70% of US students don't understand the equal sign



#2

Perhaps RPN is more popular than we thought? ;)

Story here.


#3

They probably never heard of "All animals are created equal." Maybe there is some connection. Who knows.

#4

That's probably due to scholarships mostly going to football aces in the US. Or is that just a myth like germans always waering leather trousers?

On a second thought, the example given in that article might indicate that these students have an approach to that equation like they would have when ordered to type something into their calculators. But then, why only in the US?

No, it's clearly football.

#5

:)

#6

Equal sign? You mean this thing? --> =

That my friend, is an assignment operator.

TW


#7

This is an equal sign: =

This is an assigment operator: =

These are railroad tracks: =

Edited: 11 Aug 2010, 4:28 p.m.

#8

Quote:
Equal sign? You mean this thing? --> =

That my friend, is an assignment operator.

TW


Right on baby!!
#9

Yeah, the question is ambiguous, so who is to blame? The students, obviously.

#10

Considering that around 1/3rd of US college graduates believe in creationism, stats like that no longer surprise me!

Dave.


#11

Even scarier is that 2/3 don't. But then politics and religion are two topics that don't belong here anyway.


#12

Good science and good religion coexist well. Bad science and bad religion are an anathema to each other.

Politics doesn't go with either.

IMO.


#13

Quote:
Good science and good religion coexist well. Bad science and bad religion are an anathema to each other.

IMO.


...and atheism is not an excuse to act like a jerk.


Juan

#14

Martin; You sound like a candidate for conversion to the Baha'i faith. They believe that god expects people to use science to study his universe, education is obligatory, and that work is worship, among many other sensible things. I'd be one, but i just can't get myself to believe in their god either.


#15

AFAIK buddhism originally had a very similar approach. But during 2600 years these clear ideas became heavily decorated ...

#16

Dave.

Edited: 13 Aug 2010, 2:04 a.m.


#17

I want that t-shirt.

I've educated my daughters on the noodly goodness of the FSM....
Much to their enjoyment.


- Pauli


#18

Quote:
I want that t-shirt.

You're in luck, it's FOR SALE.

Dave.


#19

You've made my weekend.

And probably my daughters' week :-)

- Pauli


#20

May you be touched by his noodly appendage!

Dave.

#21

Ordered one for me and some for my children :-)

- Pauli


#22

Quote:
Ordered one for me and some for my children :-)

More muffin money for me :-D

The Zazzle shirts are all top quality, no made in China stuff.

I might draw some more FSM designs, I just can't get enough of His Noodleness!

Enjoy.
Dave.

#23

Two gods, pictured at the place of creation:

Dave.

#24

Quote:
...Baha'i faith. They believe that god expects people to use science to study his universe, education is obligatory, and that work is worship, among many other sensible things.

db, These are all principles that can be gleaned from the Judeo-Christian scriptures, as well, although not what you usually hear emphasized, or perhaps not stated exactly so.
#25

Quote:
Considering that around 1/3rd of US college graduates believe in creationism, stats like that no longer surprise me!

Dave.


Maybe they're just being good engineers and wondering how many generations it took to evolve fully functional reproductive organs?
#26

Does anyone know how I can obtain a printout of the original article in the Psychological Journal?


#27

This link was available from Slashdot:

Link to Story

--Sancerre


#28

Sancerre:

Thank you for your help. Here are couple of interesting quotes from the reference:

Quote:
Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering, according to their research.

As a fan of algebraic I loved that quote. But I read on and here is another quote that troubles me:
Quote:
“Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer,” he explains. “So the work would look like 4+3+2=(9)+2=11.

“This response has been called a running equal sign—similar to how a calculator might work when the numbers and equal sign are entered as they appear in the sentence,” he explains. “However, this understanding is incorrect. The correct solution makes both sides equal. So the understanding should be 4+3+2=(7)+2. Now both sides of the equal sign equal 9.”


So, here are some questions that may display my lack of familiarity with the way algebra is taught these days:

Do modern students really solve problems in algebra that way?

