kitchen table math, the sequel: equation

Tuesday, May 10, 2011


from Science Daily:
Texas A and M University researchers ... have found that not fully understanding the "equal sign" in a math problem could be a key to why U.S. students underperform their peers from other countries in math.

"About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students," note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M.


The problem is students memorize procedures without fully understanding the mathematics, he notes.

"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...


The Texas A&M researchers examined textbooks in China and the United States and found "Chinese textbooks provided the best examples for students and that even the best U.S. textbooks, those sponsored by the National Science Foundation, were lacking relational examples about the equal sign."

Students' Understanding of the Equal Sign Not Equal, Professor Says
August 11, 2010
They had me until that last line.

How do Everyday Math and Terc teach the equal sign?


Catherine Johnson said...

A friend of mine had an argument with a math teacher who didn't know the difference between and expression and an equation.

Hainish said...

I remember learning (in grade 6) that the equal sign is like the fulcrum of a balance, and that whatever was on one side had to work out to be the same as whatever was on the other. The analogy served me well all through algebra.

Hainish said...

The article concludes:

"The two researchers suggest using mathematics manipulatives and encourage teachers 'to read professional journals, become informed about the problem and modify their instruction.'"

Parker said...

I told my spouse the other day that I was convinced that many of my chemistry students did not understand what the equals sign meant and therefore they couldn't handle equations. It was just a hunch, but maybe I wasn't crazy. When they rearrange an equation they tend to just move letters around.

I don't know how we could screw up teaching this stuff.

Text Savvy said...

I wrote this up a while ago:

I bristle at the suggestion that this result is due to focusing too much on procedure. I suppose that may be part of it, but, quite simply, students are taught (directly and indirectly) in the U.S. that the equals sign means "answer comes next."

It's not really about procedure vs. concepts or hands-on vs. pencil work. And it enrages me every time I see folks bring every discussion back to the same ol' dichotomies.

Truth is that they are taught incorrectly.

Catherine Johnson said...

The analogy served me well all through algebra.

I saw that visual analogy for the first time as an adult, and I always wondered whether it worked for kids!

Michael Weiss said...

This is really, really, really not new. Kieran (1981) did a pretty good analysis of the phenomenon 30 years ago ( and dozens (maybe hundreds) of follow-up studies have documented the persistence of the phenomenon in middle school, high school, and college. A 2006 study by Knuth et al showed that students who predominantly regard the equals sign as an operator (rather than a relationship) do worse at learning basic algebra.

Take a look at Singapore Math, any grade, any page. How many times do you see the equals sign used to separate "question" from "answer", with the former on the left and the latter on the right? How many times do you see it used any other way?

Text Savvy said...

I should mention that--as far as I know (I'm too lazy to re-look up this study)--the sample size for the U.S. was ridiculously small (like less than 3 dozen). I could be wrong, but if I'm right, then one really shouldn't be waving this research around to show anything.

Also, I just want to plug this site:

I go there to ask for research papers all the time (if you don't know, you have to pay some kind of ridiculous price [like $31 a pop] to download a single research article).

The good thing about References Wanted is that you simply sign up and ask. And someone with an institutional subscription sends you the PDF paper you're looking for.

I've probably requested two dozen articles, and I've received 23. It's a great site.

Anne Dwyer said...

Here is a typical example I see of students not understanding the equal sign.

Take a simple expression like this:


Students will do this:


They know they need to subtract something....but they don't subtract from each side of the equation. No matter how many times or ways you teach this, the problem persists.

The other things they typically do is move the equals sign.

For example, take a simple equation like this:


Students will subtract 5 from both sides. But instead of getting this:

x-3=0, they will simply move the equals sign and write x= -3.

Once they have internalized these mistakes, there is not enough time in a one semester pre algebra class to break them of bad habits.

Text Savvy said...

This is not just about the equals sign. Students are taught that operations are left-to-right (cinematic) affairs.

So 6 + 9 is a little short movie of someone picking up 6 rocks and then later grabbing 9 more. Makes total sense. Repeat for 3 years.

Then they get to a + 9 = 15, and, consistent with how they've been taught, they haven't the slightest clue what the hell you're talking about, because a to them is like a section of the movie that's been cut, and now you're asking them what happened during that scene.

ChemProf said...

