When I asked my 8 1/2 year old daughter to read that line aloud and find the mistake in it, she said:
"Three times two equals six, (pause), plus one equals seven. What's wrong with that?"
I pointed out that I noticed she paused between the six and the plus sign, and I asked her why she had done that. She looked at the page again and laughed. "Oh, I see it now!"
When I asked her how it ought to be written, she suggested "3 x 2 = 6, 6 + 1 = 7". Use a comma to separate the two operations.
But it is worth considering the origins of this error. When we speak mathematics orally, we are likely to say precisely what this book puts in print, and we are not wrong to do so. But we use prosodic elements (like pauses) to separate the parts of our speech into distinct phrases. Those pauses don't get recorded in print.
Maybe some of you who know more than I do about the differences between spoken English and written English can make some more out of this.
Her older sister (11 years old) commented that when you enter sequential calculations on a calculator, the button-push sequence is precisely the same as what is printed on the page here (except that the "6" and the "7" are not button-pushes but screen-readouts). Calculator use, as a performance that occurs over time, corresponds to the way we speak more than it does to the way we write.
Another way of saying this: There is no standard mathematical symbol whose meaning is "and then...". So the equals sign is (mis-)appropriated for that purpose.
Computer programming notation adds to the confusion. I'm trying to teach my son that
x = x+1
in algebra is a statement, but that in mainstream programming, it is a command. In algebra, it is telling us ABOUT something. It's not telling you to do anything. In programming, it is telling the computer to DO something. In algebra, it's making a claim that could be true or false that, in this case must be false, because there is no value of x that would make it true. In programming, commands aren't true or false, they tell the computer to do something that, in this case, CAN be done for any x. In algebra, the statement can be reversed (x+1 = x), and you would have an equivalent statement. In programming, you couldn't reverse it, because it would mean something totally different that, in this case, wouldn't make sense.
And so on.
And I don't have the heart to tell him that some of what I just said isn't true in mainstream programming languages like C. In C, these commands are called, yes, "statements," and they do have true or false values that have nothing to do with the meaning of true and false in either math or English. And it gets worse, but that's enough.
The notational problem in this little cartoon is just the beginning of wild notational craziness that is rampant in all of STEM.
You can (and should) clean up items like this one, but STEM training requires repeated discussions over the years about the differences between the underlying concepts (semantics) and the various notational attempts to represent them (syntax).
The silver lining of these inconsistencies is that they create opportunities to discuss underlying concepts separately from the notation.
6 does not equal 7. Teaching kids that equals means what the teacher says it does is the effect of this for many students. It´s a disaster.
I have another problem with it, one no one else saw.
Teaching children that 4X3 is what you do when its 40 *3 means you've eradicated the meaning of the place value.
To most of you, this may seem nitpicking. But if you can't tell a kid that it's 200×3 and 40×3 then how will they know the distributive property is true? How will then understand how to make sense of 243×23 where that 2 in 23 means 20, and that's why we more over a column?
You shouldn't carry the 1 either. You should recognize that 1 is 100. because 40×3 is 120.
Kids really think teachers make up math stuff and there are no rules. ExampleS like this are why.
Michael, you may he right that it's about speaking but I think it's deeper.
First, most people have no idea what equality is. they don't understand the relation.
They think arithmetic is about "getting an answer" so the = sign means "get an answer".
The informal, casual, and incorrect way we teach math adds to this confusion that these symbols don't have real mathematically precise meanings.
The "get an answer" idea is why many kids can't solve 7=3+? even when they can solve 7-3=?
It's why they can't hold terns in their head (or even accept them on paper) such as (8+3) because it needs to immediately be simplified. It's why they don't understand algebra.
It a teacher has taught equality properly then they wouldn't needto have written anything but 3x2+1 =7. That 3*2 is 6 wouldn't have needed that extra Step if 6=3x2 had been taught to show what equal means.
I agree with everything you have said, Alison. But the question for me is, why don't people understand what equality is? You have named the problem; I am puzzling over the epistemology.
