Comments

Here are my comments on J.D. Fisher's Fraction lesson.

I don't see the problem as teaching fractions one way or another. I tend to be very pragmatic when it comes to teaching. I think one needs to show students various ways to solve problems/equations using any legal arithmetic/algebraic method available. It seems like curriculum developers and pedagogues think way too much and expect way too little from kids. They (and the kids) should just dive right in and get to work. It does help, however, if the text is good and the teacher knows more than nothing about the subject.

The students need to be able to look at an expression like

45 5 X 9

and see that since division and multiplication have the same precedence, it doesn't matter which order the arithmetic is done. Then, (assuming that they really know their times tables) they will do the 45 9 step first. Doing the problem in different ways (and lots of practice!!!) makes sure that they really understand what they are doing. I distinctly remember my algebra course (very long ago) and the struggles I had proving to myself that I get the same answer doing the problem two different ways using legal algebraic manipulations. There's nothing like practice.

-- SteveH - 17 Jun 2005

A couple of points about my post.

First, the word "lesson" has been used here twice now to describe my post, but my post is NOT a lesson.

Second, what I consider unfortunate is that the idea of fractions as ratios holds (in text) until about the sixth grade. That is, textbooks ignore fractions as division expressions until this time. I am not arguing that we replace fractions-as-ratios instruction with fractions-as-division instruction. What I am saying is that students need to ALSO see that fractions can be perceived simply as division expressions.

-- JdFisher - 17 Jun 2005

I understand. I hesitated when I used the word "lesson". I also understand that you were not advocating either/or. Consider my comment as an additional voice for common sense over pedagogy, which apparently is the reason for the emphasis on ratios before sixth grade.

-- SteveH - 17 Jun 2005

Steve, that expression:

45 5 X 9

isn't so obvious. If the expression looks just like that, then it does matter what order you do things in.

If you do the 45 5 first, your answer is 81. If you do the multiplication first, it will be 1.

I just don't remember this rule. It depends on what order (left to right or right to left) the standard procedure dictates. Division and multiplication (like addition and subtraction) do have equal precedence, but order of operations still matters.

Does anybody know the answer?

-- CarolynJohnston - 17 Jun 2005

Order of operations tells us that, although division and multiplication have the same precedence, they must be performed in order from left to right.

I assumed that what Steve meant was that students do not need to worry about order--they can simply do the operations from left to right.

I may be wrong about that, however. Steve?

-- JdFisher - 17 Jun 2005

Sorry, but I have a hard time with equations in this text format. I would never write it like that to give to a student. Any ideas of a good way to get equations into these messages (And, I take certain things for granted, so I have to be careful.)

45 5 X 9

The 9 is NOT in the denominator unless there are parentheses. So, when I say that you could do the multiplication first, I mean 45 X 9, not 5 X 9. Of course, I have to learn to think more like a student. I was always taught that you have to make it clear with parentheses what is in the numerator and what is in the denominator. If there are no parentheses then it is not in the denominator. (unless it is written so that you can clearly see that something is in the numerator or denominator) As for the left-to-right rule, this isn't necessary and the students shouldn't be a slave to doing it this way. As JdFisher said in his commentary, students need to be able to look at an expression and see all of the legal things that can be done to simplify the expression. Doesn't everyone remember the algebra problems that wanted you to "simplify" the expression. I remember thinking: what the heck does simplify mean? Does that mean a factored form or an unfactored form?

For computer language compilers, the operations are done from left to right if they have the same precedence(presumed parentheses). (Optimizing compilers might do somthing different.) I'm so used to seeing computer equations like:

X = 45 5x 9

if you want the 9 in the denominator, it would be

X = 45 (5 x 9)

Perhaps I will use computer notation in the future. It's more exact. But, that raises an interesting issue. For teaching, what style, format, or rules should you use for describing these things. I never like to use the 'X' for times with my son. I like to use '*' which is closer to the dot they use in later math. I also don't like using the diagonal '/' for divide. Perhaps JdFisher^? has been all through this already. Any comments?

-- SteveH - 17 Jun 2005

Well, of course it's straightforward if you are using parentheses. But if there aren't parentheses, then what do you do if faced with

45 9 x 5?