Is the use of adjacent opening and closing parentheses without anything enclosed a standard procedure? It isn't acceptable when using the solver in the hp-35s and will result in a syntax error. The syntax error can be avoided by enclosing a variable inside the parentheses, or by simply replacing the () with a variable.

I have stumbled on some even stranger examples of students using the running equal sign technique which suggest to me that the reason students in India, China, Turkey and elsewhere who do better do so only because they haven't been allowed to use the running equal sign technique.

Palmer


#29

Quote:
Do modern students really solve problems in algebra that way?

No. At least my students don't. The example quoted might be used to teach the student that the equality must be "balanced." That is, the value to the left of the equal sign must be the same as the value to the right of the equal sign. So they would fill in the parentheses with the value that would make that true. That's fine for teaching how things must be balanced, but it is not "solving" an algebraic equation.

An "algebraic equation" must contain a variable, something to solve for. The example shown is not an algebraic equation.

I wouldn't read too much into this example. Kids are learning algebra today the same way we learned it years ago, at least in most schools.


#30

I'll second what Don said. I also teach algebra the old fashioned way, with x's and y's. I think the problem may illustrate the idea of equations needing to be "balanced", but I don't think it's a good problem.

As far as the running equal, that's something I've not really seen at all.

Brian


#31

Again, the question is ambiguous. There is no established meaning for (), so it's up to the student to guess what the question is asking. Really, the correct interpretation of the results is not "Students don't understand the equals sign" but rather "Students don't understand what the meaningless "()" symbol means ... but why should they?"

I recently saw another study -- was it linked here? -- that ALSO said that students don't understand the equals sign. But the test-makers in that case meant it to mean something completely different! There example was

a=5
b=2
a=b

So now what?

Of course, the answer is supposed to be that now a=b=2, but if you think "equals" always means "you must balance", then those statements made no sense.

In my opinion, teachers often do a very poor job of asking unambiguous questions. One example from my education I remember was when I was asked in an oral example about things traveling backwards in time. I had no idea what was being asked. I ventured, "Are you talking about tachyons?" That seems to me, then as now, to be a reasonable interpretation of the question. Then they rephrased the question, "No, I mean does time have an arrow?" Well, phrased that way, I immediately knew they were talking about entropy. Although in that case, it wasn't so much a question testing my understanding of entropy so much as a test of me having read A Brief History of Time, and knowing that specific phraseology from it.


#32

Yes, context is important. In this example, the context seems to be more related to programming than to algebra and the = is interpreted as "is assigned the value" .

Brian


#33

Quote:
4+3+2=( )+2

I think I would agree with crawl. Just that statement with no other directions would be ambiguous as best, meaningless at worst. If this was a problem in a math textbook or workbook and the directions for the problem stated: "In the following problems, enter a value within the parentheses that makes both sides of the equation equal, such as 3+5+1=(4)+4+1", then that would be reasonable.

In the absence of a clear instruction like that, the problem is meaningless and it would be impossible to draw any valid conclusions from the answers provided by the students.

That's my opinion.

Edited: 15 Aug 2010, 6:11 p.m.

#34

I vote for the KR "C"´s implementation:

= is attribution, so 'A=2' is not "A is equal to two", instead it is "Attribute 2 to variable A";

== is a relational operator, so 'A==2' is not "Attribute 2 to variable A", instead it is "A is equal to two", and a 'TRUE' or a 'FALSE' flag returns.

No doubts about, no double interpretation. But hey, when the equals sign was created there was no such thing as a programmable computer nor programming languages, right?

Cheers.

Luiz (Brazil)


#35

Hello, Luiz!

Quote:
I vote for the KR "C"´s implementation:

= is attribution, so 'A=2' is not "A is equal to two", instead it is "Attribute 2 to variable A";
== is a relational operator, so 'A==2' is not "Attribute 2 to variable A", instead it is "A is equal to two", and a 'TRUE' or a 'FALSE' flag returns.


Pascal totally avoids the ambiguity by using := for assignment. So does the LET instruction in BASIC (when it's not omitted).

Cheers,

Gerson.