I think at a deep level it has to do with the fact that speech is natural and writing is unnatural. Speech (math done orally) is deployed over time and is therefore asymmetric; written math is atemporal (neither the left-hand-side nor the right-hand-side of an equation "comes first") and is symmetric.
But we learn math orally first. "You have three Legos. Do you want another? Now you have four Legos." Later we learn to record that sequence of events as "3 + 1 = 4", which also (not incidentally) drops out the word "legos" and thus deals with the abstract numbers. At this stage the equals sign does mean "make an answer". That's the developmentally appropriate meaning of the equals sign for, let's say, a 6-year-old who is just learning to transcribe speech into written symbols.
That the equals sign means "identity" rather than "make an answer" is something that has to be explicitly taught, precisely because it does not come naturally.
It a teacher has taught equality properly then they wouldn't needto have written anything but 3x2+1 =7. That 3*2 is 6 wouldn't have needed that extra Step if 6=3x2 had been taught to show what equal means.
Okay, this I disagree with. :)
The reason for even having those annotations on the side of the page (and remember, this is not a "teacher", this is a workbook) is to illustrate the part of the algorithm that is not written down, i.e. the part that you normally do in your head. And when you do it in your head, the "6" is an important intermediate stage in the calculation. You think the "6": "3 x 2 is 6, then I add 1 and get 7." If you are trying to record that two-step thought process for pedagogical purposes so that students will learn what to say to themselves (not what to write down), omitting the intermediate result erases a crucial piece of information.
The problem then becomes, how do you transcribe the two-step process into written text and also conform to the conventions of mathematical notation? This book gets that transcription wrong.
Some of my PSTs suggested writing the work this way: 3 x 2 ---- 6 + 1 ---- 7
which is interesting, because it accurately records the two-step process but avoids the use of an equals sign. Instead it uses a horizontal line as a "do-something" symbol. Which is really want is needed here. However, there doesn't seem to be a way to indicate "do something" when writing the computations horizontally.
I never understood the problem with the "running equals signs" until someone here at KTM explained it to me. Since then, I've made it a point to write serial equations to avoid any possibility of confusing my students.
I don't think anyone calls it "assignment" any more. Wasn't it Algol that used ":=" instead? I vaguely remember writing some Algol programs.
How about C's
i++
or i += 1
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Then there are things like:
x = ++i * 5;
which is different than
x = i++ * 5;
Some programmers thought it was a badge of honor to write compact code.
Michael, these are musings, so I'm not wedded to them. But...
It's not a requirement that the oral introduction be of the value changing over time. Singapore's Primary math teaches the number stories and number bonds atemporally for this reason. There are 6 children on the playground; 2 are swinging, 4 aren't. 5 are wearing hats, 1 is not. 3 are boys, 3 girls. the sixness is there as a constantly and the part-whole model intuitively supports equality before anyone ever sees an equal sign. Yes, this is being taught, but it leads me to think were teaching "get an answer" more than it's a default, too.
Second, I'm not convinced that the language of "three times two is 6 (micropause) plus one is seven" is natural. It's natural that English is sloppy, yes. But when I've said such sentences to parents taking my classes or even some elementary teachers, they can't parse it or keep up. They don't naturally keep a running total in their heads. This sloppy abbreviation might work for those comfortable with math, but it's not clear that's inherent.
--If you are trying to record that two-step thought process for pedagogical purposes so that students will learn what to say to themselves (not what to write down), omitting the intermediate result erases a crucial piece of information.
Except for pedagogical purposes, every else is wrong, too. Pedagogically and mathematically, the workbook should be showing the student 200*3 + 40*3 + 3*3
if each of those products is written out, right to left, underneath the computation so the work shown is (properly justified)
9
120
600
then this problem would go away, and the Verbal internal dialogue wouldn't need the confusing running total to involve equal signs at all.
You'd say "the product of 3 and 3 is 9. The product of 40 and 3 is 120. the product of 200 and 3 is 600.(If necessary, you say the sum of 9 and 0 is 9. of 20 and O is 20. the sum of 100 and 600 is 700.) the sum of the above is 729.