My recollection is that, at some point, a standard way of parsing ANY mathematical expression with or without parentheses was taught in school.

But it sounds like we have some consensus: order of operations goes like this:

1. do exponentiations (powers) first 2. Do multiplications/divisions in order from left to right 3. Do additions/subtractions in order from left to right

-- CarolynJohnston - 17 Jun 2005

Step 0. Do the operations inside of the parentheses first.

Be careful not to become a slave to a procedural process. You have to see all of the possibilities. For example, if you have the following:

x = 23x 5 3 + 5 2 x (4-1) - 4 3 x 2

It's not necessary to do all of the operations using the above steps in sequence. Each term of the right-hand side expression can be done separately. Having written a compiler parser ages ago, they only work with two consecutive symbols at a time. Most compilers first convert the usual prefix notation of equations to a more convenient postfix (reverse polish) form, to eliminate the need for parentheses and recursion, if I recall. (The difference between TI and HP calculators, or am I dating myself?)

Just look at any two consecutive symbols/operators. These precedence steps apply to the order of two consecutive symbols/operators only, not the entire expression.

If you have two consecutive multiplication operators together, do them in any order that is convenient.

... a multiplication followed by a division or vice versa, do them left to right.

.... an add followed by a subtract or vice versa, do them in any order that is convenient

.... an add or subtract followed by a divide, multiply, or exponentiation, do the divide, multiply, or exponentiation first.

... Calculate the operations in each term before adding or subtracting them in any order.

... if you encounter a paren do what is inside of the parentheses first.

... You can assume that the whole expression in a numerator or denominator is surrounded by parentheses. I always drilled this into my students' heads (way back when I taught college algebra - is that an oxymoron?) This rule helps prevent situations like

(3+4) 3

It's clear that you do the 3 + 4 before the divide. If this was written with the horizontal dividing line, (I wish there was some way to do this kind of display here) with no parentheses around the 3 + 4, then students would try to cancel (that vague, wonderful non-rule) the two threes to get an answer of 1 + 4 = 5, or some such wrong result.

... you can assume that the whole expression in an exponent is surrounded by parentheses.

... I even had students put parentheses around terms, like this

y = (3 x 4 (5-2)) - ((4 + 5) 2)

... if one of two consecutive operators is an exponent, you do the exponentiation first.

Let's say you have:

X = 3 x x^4 + 5 I hope the "hat" exponentiation appears correctly.

Look at the first two operators '*' and '^'. the exponentiation is done first. Since the 4 + 5 is not in parentheses, then the exponent is just 4, not (4+ 5). If you want 4 + 5 to be the exponent, you have to put it in parentheses. Some of this explicitness is required for writing math equations in simple text form. The rule was that if there is any doubt (for the human and not the compiler), put in parentheses. But this understanding/rigor is useful for expressions in their regular form. I always made a big deal about forcing students to show me where the assumed parentheses were ... and everyone knows that you have to do what is inside of the parentheses first. This avoids incorrect operations between numbers/variables inside and outside of parentheses.

Try picking any two consecutive operators in an expression and see if you can figure out which operation is done first. Students need to be able to do this to be as flexible as possible. I'm just writing this off the top of my head. Someone must know a more complete write-up for this somewhere.

-- SteveH - 18 Jun 2005

Now, I'm getting the sense that we don't know the textbook order of operations, which is summarized by the mnemonic, Please excuse my dear Aunt Sally. That is, (1) Parentheses, (2) Exponents, (3) Multiplication and Division [from left to right] (4) Addition and Subtraction [from left to right].

This is what is taught, and it does not, I believe, conflict with anything said above. For a brief explanation of why we use these order of operations (by the way) see this:

http://mathandtext.blogspot.com/2005/05/order-of-operations.html

Other than that, I'm interested to hear what problems Catherine has run in to with fractions as fractions.

-- JdFisher - 18 Jun 2005

JD,thanks! I have never heard that mnemonic, and I think it does capture everything (I was assuming parenths came first, but as Steve says, we have to learn to think like students again).

What is it they call that state? Open mind? New mind?

