posts from MSMI2010.

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## 15 comments:

This is a lesson on what "equals" means, right?

I gather that he would recommend teaching students to write the remainder as a fraction as soon as possible? That writing it with the R gives the mistaken impression that 53/3 and 87/5 have the same quotient? (17 2/3 is not the same as 17 2/5, so this is more obvious when the remainder is written as fraction.) Or what was his point?

It's a lesson that writing division with remainder as "Something R something" is wrong--because the remainder portion isn't well defined. Not all (maybe not most) of your students will really understand your shorthand. It will be impossible for them to properly recognize that the remainder is really a fraction expressed in terms of the divisor. Some of them will instead be led to believe that they can just claim various other numbers are equal because they both had the same number R number.

If you wish to properly express a number in terms of its quotient with remainder, then you write it as:

x = aq + r

The bigger lesson is that shortcuts for adults who understand are dangerous for students trying to learn. Teaching division with remainder properly requires practice with mathematical precision and mathematical understanding.

No, Wu would not suggest writing the result of division with remainder as a fraction. He insisted that doing so was unacceptable, because when the students are first taught division with remainder, they have yet to see a fraction defined.

His points were 1) that division with remainder is usually taught wrong; 2) textbooks are filled with errors like this that make it impossible for students to see that math is coherent, precise, and sensible; 3) abuse of notation translates into abuse of students, because this is an assault on their ability to make sense of mathematics.

Wu would say that the only correct way to write division with remainder is as a proper equality in the form x = aq + r.

In the case of 87 divided by 5, the quotient is 17 and the remainder is 2. That is 87 = (5 * 17) + 2.

We call 87 the dividend, 5 the divisor, 17 the quotient, and 2 the remainder.

Wu would say that the only correct way to write division with remainder is as a proper equality in the form x = aq + r.Or (x - r) / a = q.

Or (x - r) = a*q.

I think the real problem is not the notation "

nRm" but the use of the equals sign. (So bky and I are on the same page.) The problem is that "nRm" does not designate a unique real number, and therefore it can't be "equal" to any rational number.But that doesn't mean the notation is meaningless; it describes an

ordered pairof positive whole numbers that may be very meaningful in a real-world context. What's more I dare say that in such contexts Wu's argument falls flat.For example, consider the following two story problems:

"53 students are going on a field trip, with parent volunteers driving. If each parent can take three passengers with them, how many full cars will there be, and how many students will be in a not-full car?"

"87 students are going on a field trip, with parent volunteers driving. If each parent drives a minivan and can take five passengers with them, how many full minivans will there be, and how many students will be in a not-full minivan?"

Notice that the form of the question calls for

twonumbers -- one corresponding to the quotient (the number of full cars) and one corresponding to the remainder (the number of extra passengers not in a full car).The two problems have the same solution: 17 full vehicles, 2 extra passengers. That does not mean, of course, that 87 / 5 = 53 / 3. But it does mean that the problems 87 / 5 and 53 / 3 have the same quotient, and the same remainder. And there is nothing wrong with saying that, because it is true, and it may be a perfectly reasonable thing to care about.

The issue is not what is correct for us adults who know a lot of math.

The issue is what's correct for a 3rd or 4th grader who *had not been taught fractions yet*.

So no, you still can't use (x - r) / a = q.

Yes, the division form is mathematically correct for experts, but it is incorrect to teach because the students have not had / defined. Context here matters.

Now: to the issue of whether n R m is well defined, or the issue is the equals:

Had one written

37 divided by 5 # n R m and then tried to explain to a 4th grader that # expressed a relation taking the integer division with remainder to ordered pairs, would you be satisfied? It solves your equality problem, right?

But of course, it doesn't satisfy the issue of how to make sense of division with remainder to children without misleading them, because it doesn't help them understand with the building blocks they already have. And what they've got is counting, addition (repeated counting), multiplication (repeated addition), and some subtraction. They don't have functions, ordered pairs, relations.

So there is much wrong with saying things that are true to you because of your knowledge but that have no definition or meaning for the students.

One of the quotes from Wu that day, in reference to a parent group challenging the common core was:

The irony - mistakes of the past have now become the norm.Seems to apply here, too.

But of course, it doesn't satisfy the issue of how to make sense of division with remainder to children without misleading them, because it doesn't help them understand with the building blocks they already have.Which is why the understanding AND pedagogy are important for teachers. How do you take this information to be true, then hand students a math textbook that uses the agreed upon notation of n R m?

Is it asking too much of an elementary teacher to tell them:

"Well, we know the book is wrong in some instances, but it's what we've got so use it."

or

"Since you know the book is wrong, devise your own lessons, materials, worksheets,assessments." for those concepts.