Yes, and so did Pascal. Some languages like Haskell even use the (or at least a) real math assignment operator, "<-" instead of the math equality symbol.
But most mainstream languages use "=" for assignment these days.
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Yes, but I was teaching both algebra and programming simultaneously to a ten-year-old. The meaning of mathematical notation had not had years to gel before programming was introduced, so I had to carefully explain the different semantics denoted by identical notation.
I wasn't making a comment that pointing out this distinction shouldn't be necessary. In terms of the general discussion, the issue is that there are many of these concepts that can trap or slow down a student. You can try to prevent them with the curriculum, but a real, live teacher can, and should, make all of the difference. The direct involvement of a teacher (and not just a group of peers) can save a lot of time and frustration. Then again, some teachers think that struggle is necessary. I prefer to find the book or source or teacher that makes learning as easy as possible. If I learn something easily, I never think that I don't know it well enough. I just wonder why the other sources make it so difficult. Sometimes I will drop a source because I can just tell that there must be an easier path.
For the example in this thread, the "equation" 3X2=6+1=7 is wrong and the authors should have known not to include it. Even though kids should not have a problem with it (by this time), one should not do it. However, a bigger question is what material precedes or supports this explanation? Do they start with partial products? Is this just the beginning transition to the standard algorithm? My first reaction is that this explanation is not careful enough. Then again, I'm seeing it out of context.
Also, students can have these problems even if their math materials never mention them. That is a bigger problem. You can't possibly teach all of the things that can go wrong in students' heads. Even if a textbook is technically correct, kids will still fill in the blanks incorrectly. The teaching process has to be designed to make those corrections, and homework sets usually catch them. I've always thought that the homework feedback loop was critical.
"However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on."
This is a fair observation for most programming students, *but* the mistake in C-like languages that does get made is using '=' to test for equality instead of '=='.
If C had simply used ':=' as God (and Nikolas Wirth) intended, then '=' could be the test operator and '==' wouldn't need to exist.
In some sense, the real issue here is that '=' *has* a meaning in a mathematical context. There was no need for C-like languages to reuse the symbol with a different meaning.
But, yeah, if this proves to be a large hurdle for a programming student, that student is probably not going to make it as a programmer.
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Then there are things like:
x = ++i * 5;
which is different than
x = i++ * 5;"
I don't think that '++i' and '+=' would cause problems because of a conflict with math. These operators don't exist (I don't think...) in a math context, so there isn't any clash and you just need to understand how they work when programming.
Having said that, '++i' and 'i++' actually made sense in the late-1960s and early 1970s. Modern languages should just omit them.
Things like '+=' cut down on one form of error, even today. I'd expect that learning to use '+=' and friends is a very small item.
The underlying assumption here is that you are learning/teaching and imperative programming language. If you don't want to do *that*, then most of this goes away.
Perhaps I should mention that this comes from a workbook, not a textbook -- the kind of thing you find on the magazine rack of your local drugstore for $2.99. It's not a curriculum, or part of a curriculum, and isn't used in schools. I can upload the cover later if it helps. I think the presumption is that any kid using this book has been taught what to do in school, and this book exists solely to provide some supplemental practice.
27 comments:
fabulous post! thanks so much for putting this up
(and, uh, no: it's not just you ....)
So I can feel upset, too, what is the problem here?
-Mark Roulo
Here's a hint.
$35 dollars to purchase just that article. That's crazy.
Luckily the title of the article was enough to clue me in.
If I recall from the MSMI Singapore conference correctly, this is just the sort of thing that makes Allison's head explode.
UGH. Ugh ugh ugh. (It's not just you.
SPOILER ALERT:
3 x 2 = 6 + 1 = 7
When I asked my 8 1/2 year old daughter to read that line aloud and find the mistake in it, she said:
"Three times two equals six, (pause), plus one equals seven. What's wrong with that?"
I pointed out that I noticed she paused between the six and the plus sign, and I asked her why she had done that. She looked at the page again and laughed. "Oh, I see it now!"