#36

I found the following discussion at this site: http://castingoutnines.wordpress.com/2010/08/11/student-misunderstanding-of-the-equals-sign/"

Quote:
The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation for x, you might see:

3x - 2 = 10 = 12 = x = 4

The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4? — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all.


Do some mathematics teachers really allow stuff like that? We would have discouraged it when I was teaching algebra by grading not only the solution but also the solution process.

Palmer


#37

Quote:
Do some mathematics teachers really allow stuff like that?

Good God no!!!

As I teach algebra, the equals sign separates the two sides of an equation, and there better be one per each line when you re-write the modified equation, and each line is a step of solving the equation. It better look something like this:

3x-4=8
+4 +4
3x =12
/3 /3
x =4

#38

FYI, we were taught this way some decades ago to solve such problems:

3x-4=8    | +4
3x =12 | :3
x =4
Unambiguous, one line per step, easy to follow, economic. But only possible where you write a real 1. Sorry d8-)

#39

I was taught exactly like Don showed, 47 years ago. But I agree, Walter, your system is certainly more efficient.

#40

Walter, I kind of like that style, but what do you mean by "only possible when you write a real 1"?

Don


#41

Quote:
Walter, I kind of like that style, but what do you mean by "only possible when you write a real 1"?

I didn't originally make the effort to 'get the reference' at first, but I see it now.

Europeans write the numeral "one" with a 'streamer' connecting to the top. Americans tend to write only a vertical line segment, which could be confused with the vertical bar in Walter's notation.

-- KS


#42

So he's really referring to the delimiter between the equation and the steps you take to solve it, here.

Like I say, I kind of like this approach but it's not taught this way in the US and, as a teacher, I would be reluctant to teach it to my students when the next math teacher they have may not understand it.


#43

Don,

there is a single vertical stroke in each line, no continuous line over the whole calculation. Behind that stroke stands the operation to be applied to both sides of the equation in the respective calculation line, resulting in the next line. The stroke will stay close to the right end of an equation. So particularily in the process of reducing long formulae, the stroke will wander to the left from line to line. Another nice thing is you may use this notation in matrix algebra as well.

And Karl is right, at least in good ol' Germany people write 1 like printed here, but without the horizontal bar at the bottom. OTOH, a handwritten 7 often gets an extra horizontal stroke through its middle.


#44

Thanks Walter and Karl. Yeah, my wife puts horizontal bars through her sevens, and she's not even German!

Don


#45

The "1" is written like this:

It took me some time, however, to find a quotable sample for the "7" in the internet, but here it is:

Edited: 19 Aug 2010, 3:22 a.m.

#46

Quote:
And Karl is right, at least in good ol' Germany people write 1 like printed here, but without the horizontal bar at the bottom. OTOH, a handwritten 7 often gets an extra horizontal stroke through its middle.

So I was taught, here in Italy, back in the days... :)

Greetings,
Massimo


#47

The Continental Europeans and French Canadians among others (I was educated in French in Quebec), and presumably most non-anglophone Americans (North and South), generally do not confuse the 1 and 7 in hand-written form, by adding the horizontal line across the seven.

In a high school math book, I recall learning that there is apparently a historical reason why this is so, as well as why the '1' is written with the upward slash almost like an inverted 'v' but without the horizontal line at the base; if you write all numbers from 1-9 'properly', using straight lines, you should have a number of acute + right angles equal to the number itself. A little creativity is required for the 9, as I recall... This explains the horizontal line in the 7. Try it! The zero is no exception, of course; no angles at all in a circle.

Jeff Kearns

p.s. Not sure about the Brits, Aussies, Kiwis, etc... Dave J.?


#48

Why numbers are written that way

#49

That's because less and less students have heard about Logic these days. I used to explain the = sign with the Leibniz definition: a=b iff every (and all) property that a has, b has it too. Tarski has a nice elementary book called Introdution to Logic (very cheap now), that provides just enough background in Logic for any Maths student. (I'm a physicist and sometimes have a hard time trying to cope with the poor formal reasoning in my mates' "proofs").

Edited: 17 Aug 2010, 8:58 a.m.


#50

Thanks for the tip. I picked up a copy and am making my way through it.

Regards,

John


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