What about the students? If a teacher says, well, "let me explain this lesson in a different way than the textbook", will students eventually distrust the text? (Not necessarily a bad thing, but does it undermine learning?)

Just thinking...

Good questions. By the way, Singapore explains division (in Textbook 3A in Primary mathematics (US edition and Standards Edition) in terms of the n R m notation.

"Division-with-remainder assigns

a pair of numbers --the quotient and the remainder --to a pair of given numbers (the dividend and the divisor), and as such, it is not an arithmetic operation."

That's from Wu.

But if you're a 4th grader, or even a 7th grader, what you have been told, and all you've ever seen, is arithmetic. A teacher can't just spring this on them, as if they can connect up these dots or gloss over the details of why suddenly the rules of math have changed without undermining the student's knowledge and trust.

i'm with the ones that say

that the "R" notation all by itself

isn't "wrong"... but that using

it in *equations* is a bad idea.

"17 R 2" simply isn't the name

of any number (whereas 53/3

*is* the name of a number,

one that's also known, e.g.

as seventeen-point-six-bar).

we very much need to keep stressing

"the equality meaning of the equal sign"

(i kid you not; see this ramble

by me for a quote from what is

evidently a large and growing

literature on this kind of

"big duh moment").

it's not helpful that many apparently

effective users of math are sloppy

about "=". and in general, we need

better fences between slangy-shorthand

notations and formal meant-as-correct

writing. i have very little idea what

can be done about it. kings always

want the royal road to geometry;

executives always want the executive

summary. have fun fighting city hall

but in the meantime when you say

"=" be sure you mean "the objects

named on the left and right are

the *same* object" (though usually,

of course, having different "names").

otherwise we give up stuff

we can't afford to lose.

like "transitivity":

if A = B and B = C,

then A = C.

pretty important stuff

in *my* moral universe;

i'm *never* giving it up.

not sure if this was wu's main point,

of course. but it sure looks like

the main point to *me* (as of this second).

Can I just suggest that deciding approaches are "right" or "wrong" may be more harmful than the problem you are trying to correct. Perhaps it would be more useful to ask the student to explain in what context they are equal (the same answer). I wold like to offer that there are a great number of applied contextual situations in which 53 divided by 3 is exactly the same result as 87 divided by 5. In mathematics we have all different ways to represent that two things are "alike", ranging from similarity (which has many levels of application itself) to equality, congruence, isometry, homeomorphisms, and on and on.

I am not suggesting that anyone writes out, or even tells a student that 53/3 = 87/5, but we might create a class of ordered division pairs that give the result 17R3, and they may even have interesting geometric properties.

No, it is critical that teachers learn that certain approaches are *wrong*.

This is because mathematics is about the truth you can derive from the other truths you know. If a student is taught false things, they will derive more false things. The math they "know" will fall apart. They will be unable to make sense of school algebra and school geometry, because they won't know what's true or understand what it means.

The problem here is with giving 4th graders something other than arithmetic. 4th graders have only been taught arithmetic: you take whole numbers, do operations on them, and get back *a single whole number*.

Ordered division pairs are a complete violation of that. You can't prep that for a 4th grader in the context of teaching them division--all you will do is make them fail to understand division. That's why you can't even appeal to fractions yet--they haven't been defined.

Can older students make sense of it? Sure. Can students who have been taught about relations beyond arithmetic make sense of it? Can they learn interesting geometric properties? If they've been taught geometry. But you must never forget that students in math have ONLY WHAT YOU'VE TAUGHT THEM as knowledge. You can't start building on what you know--you must build on what they know.

So, to be more explicit: 4th graders don't know about congruence, isometry, or homeomorphism. They have not been defined. Most haven't even been taught about equality correctly. You can say what's wrong here, because you can look at what they've been told and realize appealing to any of the concepts you listed is illegal. You must introduce these concepts based on what they already know. New definitions must be grounded on previous knowledge. That's how math works, and there's no way to get around that.

we very much need to keep stressing"the equality meaning of the equal sign"

I want to endorse this on the basis of personal experience teaching myself math & helping others learn math (or teach themselves math...)

With students, you continually have to remind them that = means = --- or, maybe more accurately, you have to continually remind them what the implications of this are.

I think the issue has something to do with Dan Willingham's "infelxible knowledge." When you are a novice, it's unnatural to see two different sets of numbers & variables, one on either side of an equals sign, as 'the same' in the sense of being equal.

The meaning of the equals sign has to be hit on over and over and over again, I think.

btw...we had a math teacher here who did not know the difference between an equation and an expression. A friend of mine had an argument with the teacher about it.

I don't know who won.

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