When I asked her how it ought to be written, she suggested "3 x 2 = 6, 6 + 1 = 7". Use a comma to separate the two operations.
But it is worth considering the origins of this error. When we speak mathematics orally, we are likely to say precisely what this book puts in print, and we are not wrong to do so. But we use prosodic elements (like pauses) to separate the parts of our speech into distinct phrases. Those pauses don't get recorded in print.
Maybe some of you who know more than I do about the differences between spoken English and written English can make some more out of this.
Her older sister (11 years old) commented that when you enter sequential calculations on a calculator, the button-push sequence is precisely the same as what is printed on the page here (except that the "6" and the "7" are not button-pushes but screen-readouts). Calculator use, as a performance that occurs over time, corresponds to the way we speak more than it does to the way we write.
I didn't catch it. I guess I was just wondering what grade the 70's Michael Jackson was in.
ABC 123
http://www.youtube.com/watch?v=LgpoMxmuAa0
Maybe we can put the standard algorithm to music. You know ... different learning styles.
Another way of saying this: There is no standard mathematical symbol whose meaning is "and then...". So the equals sign is (mis-)appropriated for that purpose.
Computer programming notation adds to the confusion. I'm trying to teach my son that
x = x+1
in algebra is a statement, but that in mainstream programming, it is a command. In algebra, it is telling us ABOUT something. It's not telling you to do anything. In programming, it is telling the computer to DO something. In algebra, it's making a claim that could be true or false that, in this case must be false, because there is no value of x that would make it true. In programming, commands aren't true or false, they tell the computer to do something that, in this case, CAN be done for any x. In algebra, the statement can be reversed (x+1 = x), and you would have an equivalent statement. In programming, you couldn't reverse it, because it would mean something totally different that, in this case, wouldn't make sense.
And so on.
And I don't have the heart to tell him that some of what I just said isn't true in mainstream programming languages like C. In C, these commands are called, yes, "statements," and they do have true or false values that have nothing to do with the meaning of true and false in either math or English. And it gets worse, but that's enough.
The notational problem in this little cartoon is just the beginning of wild notational craziness that is rampant in all of STEM.
You can (and should) clean up items like this one, but STEM training requires repeated discussions over the years about the differences between the underlying concepts (semantics) and the various notational attempts to represent them (syntax).
The silver lining of these inconsistencies is that they create opportunities to discuss underlying concepts separately from the notation.
6 does not equal 7. Teaching kids that equals means what the teacher says it does is the effect of this for many students. It´s a disaster.
I have another problem with it, one no one else saw.
Teaching children that 4X3 is what you do when its 40 *3 means you've eradicated the meaning of the place value.
To most of you, this may seem nitpicking. But if you can't tell a kid that it's 200×3 and 40×3 then how will they know the distributive property is true? How will then understand how to make sense of 243×23 where that 2 in 23 means 20, and that's why we more over a column?
You shouldn't carry the 1 either. You should recognize that 1 is 100. because 40×3 is 120.
Kids really think teachers make up math stuff and there are no rules. ExampleS like this are why.
Teaching children that 4X3 is what you do when its 40 *3 means you've eradicated the meaning of the place value.
Actually, that's one of the main things I didn't like about it.
I didn't realize how place value related to the standard algorithm until I was an adult, trying to re-teach Chris!
No one ever told me, and I never noticed.
Michael, you may he right that it's about speaking but I think it's deeper.
First, most people have no idea what equality is. they don't understand the relation.
They think arithmetic is about "getting an answer" so the = sign means "get an answer".
The informal, casual, and incorrect way we teach math adds to this confusion that these symbols don't have real mathematically precise meanings.
The "get an answer" idea is why many kids can't solve 7=3+? even when they can solve 7-3=?
It's why they can't hold terns in their head (or even accept them on paper) such as (8+3) because it needs to immediately be simplified.
It's why they don't understand algebra.