This all reminds me of a student of mine years ago. We were working through some algebra, and wound up with something like 8.5 = 2.7x. I asked her what she should do next, and she said it couldn't be solved. Finally figured out that she couldn't solve it because x was on the right side of the equals sign. I spent some time trying (and failing) to convince her she could solve it either way, then gave up and taught her that she can always turn around any equality. This was a college sophomore, too! Although often my sophomores are more math challenged than my freshmen, which is why they are taking my class in their second year.

Allison said...

There are two related concepts that students get wrong, and both contribute to their complete misunderstanding of equations, equalities, and expressions.

The first concept they misunderstand is that of a mathematical expression.
3 + 4 is a mathematical expression. It's a perfectly valid one. You DO NOT NEED TO DO SOMETHING TO IT. It's true, it exists, and it requires nothing of you. But the US books are notoriously awful in this regard, and teach their students that such an expression is basically incorrect or incomplete, and must be immediately evaluated down to a number, though usually they misuse the word "simplify" to indicate this. (evaluation is not the same as simplifying; evaluation is its own concept, and sometimes, leaving things as expressions until later in the computation is what is needed to simplify the problem.)

As such, students feel the need to perform operations rather than just holding the concept of an expression in their head.

Since they don't understand expressions, they can't possibly understand what an equation is if that equation contains expressions.

Then, they are mistaught the relationship between equalities and equations.

Here's Wu on expressions:
"Let x and y be two (real) numbers. Then the number obtained from x and y and a fixed collection of numbers by the use of the four operations +, ", !, ÷, together with nth roots
(for any positive integer n) and the usual rules of arithmetic, is called an expression in x and y. E
An expression in other symbols a, b, . . . , z is defined similarly"
Still with x and y as (real) numbers, we may wish to find out all such x and y for which two given expressions in x and y are equal. For example, are there x and y so that x^2 + y^2 + 3 = 0?
(No.) Another example: Are there x and y so that 3x - 7y = 4 ?
(Yes, infinitely many.)
In each case, the equality of the given expressions in x, y is called
an equation in x and y. To determine all the x and y that make
the expressions equal is called solving the equation."

I'll show disastrously bad examples from Saxon and Everyday Math in another comment, but overall, students are never taught what Wu says. I'll say what's great about Singapore's Primary Math in another comment too.

You can assume that when most students see an equation with an unknown they just think that means "compute the value", when unless that's specifically asked for, that's not what an equation is saying--it just simply is an equation.

Most students do not understand that the equation like "given the domain of the reals, x = 3 means the values of x over the reals that satisfy x being the same as 3". They think it means "x is assigned the value 3."

Wu, btw, dislikes the metaphor of the balance, because he thinks math by metaphor is not math. It misleads students into sloppy thinking. I understand where it comes from, but think by 6th grade, we'd be better off understand why it's legitimate from the axioms that we can compute with numbers in this way.

Allison said...

So, what does Singapore's Primary Math do right? They start in 1st grade teaching the notion of equality in a way that promotes accepting expressions without having to evaluate them.

They show a picture of 7 kids on a playground, and the picture lends itself to students seeing that in those 7 all of the addition and subtraction facts before those concepts are taught.
So, 7 kids on the playground, 6 are playing while 1 is sleeping; 2 are girls and 5 are boys; 3 are wearing hats and 4 are not. This encourages thinking about all of the addition and subtractions facts as a unit, which encourages thinking about equality--7 is the same as 3 and 4 and is the same as 2 and 5, etc. Also, you don't need to "solve" 2+5 to get 7, as you just hold in your head that 2+5 = 7 means 2 and 5 is the same as 7. Just as 2 is the same as 7 - 5. This also encourages thinking about expressions as something perfectly valued, so when you get to algebra, you are okay with expressions.

Now, I think Michael above was unhappy with Primary's math's use of equals signs, but I think the instructions are relevant. Here's an example from Singapore Math. This is the third problem set in "Unit 3: Fractions" in the 5th grade Singapore Math Primary Mathematics Workbook 5A, pp. 53-54:

Add. Give each answer in its simplest form.


(a) 7/8 + 3/4 = 7/8 + ?/8 =

(b) 2/3 + 4/9 = ?/9 + 4/9 =

(c) 4/5 + 3/10 =

(d) 3/4 + 7/12 =

(e) 5/6 + 2/3 =

(f) 1/2 + 9/ 10 =

Note that it *says* ADD, and simplify your answer.