Call me Blank Mind. Or maybe just Never Mind.

-- CarolynJohnston - 18 Jun 2005

I actually do know the textbook order of operations, minus the exponents, which we haven't gotten to yet.

I've had vast quantities of practice at this point. (The page of Singapore Math problems were a snap, proving the Russian maxim that 'repetition is the mother of learning.')

The bicycle problem was another whole story, though at Six Flags yesterday, I could see that the Round-Up ride has the same form, which was a terrifically satisfying moment of -- and I hate to say it -- 'making connections.'

Sigh. It depresses me that good language has been made to mean such stupid stuff. I was thinking yesterday that we should get more writers acquainted with fuzzy math. Mathematicians immediately loathe it because they're mathematicians. Fuzzy math is in many ways contemptuous of mathematics and, certainly, of mathematicians.

But it's possible that I loathed it at once because of the Orwellian language. I have a visceral reaction to the prose I see in constructivist documents.

As we were waiting in line, people kept saying the Round-Up 'wasn't going very fast,' because the big rotating sphere in the center wasn't going very fast.

Meanwhile the people aren't in the center of the disk; they're on the outside, and they're going fast.

Aren't they?

OK, how did I get off on Russian Math at Six Flags?

+ + +

Back to Jd.

I wouldn't be able to use <nopJ.D.'s expression to explain the concept of fractions-as-division to Christopher because of the associative property of multiplication.

I tried to show him one time that you can make a division problem into a multiplication problem:

9 3 is the same thing as 9 x 1 3.

He probably knows that, but in the context of simplifying an expression like <nopJ.D.'s, he doesn't recognize it. He would need a couple of quick sessions of direct instruction first, and then a lot of practice.

Jd's explanation would look like to him like it violates the associative property.

11 3 x 3 x 3 = 11 x 3

Christopher would see this and think he had to divide 11 by 3 first, then multiply the quotient by 9.

To make this clear, say the expression were:

11 dividedby 3 x 6 x 4 x 2

Christopher would see at once (let's hope) that he could multiply the 4 by the 2, and then multiply the 6 by 8.

He would not see that he could first divide the 6 by 3.

-- CatherineJohnson - 18 Jun 2005

btw, this happened to me all the time while reading Liping Ma's book -- and to Ed.

I'm too lazy to look it up now, but a couple of times she had demonstrations like Jd's that neither of us could follow at all, and that both of us thought were wrong.

The first time it happened, I thought the book had a misprint, but the second time I knew I just wasn't understanding.

That was back when I was first getting to know Carolyn, and either Carolyn or my neighbor, Laura, who is a statistician, had to explain it to me.

-- CatherineJohnson - 18 Jun 2005

I'm thrilled to learn Please Excuse My Dear Aunt Sally!

I'd read it somewhere else, and didn't have a clue what it meant.

I'm going to teach it to Christopher today.

-- CatherineJohnson - 18 Jun 2005

Jd, I have a question.

I'm confused by the idea that fractions are initially taught as ratios.

The Singapore Math books don't get to ratios until Book 6 (7th or 8th grade for us), and I can't remember at the moment how much work Christopher has had with fractions-as-ratios.

It's not too much.

All of the texts I've seen so far start with fractions as division-of-one-whole.

The denominator is the number of parts the whole has been divided into; the numerator is the number of parts you 'take.'

The Russian Math book says the numerator is the number of parts you 'have,' I believe. (And this is a 6th grade book.)

Next the books move on to division of a group of things.

John had $20 and spent 3/4 of it.

Are these ideas part of the concept of ratio? I only know 'ratio' as the classic 2-boys-for-each-girl idea.

+ + +

On a related subject, the jump from 3/4 as one-whole-divided-into-4-parts-and-you-take-3 to 3/4 as 3-parts-divided-into-4-parts is huge, and is quite confusing. I was utterly bewildered by this lesson in SRA Math Grade 6 this year, just as I was bewildered by this concept when I first encountered it in Wickelgren.

-- CatherineJohnson - 18 Jun 2005

Steve -- take a look at the interview with Ingrid Wickelgren at MathRefs.

