The equal sign problem

An interesting piece of research has just come out on the misconceptions with the equal sign (=).

Students' understanding of the equal sign not equal

According to the research, US students exhibit this misconception much more often than students in other countries. It has to do with thinking of the = sign as an operator. Kind of like thinking that = means "to do" the operation.

For example, a student with that misconception tends to solve the problem

7 + 6 = ____ + 2

by adding 7 + 6, and placing the answer on the empty line.

The correct way is of course to think of the equality: 7 + 6 equals 13, so the other side has to equal 13 too. 11 fulfills this little equation:

7 + 6 = 11 + 2

I have known of this problem for years, and have tried to include problems in my Math Mammoth books to help children NOT to develop this wrong idea. For example, children solve


200 + 50 + 6 = ____ + 200 + 50 in the place value section.

Or, I use problems where they have to put either <, >, or = in between (one the line here, but I like to use boxes in the books):

20 + 9  _______ 90 + 2

8 + 6   ________  7 + 7


Or, just simple missing addend problems from the very start (1st grade):

3 + _____ = 5.


Remember, students exhibiting the misconception would add 3 and 5, and put 8 on the empty line. But we can teach kids to think of this problem as "3 and how many more makes 5". It is a starting point in understanding the equal sign in the correct way.

Then they should also (later on) encounter the same problem this way:

5 = ____ + 3

And other variations.

Comments

KennisonDF said…
Identity ≠ Equality

Mathematics teachers should be instructed to treat as a widespread misnomer the English translation of "=" as "equals".

It is crucial that the student not confuse the concept of similarity or equivalence – ‘the same in some respect(s)’ (which constitutes joint membership in a set or class) – with the concept of identity – ‘the same in all respects’ (which constitutes being one and the same object).

“(3 × 2)” and “6” are two different names (numerals) for the same object (the number 6). These two names are similar names, in that they each name the same object; they are equivalent names, in that they each take the same object as their semantic value. However, they are not identical names.

We can represent such facts with metamathematical equivalency statements about the numerals (the names) that we use to refer to numbers:

_ “ “(3 × 2)” names the same number that “6” names ”,

_ “ “(3 × 2)” is equivalent to “6” ”,

_ “ “(3 × 2)” ≡ “6” ”, or

_ “ “(3 × 2)” equals “6” ”.

Usually, however, we simply represent such facts with mathematical identity statements about the actual numbers (the objects) themselves:

_ “(3 × 2) is the same number as 6”,

_ “(3 × 2) is identical to 6”,

_ “(3 × 2) = 6”, or

_ “(3 × 2) is 6”.

Therefore, whenever vocalizing mathematical expressions (identity statements or equations), it is preferable to use language which will not contribute to the student's confusion:

_ Vocalize the expression “1 + 1 = 2” as “one plus one is two”.

_ Do not vocalize “1 + 1 = 2” as “one plus one equals two”.

It is indeed unfortunate for the student who learns Mathematics as a second language, relying on English (his first language) as the metalanguage, that "is" is used to express three of the most important concepts that lie at the very foundation of his conceptual grasp of reality -- the concept of identity, the concept of equivalence, and the concept of class membership.
Chick Hatchers said…
I had never thought of the problem that you address here and after having taught each of my kids math, they don't have difficulty with the understanding of an equal sign. Interesting that some kids do...
Connie said…
If children have this misconception, it is the fault of the instructor. I have never, in all my years of tutoring and teaching, seen a child of any age exhibit such a misunderstanding of the equal sign.

I'd be interested in reading the research, not simply a blog about the research.
Maria Miller said…
Here's a link to Robert Capraro's homepage (one of the researchers). I couldn't find anything yet, but maybe he will eventually write about it or list his report once it's published in some publication. The University of Texas did release a news item about it (here's the link).
Anonymous said…
Maria,
You will make permanent, deadly enemies by calling Texas A&M the University of Texas! LOL
Anonymous said…
I have been a math tutor for many, many years, and encounter this all the time! So many students really don't understand what the = means. I have found it all depends on the teacher. It's so easy for me to tell which students have a teacher who is afraid of math )or who just totally doesn't understand math at all) versus a teacher who truly understands it.
B & K said…
When I first read your post (and the comments) I thought, 'How odd; I've never run across this with our kids.' Then I read the Texas A and M article, and my guess is that using calculators at young ages may be what is causing the problem. I've home educated all 6 of our kids and never let them use a calculator until they were in Saxon Advanced Math; and then, only the youngest 3 actually used one ~ well, the 6th one still isn't using one (she's not in Advanced Math yet); and won't be allowed to until she shows she can do the work without a calculator. If this is such a problem w/ kids today, an inexpensive fix would be to let children learn to do their math without a calculator! What do they think people did before the calculator was invented? Actually, my college roommate's older brother was using a slide rule for his calculus class in 1972 [am I showing my age here? ;-)]! One had to be able to understand the basic concepts in order to use a slide rule. Just my two cents worth! :-)
Rivka said…
Thanks for this post. I was recently surprised to see my child struggle with problems in this format - we worked with games to help her understand this type of construction.
Fonntri said…
H, there are many difficulties with the meanings of signs (and words) - not all on the child's side.

One 'simple' meaning for some uses of the = symbol is to read the statement as the value of the expression on the left hand side of the equals symbol evaluates to the same numerical vale as the expression on the right hand side. (NOT so easy but neither is equality). Fonntrí
Fonntri said…
Hi,
symbols (including words, of course) are tricky things for most people - possibly all.

The equals symbol is used for many purposes. In the examples shown, it just may be helpful to paraphrase (from time to time) as

the numerical value of the expression on the left hand side of the equal sign is the same as the numerical value of the expression on the right hand side of the equal sign.
(you have to evaluate the expressions : 3 is not the same thing as 1 + 2 - in 'ordinary' arithmetic they evaluate to the same value)
It is fraught with many difficulties in spite of its apparent simplicity.
Fonntrí.

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