This is much much much better than US texts. Here's what Saxon says in a comparable lesson about adding fractions with unlike denominators:
10. 1/4 + 1/8
11. 3/4 - 1/2
12. 7/8 - 3/4

It doesn't SAY ANYTHING AT ALL. It doesn't tell them to add. It doesn't tell them the above are expressions. It doesn't say evaluate or simplify the expressions. It just assumes that an expression will be "simplified" or evaluated. But that's a terrible assumption, and students should never be taught that 1/4 + 1/8 NEEDS to be evaluated while it's just sitting there. The above "problems" aren't problems at all.

Everyday Math is bad in other places, too. I'll get to that in another comment.

Anonymous said...

I just had our 5th grader--nearing the last few weeks of 5th grade--sitting next to me with her Everyday Math workbook. A problem asked her to write 5/8 as a decimal. She had no clue how to do it.

When I told her she had to divide 5 by 8, she still had no clue how to do it, because 5 was smaller than 8. At which point, more of my hair turned gray and I showed her the long division. She said they'd done that briefly once sometime this year. More gray hairs.

Oddly, I've been working with her on pre-algebra and she understands the equals sign pretty well.

Basic...or should I say everyday...math is still a problem!

Anonymous said...

I taught math to struggling college students for over a decade. To them an equals sign means, "and then...."

It is the symbol that you put at the beginning of the next line to indicate that you are still working on the problem.

For example, calculus students will write:

f(x) = x^2
= 2x

Anonymous said...

Two quick points:

(1) about the Everyday Math assignment to write 5/8 as a decimal: my guess is that the kids are supposed to know that 1/8 = 0.125 and then multiply by 5. Don't they tend to emphasize knowing certain fraction-decimal equivalences and ignoring the general case?

(2) when I am teaching my kids things about adding fractions and it comes to something like 2/8 + 6/7 I like to tell them to find a numerator and a denominator for this number; the point being that it is already a number and now we want the simplest represetation.

lgm said...

>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.

My 6th grade teacher would have this fixed in one week - using the traditional teaching method of giving feedback through grading homework. My sons' 6th grade teacher was one of those drawing a $100K salary with Tier 1 retirement...not one single homework or test paper did he give feedback on in their years. It's all about the long as the kid is passing, that's it. They have to get to the '4' on their own. But researchers can't say that...that'd cut off the money train.

Crimson Wife said...

I'm having trouble following this discussion (which is probably a testament to the poor math education I had growing up). What exactly is the problem with the "running equals signs"?

4 + 3 + 2 = (7) + 2 = 9 is how I would personally solve the problem. Is there something important about writing out 2 separate equations that I'm missing?

Bonnie said...

I was one of those kids who didn't get equality, or solving equations by adding/subtracting from both sides. I had a teacher who taught everything as a recipe, and I simply could not learn that way. I felt panicked and confused because I couldn't memorize the procedures without understanding why they worked. Finally, my father (a physicist) sat me down and started drawing pictures, and (gasp!) using words to explain all the concepts that I had not understood. Once I had the basic concepts down, and could explain them to myself in English, I was able to jump back in, and ended up with an A in algebra, and eventually a bachelors degree in math.

I think our schools spend way too little time making sure the students get the basics, really GET the basics and haven't just memorized something. We spend way too much time memorizing procedures. I see it all the time at my university. Most students come in having NO CONCEPT of the number system, which of course is a real problem when we need to teach binary and hexadecimal. I really need to go back and reteach the base-10 system, but there is no time in the semester.

When I teach intro to computer science, I spend a lot of time drawing pictures and making my students draw pictures, so they will understand what is happening. We do diagrams of values jumping into variables (remember, in CS, a variable is different from the mathematical concept), we do call stacks, we do diagrams of how things line up in memory, and we do lots of diagrams of number systems. I use the bundles of crayon diagrams, which I seem to recall I stole from my son's Singapore math book.

I know a lot of people here hate that word "ownership", but I think that is what is needed - kids need to "own" the concept of equality, and they need to "own" the concept of the base-10 number system - NOT memorize it, but KNOW it upside down, rightside up, and sideways. A kid should be able to explain it in English, draw any kind of diagram about it, and maybe even write a song about (OK, just joking) because he or she should just feel completely comfortable with the concept.