Your concern about confusing notation is what she's talking about there, I think.

Her dad (Wayne Wickelgren, author of MATH COACH) always said that the fundamental reason math is so difficult for children to learn is that it's too 'similar'--everything looks alike.

She puts it much better than that.

-- CatherineJohnson - 18 Jun 2005

Steve --

I just looked it up.

Here's the salient passage:

Why do so many kids have such trouble with math?

It's the nature of the subject. The brain has trouble with math, not so much because there are so many facts to learn, but, surprisingly, because the facts are so similar. For example, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3, they both contain +, and they both contain single-digit numbers. To a child, the facts overlap in the brain, creating a blur that confuses them and makes it difficult remembering any single answer. It's like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station's music or speech. When the child sees 3 + 5 = __, all the arithmetic facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.

In every area of math, a relatively small number of basic concepts are used to express a large number of more advanced concepts or facts. But as each basic concept activates many other facts in the brain, the facts interfere with one another.

Is there a way around this problem?

Yes! Create a new idea that binds facts into chunks, making the facts hang together in logical ways, like notes in a song. In chapter six, we lead readers through a method of teaching basic arithmetic that primes children to create chunks.

-- CatherineJohnson - 18 Jun 2005

A couple of ideas to respond to here.

First, regarding Catherine's comment about the associative property of multiplication: I'm not sure I understand the connection here to the associative property of multiplication [i.e., a x (b x c) = (a x b) x c]. However, if Christopher were to think that he should first divide 11 by 3, then he would be absolutely correct. The idea here is to apply one's knowledge of inverse operations, which, as I mentioned in my post, is a prerequisite for understanding the thought process behind this technique.

That is, one should see that a number multiplied by three and then divided by three is just that number: a x 3 3 = a. Given the expression 11 3 x 9, one can think of 9 as 3 x 3 and rethink the expression as 11 3 x 3 x 3. Then we actually do the operations in order; we just use reasoning to jump ahead a bit. We see that 11 3 x 3 is equal to 11, because of inverse operations. What's left is to simply multiply 11 by 3.

I should make a couple of points here: (1) You made the comment "I wouldn't be able to use JD's expression to explain the concept of fractions-as-division." This is not my intention either. I am outlining some of the possible benefits of students' seeing fractions this way before they reach the sixth grade. I am not suggesting that these kinds of examples be used to introduce this idea. (2) "He probably knows that, but in the context of simplifying an expression like Jd's, he doesn't recognize it. He would need a couple of quick sessions of direct instruction first, and then a lot of practice." I would never ask anyone to assume that any new way of thinking about something in mathematics can be apprehended instantly, with little effort. Indeed, the argument one hears often from teachers is "My students don't understand this." My reply: "Of course they don't. You haven't taught it to them yet." The idea that new learning requires a bit of strain at first is not, to my mind, an argument for its complete 'unteachability'. (3) My intention was to show a variety of ways one could apply the idea of fractions as division expressions to solve some problems involving the multiplication of fractions more efficiently. Students and teachers versed in the three-step method would be surprised, I think, to know that some of these kinds of problems (e.g., 45 5 x 9) can be solved much more quickly, and even mentally. It remains my opinion that this alternative method of thinking about the multiplication of fractions, given the right instruction, can be extremely beneficial for students.

Catherine also asked about the idea of fractions as ratios. I don't have my Singapore books in front of me, but from what I remember, these books, along with many commercial basal texts, introduce the idea of fractions as a comparison of the numerator and denominator. Thus, they say that, for example, 1/4 is 1 shaded part compared with 4 equal parts. Although the models may show a "division of one whole," this explanation of fractions perceives them as ratios--comparisons of two numbers. The 1 is identified not as the whole, but as the quotient. And the 4 is turned into a kind of dividend, when it is really the divisor.

-- JdFisher - 18 Jun 2005

Check out Hung-Hsi Wu's web site. He's a mathematician at Berkeley who has been quite involved in math ed as well as the math wars and has written some great papers available from his web site. He has a paper on order of operations located at http://math.berkeley.edu/~wu/order3.pdf. He also has a draft chapter of a book he's writing on math for teachers on fractions. Excellent discussion. It is located at http://math.berkeley.edu/~wu/EMI2a.pdf. He gets into how ratios at the end, and says that they should be taught carefully so kids don't think that ratios are fractions. They are not, though they can be manipulated as such.