It a teacher has taught equality properly then they wouldn't needto have written anything but 3x2+1 =7. That 3*2 is 6 wouldn't have needed that extra Step if 6=3x2 had been taught to show what equal means.
I agree with everything you have said, Alison. But the question for me is, why don't people understand what equality is? You have named the problem; I am puzzling over the epistemology.
I think at a deep level it has to do with the fact that speech is natural and writing is unnatural. Speech (math done orally) is deployed over time and is therefore asymmetric; written math is atemporal (neither the left-hand-side nor the right-hand-side of an equation "comes first") and is symmetric.
But we learn math orally first. "You have three Legos. Do you want another? Now you have four Legos." Later we learn to record that sequence of events as "3 + 1 = 4", which also (not incidentally) drops out the word "legos" and thus deals with the abstract numbers. At this stage the equals sign does mean "make an answer". That's the developmentally appropriate meaning of the equals sign for, let's say, a 6-year-old who is just learning to transcribe speech into written symbols.
That the equals sign means "identity" rather than "make an answer" is something that has to be explicitly taught, precisely because it does not come naturally.
It a teacher has taught equality properly then they wouldn't needto have written anything but 3x2+1 =7. That 3*2 is 6 wouldn't have needed that extra Step if 6=3x2 had been taught to show what equal means.
Okay, this I disagree with. :)
The reason for even having those annotations on the side of the page (and remember, this is not a "teacher", this is a workbook) is to illustrate the part of the algorithm that is not written down, i.e. the part that you normally do in your head. And when you do it in your head, the "6" is an important intermediate stage in the calculation. You think the "6": "3 x 2 is 6, then I add 1 and get 7." If you are trying to record that two-step thought process for pedagogical purposes so that students will learn what to say to themselves (not what to write down), omitting the intermediate result erases a crucial piece of information.
The problem then becomes, how do you transcribe the two-step process into written text and also conform to the conventions of mathematical notation? This book gets that transcription wrong.
Some of my PSTs suggested writing the work this way:
3
x 2
----
6
+ 1
----
7
which is interesting, because it accurately records the two-step process but avoids the use of an equals sign. Instead it uses a horizontal line as a "do-something" symbol. Which is really want is needed here. However, there doesn't seem to be a way to indicate "do something" when writing the computations horizontally.
I never understood the problem with the "running equals signs" until someone here at KTM explained it to me. Since then, I've made it a point to write serial equations to avoid any possibility of confusing my students.
3 x 2 = 6
6 + 1 = 7
"x = x+1"
I don't think anyone calls it "assignment" any more. Wasn't it Algol that used ":=" instead? I vaguely remember writing some Algol programs.
How about C's
i++
or i += 1
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Then there are things like:
x = ++i * 5;
which is different than
x = i++ * 5;
Some programmers thought it was a badge of honor to write compact code.
Michael, these are musings, so I'm not wedded to them. But...
It's not a requirement that the oral introduction be of the value changing over time. Singapore's Primary math teaches the number stories and number bonds atemporally for this reason. There are 6 children on the playground; 2 are swinging, 4 aren't. 5 are wearing hats, 1 is not. 3 are boys, 3 girls. the sixness is there as a constantly and the part-whole model intuitively supports equality before anyone ever sees an equal sign. Yes, this is being taught, but it leads me to think were teaching "get an answer" more than it's a default, too.
Second, I'm not convinced that the language of "three times two is 6 (micropause) plus one is seven" is natural. It's natural that English is sloppy, yes. But when I've said such sentences to parents taking my classes or even some elementary teachers, they can't parse it or keep up. They don't naturally keep a running total in their heads. This sloppy abbreviation might work for those comfortable with math, but it's not clear that's inherent.
--If you are trying to record that two-step thought process for pedagogical purposes so that students will learn what to say to themselves (not what to write down), omitting the intermediate result erases a crucial piece of information.
Except for pedagogical purposes, every else is wrong, too. Pedagogically and mathematically, the workbook should be showing the student
200*3 + 40*3 + 3*3
if each of those products is written out, right to left, underneath the computation so the work shown is (properly justified)
9
120
600
then this problem would go away, and the Verbal internal dialogue wouldn't need the confusing running total to involve equal signs at all.