Anonymous said...

Crimson Wife, you are an A student. You look at 4+ 3 + 2 = () + 2 and realize that the () must be 7, and then 7+2 is 9.

The non-A student looks at 4+3+2= and says "I must now write down the answer in the next available space, because that is what the command "=" means" and he writes down 9 in the ().

SteveH said...

"As such, students feel the need to perform operations rather than just holding the concept of an expression in their head."

Simplify is the bane of all math students. Do I factor? Do I expand? Worse is the over-emphasis on order of operations as if one is writing an expression as a line of code. Math is not compiling and executing code. Math is not the linear, left-to-right expression on their calculator. I see expressions and equations as two-dimensional images, not linear text. Order of operations is important only on a local basis, not the entire equation. I can move things up and down and back and forth as long as I know the basic rules.

Back when I taught college math, I liked to take rational terms and move the factors up and down and around. I would take 'x' in the deonminator and move it to the numerator and change the exponent to -1. They thought that was illegal.

I remember writing things like




It's got to be illegal to put the constant as the last factor, isn't it?

I remember those problems much more than any with an equal sign. Many also had basic sign problems in adding, multiplying, and dividing. Some could not understand how

-(1/2) = (-1)/2 = (1/(-2))

But watch out for


you can't take that minus by the 2 and move it somewhere else. The rule isn't about moving things around. You have to know the real mathematical rules. I also would forbid my students to talk about cross-multiplying.

"... but think by 6th grade, we'd be better off understand why it's legitimate from the axioms that we can compute with numbers in this way."

I agree. It doesn't have to be very formal, but it needs to start early and be systematic. They didn't do this with the traditional math when I was growing up, and it's worse with curricula like Everyday Math. Nobody wants to get too formal too soon, but I don't see when the job gets done.

I distinctly remember that it took me until Algebra II before I got it all figured out. Before then, I could approach some problems two different ways and get two different results. I don't think that anyone taught me the basic rules (and all of their variations and implications) in any systematic fashion. I knew them, of course, but I wasn't forced to use them.

For example, everyone knows that

A/1 = A

but I would always emphasize that many times, we want to do this:

A = A/1

(You can see an equal sign as a substitution.)

Likewise for

x = x^1

Students know these rules, but they don't really know how to use them when the equations get complex.

They might learn the rule that

x^m/x^n = x^(m-n)

but they get confused when they see


Then they get nervous if you say that this is the same as

(x) x^(-4)

Simplify mistakenly tells students that there is an official direction in which you always have to go. I've had to emphasize to my son many times that some form is standard practice and is not required.

"...have found that not fully understanding the 'equal sign' in a math problem could be a key to why U.S. students underperform their peers from other countries in math."

The "key"? No, there are lots of basic keys related to mathematical rigor and practice. If you gloss over the rigor early, you will pay for it later. In Everyday Math, I saw simplified concepts, but not mathematical rigor. The simplified concepts become problems later on.

Allison said...

Steve, you're right about the way they learn "the rules".

The most annoying one for kids in pre-alg or lower is how they teach the distributive property. This is true in Everyday Math, Saxon, and even the "discovering math" series from Singapore.

They teach that the distributive property is
a*(b+c) = a*b + a*c. I think Everyday Math teaches this as an aside, Saxon makes it explicit, but only the Discovering one actually names it the distributive property.

But NONE of the above books teach
a*b + a*c = a(b+c) IS ALSO using the distributive property.

They invoke the use of it in their various two column proos and things in examples without CALLING IT THE DISTRIBUTIVE property.

So the books specifically treat the "left" side as a starting, and the "right" side as the ending, and you have to use it that way or it's not the distributive property. So the kids never see why equality is equality, since even the books they use never admit it. It's just another example of how you keep "computing" from left to right, without thinking.

Text Savvy said...

The example Crimson Wife uses isn't strictly incorrect. But students will often write out operations the way they think about them. You can easily imagine saying to yourself, "Ten plus eleven is twenty-one, plus three is twenty-four, and minus eight is sixteen."

Using the symbols, this might translate to 10 + 11 = 21 + 3 = 24 - 8 = 16.