Although his text is designed for pre-service teachers, some feel it is not "dumbed down" enough for that audience. Hence the book that is used for pre-service teachers is Parker and Baldwin which isn't bad but surely isn't as rich as Wu's discussion.

-- BarryGarelick - 19 Jun 2005

Oh Barry--thanks.

My neighbor and I taught the Singapore Math lesson on ratios about a month ago, and I've been confused ever since over this exact point: are ratios fractions?

I don't know if I can explain my confusion; this is one of the most difficult challenges in writing about math.

I'm beginning to think it's easier to explain how to do math in words than to explain why you don't understand math in words.

I'm thinking this is part of pedagogical content knowledge: conscious, verbal understanding of where students will go wrong, what they will and won't understand.

I also think that one part of pedagogical content knowledge is conscious, verbal understanding of one's own confusions (or at least, for those of us who are not remotely expert in mathematics.)

I'm going to give it a whack.

I was utterly bewildered by this:

Say you have a ratio of 2 boys to 3 girls.

The 'fraction' aspect of that ratio is 2/5 + 3/5 (two-fifths of the children are boys; three-fifths of the children are girls).

I found this bewildering, because I kept looking at the 2/3 ratio and thinking, "How come that's not the 'fraction'?"

That may sound stupid, and it may be stupid.

But that's not my problem!

I'm trying to capture the . . . confusion.

(Now there's an ambition for you! Capture the confusion!)

Reading, here, that ratios aren't fractions is enormously helpful.

-- CatherineJohnson - 19 Jun 2005

After the Singapore ratio lesson, I actually wrote up a '7-fact family' sheet I think I'm going to start drilling Christopher on.

I created it in analogy to Saxon's 4-fact families, which have been incredibly powerful, for Christopher and for me, in drilling inverse operations into our brains.

-- CatherineJohnson - 19 Jun 2005

As to the associative property, here's why Christopher would not see that Jd's way works (I know it's not a lesson!)

(I wouldn't have seen it, either, in the past. Unfortunately, I don't remember when I figured out that it did).

associative property
(a + b) + c = a + (b + c)
(a x b) x c = a x (b x c)

Christopher and I translate this to mean that:
(2 x 3) x 4 = 2 x (3 x 4)
but:
(2 3) 4 does not equal 2 (3 4)

So take the problem:
72 3 x 3

Obviously the easy way to simplify this expression is to notice that 3 http://us.metamath.org/symbols/div.png3 equals 1, and 72 1 equals 72.

However, Christopher would not see this. (I'm going to ask him, and see if he does, but I'm pretty sure the answer will be no.)

At some point there in the way-back, I wouldn't have recognized it, either.

I would have 'applied' the associative property in the most literal, surface-feature way and thought I had to simplify the expression by moving strictly left to right.

pedagogical content knowledge

• at some point along the line I picked up on the fact that 'http://us.metamath.org/symbols/div.png 5' is the same thing as 'times one-fifth.' I figured this out without being taught, IIRC.

• just a couple of months ago, when I was wrestling with teaching Christopher, it suddenly struck me that if you attach the 'operator' to the number (I may not be saying this right) you can do the operations in whichever order you like.

When you're still in a 'literal-minded, surface-feature' state, you use the associative property to separate the numerals from the operators. (Is that the word? From the division & multiplication signs is what I mean.)

So you change:

(72 3) times 3
to
72 (3 times 3)

The 'divided by' has been detached from the 3.

If you keep the operator attached to the number, you know to simplify the expression by dividing 3 by 3 first.

+ + +


I'm thinking about whether it would work to teach Christopher not to see the numbers in isolation: 72, 3, 3.

I'm thinking I should teach him to see 'divided by 3' and 'multiplied by 3' instead of 3 and 3.

Of course, I have no idea what to call the 72, other than just . . . 72.

Multiplication and division are not so easy as N.S. would say.