You'd say "the product of 3 and 3 is 9. The product of 40 and 3 is 120. the product of 200 and 3 is 600.(If necessary, you say the sum of 9 and 0 is 9. of 20 and O is 20. the sum of 100 and 600 is 700.) the sum of the above is 729.
Wasn't it Algol that used ":=" instead?
Yes, and so did Pascal. Some languages like Haskell even use the (or at least a) real math assignment operator, "<-" instead of the math equality symbol.
But most mainstream languages use "=" for assignment these days.
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Yes, but I was teaching both algebra and programming simultaneously to a ten-year-old. The meaning of mathematical notation had not had years to gel before programming was introduced, so I had to carefully explain the different semantics denoted by identical notation.
I wasn't making a comment that pointing out this distinction shouldn't be necessary. In terms of the general discussion, the issue is that there are many of these concepts that can trap or slow down a student. You can try to prevent them with the curriculum, but a real, live teacher can, and should, make all of the difference. The direct involvement of a teacher (and not just a group of peers) can save a lot of time and frustration. Then again, some teachers think that struggle is necessary. I prefer to find the book or source or teacher that makes learning as easy as possible. If I learn something easily, I never think that I don't know it well enough. I just wonder why the other sources make it so difficult. Sometimes I will drop a source because I can just tell that there must be an easier path.
For the example in this thread, the "equation" 3X2=6+1=7 is wrong and the authors should have known not to include it. Even though kids should not have a problem with it (by this time), one should not do it. However, a bigger question is what material precedes or supports this explanation? Do they start with partial products? Is this just the beginning transition to the standard algorithm? My first reaction is that this explanation is not careful enough. Then again, I'm seeing it out of context.
Also, students can have these problems even if their math materials never mention them. That is a bigger problem. You can't possibly teach all of the things that can go wrong in students' heads. Even if a textbook is technically correct, kids will still fill in the blanks incorrectly. The teaching process has to be designed to make those corrections, and homework sets usually catch them. I've always thought that the homework feedback loop was critical.
"However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on."
This is a fair observation for most programming students, *but* the mistake in C-like languages that does get made is using '=' to test for equality instead of '=='.
If C had simply used ':=' as God (and Nikolas Wirth) intended, then '=' could be the test operator and '==' wouldn't need to exist.
In some sense, the real issue here is that '=' *has* a meaning in a mathematical context. There was no need for C-like languages to reuse the symbol with a different meaning.
But, yeah, if this proves to be a large hurdle for a programming student, that student is probably not going to make it as a programmer.
-Mark Roulo
"How about C's
i++
or i += 1
However, I don't think I ever had any student confuse this with mathematical equality. If you get stuck at that level, then you won't be happy as time goes on.
Then there are things like:
x = ++i * 5;
which is different than
x = i++ * 5;"
I don't think that '++i' and '+=' would cause problems because of a conflict with math. These operators don't exist (I don't think...) in a math context, so there isn't any clash and you just need to understand how they work when programming.
Having said that, '++i' and 'i++' actually made sense in the late-1960s and early 1970s. Modern languages should just omit them.
Things like '+=' cut down on one form of error, even today. I'd expect that learning to use '+=' and friends is a very small item.
The underlying assumption here is that you are learning/teaching and imperative programming language. If you don't want to do *that*, then most of this goes away.
-Mark Roulo
"the mistake in C-like languages that does get made is using '=' to test for equality instead of '=='."
Yes, and the single "=" will most likely always compile and run. The value will get set to TRUE (1) or FALSE (0). That's a tough one to debug.
Perhaps I should mention that this comes from a workbook, not a textbook -- the kind of thing you find on the magazine rack of your local drugstore for $2.99. It's not a curriculum, or part of a curriculum, and isn't used in schools. I can upload the cover later if it helps. I think the presumption is that any kid using this book has been taught what to do in school, and this book exists solely to provide some supplemental practice.
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