And of course this is not correct. 10 + 11 does not equal 21 + 3, and neither of those expressions is equal to 24 - 8.

This is the thorn when it comes to the running equals sign. And it's why Liping Ma tells her teachers to NEVER write two equals signs on the same line.

Anonymous said...


I'm sorry, but they don't spend any time memorizing procedures any more than they spend learning the concepts. They leave that to the parents and Kumon. You're thinking of another era.

They also spend little to no time on fractions/decimals/percentages, and practically none on division by 5th grade. That is, if we're talking about the reform curriculums.


Bonnie said...

My 5th grade son has done a lot of fractions and division during his time in school. Our district uses Think!Math - I don't know if that qualifies as "reform" or not. The teachers augment it with drill n' kill worksheets, way too many IMHO.

Before our district adopted that curriculum, teachers in the lower grades were free to do what they wanted, and that is where I saw the worst problems.

Crimson Wife said...

Text Savvy- thanks for the example! That really makes it clear to me now why writing separate equations is superior to the "running equals signs". I'll have to remember that when I'm teaching my DD in our homeschool.

Glen said...

This confusion is one of the reasons I taught my kids "two plus two IS four" instead of "two plus two EQUALS four" when they memorized math facts. Of course, it's true that two plus two equals four, but the meaning of that phrase is not what I wanted.

"Two plus two is four" means that the value of the expression (2+2) is 4. One expression and its value.

"Two plus two equals four" means that the value of the expression (2+2) is the same as the value of the expression (4). Two expressions whose values are the same.

Both of these claims are correct, but they are two different claims, and keeping them separate in the early years makes it easier for me to explain what equals means:

"2+2 = 3+1. Two plus two is how much?"


"And three plus one is how much?"


"They're both the same, aren't they? That's what the equals sign means: these two expressions both have the same value."

With this approach, they will say, "10 + 11 is 21, plus 3 is 24, minus 8 is 16." They can't write that down as a messed up equation, because it doesn't contain any equals signs, and it doesn't contain any equals signs because it is the evaluation of a single expression, not a comparison of multiple expressions.

I think that making an effort to keep these concepts separate makes it easier to extend them later, to such things as the important semantic differences between "2/2 is 1", "2/x = 1", "x/x ≡ 1", and "f(x) = 1". (Including why you must add "x not equal to zero" to the third but mustn't add it to the others.)

SteveH said...

I think there are lots of places for misunderstandings and inflexible knowledge.

If you have 2x = 5, do students know that this is the same as 5 = 2x? Is 10 = 4x the same equation? How about 10 = 2x + 5? Is 2 = 2 legal? Is it the same as 3.715 = 3.715? In that sense the seesaw analogy is best, and most curricula use that analogy, but students still have problems. So, is the educational problem the curriculum, the teachers knowledge, or the fact that schools don't take responsibility to make sure that even the basics are mastered? Although all are issues, I put the biggest blame on the third for K-6.

I've talked in the past how my son's fifth grade teacher found that many kids hadn't mastered basics like the times table or quickly adding something like 7+8. These things fall well within the balance and mastery discussion we had at a parent/teacher conference about Everyday Math. However, the fifth grade teacher didn't raise a stink with the school and with the earlier grade teachers. She just took extra time in class and offered after-school sessions to fix their problems. Blame the students. Blame engagement or motivation or parents or society. How simple does the task have to be to accept some level of responsibility? The fifth grade teacher took responsibility, but made no attempt to fix the systemic problem. Schools can't just point to the worst students or to the worst effects of poverty and lump all other students in the same category. I can't tell you how many times I've heard the excuse that public schools have to teach "all kids", as if they all have to be in the same classroom. Full-inclusion, student centered discovery with teachers as guides-on-the-side is really a pedagogical way to reduce school expectations. They just have to go through the motions.

I think that talking about issues of deep understanding lets schools off the hook. Schools use this talk to avoid explaining why kids can't tie their shoes. I don't think there are many mysterious things about equality that a little bit of hard work won't fix. Everyday Math talks about many of these things, but the curriculum jumps around so much that it makes poorly prepared teachers worse. Besides, they expect students to take responsibility for their own learning. Everyday Math institutionalizes not ensuring anything. "Trust the spiral." Schools see perfectly capable kids who don't know the times table in fifth grade and they don't look in the mirror. They don't have to ... by definition.