-- CatherineJohnson - 19 Jun 2005

OK, Investigation completed.

I wrote down the expression:

72 3 x 3

No parens.

Christopher immediately started to go left to right, dividing the 72 by 3.

I said, 'Is there an easier way to do this?'

He said, 'Oh, yes. 72 9.'

Then I showed it to Ed who thought it was confusing because there were no parens.

I asked him if there was an easy way to do it, and he said, 'Well the answer is 72.'

I asked him how he knew that, and he said, 'You're dividing and multiplying by the same number.'

(Ed is my husband, fyi. He taught a high school GED class many years ago. Fractions were the Big Hard Thing there, too.)

-- CatherineJohnson - 19 Jun 2005

I would not have seen that since you are dividing and multiplying by the same thing the answer is 72.

-- CatherineJohnson - 19 Jun 2005

I think Ed was basically going left to right, too--??

He was seeing the 72 as the Important Number:
72 3
and
72 x 3

Golly!

There are Many Ways Of Solving The Problem!

I wish I'd written down exactly what Ed said at first.

He was thrown by the lack of parens.

-- CatherineJohnson - 19 Jun 2005

btw, I owe a 'thanks' to Steve, because his discussion of 'operators'--which I admit I couldn't completely follow!--reinforced my tentative 'feeling' that you should keep the operator with the number . . .

I'm sure that doesn't make sense, but we don't really have language to describe learning-of-math.

-- CatherineJohnson - 19 Jun 2005

Here's a '7 fact family' (this is a Word document that downloads to your desktop):

7 fact families

If the math is wrong, please tell me!

-- CatherineJohnson - 19 Jun 2005

One uses the same process to solve equations. For instance, the equation a x 4 = 12 can be solved by dividing both sides by 4: a x 4 4 = 12 4; a = 12\4, or 3.

If the equation were written 4 x a = 12, the process would not change: 4 x a \ 4 = 12 4. There is no real need to 'associate' any numbers with operators, except maybe for one's own comfort, because the operations have the same precedence.

The expression a + 4 x b, however, cannot be simplified from left to right, because multiplication, which comes after addition in the expression, has a higher precedence.

-- JdFisher - 20 Jun 2005

Which, incidentally, makes the process of explaining that 9 x 1 3 is the same as 9 3 easy peasy. (Well, it's at least possible.)

If an expression contains only multiplication and division, then it is simplified from left to right. So 9 x 1/3 is simply 9 x 1 3. We find 9 x 1 first. Then divide the result by 3 (9 3).

-- JdFisher - 20 Jun 2005

9 x 1 3 is simply 9 x 1 3

Yes!

That's another example of ... inflexible knowledge for me.

It's very difficult for me to 'see' division and fraction at the same time.

I 'know' that a fraction is a division and a division is a fraction, but I don't 'see' it or 'feel' it ... as I say, we don't have language to describe learning-of-math.

At one point, when I was struggling to understand why you multiply numerators & multiply denominators, I realized that 1/3 x 1/3 is 1 divided by 3 twice.

That was a huge revelation, believe me.

I've just had that revelation again, reading your last comment.

Flexible knowledge takes a long time to develop.

-- CatherineJohnson - 20 Jun 2005

btw, one of the math warriors -- perhaps Stanley Ocken? -- has speculated that a student needs a good 8 years for division to 'sink in.'

In other words, it won't work for a student to diddle around with TERC for 8 years, and then try to make up for lost time with long division when he's 20.

He needs a long period of time for his mathematics learning to be 'gelling' inside his brain.

I'm relaying this from memory, so I almost certainly have the specifics wrong.

But I wouldn't be at all surprised if whoever made this argument is correct.

As Rafe Asquith says, 'There are no shortcuts.'

-- CatherineJohnson - 20 Jun 2005

I know I am entering the discussion late, but I just stumbled upon your treasure check.

In my neck of the woods, Ontario, Canada, we teach our students BEDMAS to remember the order of operations. Brackets, Exponents, Division and Multiplication in the order which they appear, Addition and Subtraction in the order which they appear. Not as fun as Aunt Sally!

SmartestTractor^?

-- SmartestTractor - 19 Sep 2005