kitchen table math, the sequel: Parker on the equal sign

Tuesday, May 10, 2011

Parker on the equal sign

On the subject of American students not knowing what the equal sign signifies, Parker writes:
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.
I'm curious about this -- what happens in a chemistry class when students don't understand the equal sign?


TerriW said...

What happens in a chemistry class when kids don't understand an equals sign?


Parker said...

Lots of practice and instruction on how to manipulate literal equations; often as simple as D=m/V.

Many are convinced X is special; therefore it becomes easier for a student to solve 1.2=x/2 than 1.2=m/2.

Of course, most of the students are in Algebra II and passing Algebra II somehow.

I do teach in a low income school therefore the students get a double wammy of screwing up due to the current math curriculum in the elementary schools. It is so ingrained by the time they get to high school it takes forever to deprogram them.

ChemProf said...

The other one I see a lot is students who can solve basic algebra with whole numbers (e.g. 3x = x - 2) but not with "ugly" numbers that come up a lot in chem (e.g. 2.88 x = 1.75 x - 2.33).

And yes, convincing them that x isn't special is sometimes a challenge!

Allison said...

But x is special. It's an undefined symbol. Why in the world do you get to manipulate it the way you would numbers? You can't add grams and liters or kilometers and kilograms; why can you add x and numbers?

I believe no one ever told these students that we assume x is well defined over some domain, that it is a symbol representing some type of number, and with that assumption, we manipulate it as a number. We might have told them "it was a variable" but of course, that's nuts, since it's not varying, is it?

Andy said...

I ran into a freshman at my (elite small private) college who spent several weeks confused in Chemistry because he honestly didn't know that "reactants = products" was equivalent to "reactants - products = 0".

Definitely a case of inflexible knowledge.

Bonnie said...

In programming, it means something totally different, and I spend a lot of time explaining the concept to the students. Many of them never get it. In the context of most programming languages, it is an operator, which is unidirectional, and causes an action (the right hand side is evaluated and loaded into the space designated by the lefthand side). Confusing, huh? I tell the students to always call it "assignment" rather than "equals" when they are in my class, to drive home the distinction.

Anonymous said...

Bonnie, Kernighan has a lot to answer for, making "=" the assignment operator in C. Several other programming languages of about the vintage used different symbols that were less likely to cause confusion, but he chose "=" because it was faster to type.

Bonnie said...

A lot of the dicier "features" of C came about because they were "faster to type".
But, we have to live with it now.

Allison said...

While Kernighan was sloppy, these kids aren't coming in having not learned assignment vs equality from programming. If they had a proper understanding of equality, you could explain the bad notation in one sentence without confusion. Their problem is they think elementary school and high school equality is really an assignment in the first place.

Bonnie said...

I sort of agree with Allison,because when we move onto true equality (in Java and C++, it is represented by ==), they don't get that either.

However, I don't think they get assignment either.

Allison said...


They don't get expressions, they don't get equality, they don't get assignment.

They don't understand that an expression is valid and need not be evaluated, they don't understand when "variables" are bound, and they don't understand why you do function evaluation (since they don't really know that a function can exist without being evaluated.) They don't understand assignment because they don't understand how this token stands for something in the first place--since they don't really understand "variables". They don't know what x means, they don't know they need it defined over some domain, and they don't understand what f(x) is, because they don't understand x or f.

Anonymous said...

kids... and grownups... "they"...
the mathphobe majority... don't "get"
that the kind of care-about-language
used routinely in maths is possible *at all*.
they figure the symbolism is just there
to confuse them. *all* of it.

clear language is worse than useless
in politics after all and most of life
is a heck of a lot more about politics
than about seeking clarity.

even if "they" admit math works...
that near-perfect precision is possible
and even useful in some contexts...
they sure don't believe it's possible
for *them*. nor do they *want* to.

and having *made* this decision,
the self-fulfilling prophecy kicks in.
and they never do "get it".

my take on the equality meaning
of the equal sign
was posted in
summer '09... near the height of my
highly-discursive "ramble" period.
i've put a link in its comments
back to here.


Text Savvy said...

Ah, man. I have yet to read Vlorb's posts on this topic.

I would like to say that Liping Ma didn't improve U.S. education by introducing new teaching methods; she taught us math.

SteveH said...

I tried to track down how the equal sign got to be the center of attention. I couldn't. I could probably show a correlation between test scores and understanding that a fraction means divide. In my experience, the equal sign was just one of many misunderstandings. You can blame it on inflexible or rote knowledge. This can happen no matter which teaching approach you take.

The simple Everyday Math understanding of fractions fails when it comes to manipulating rational expressions. I struggled making the transition with traditional math when I was growing up, although I didn't need help from home or from a tutor. Curricula like EM worry about very conceptual understandings, but fail at the abstract level.

These understanding problems can come at all levels. If the problem is at a basic (especially abstract) level, it makes it difficult to progress to a higher level of understanding. Much of this can be fixed by the curriculum and ensuring mastery at each level, and the rest can be fixed by prepared teachers and hard work.

I keep thinking of a programming analogy. Let's say someone writes a whole program, manages to compile it, and then runs it. When it crashes, the programmer might try to fix it with guess and check, as if the error reflects just one problem. When many students get to high school math, they are loaded with errors, many of which interact. Maybe a tutor can find and solve each individual problem, but it may be better to start over and rewrite the program. Unfortunately, that won't work with students.

Of course, the best way to write a program is to build it up, piece-by-piece, testing and ensuring correctness at each step. You isolate and encapsulate parts rather than use a whole or thematic approach. The programmer, like a curriculum, ensures that all the parts match up.

Maybe that's not such a good analogy, but I don't understand why so many try to use guess and check to find a solution to middle school or high school students who have many levels and interactions of math issues. The search for one solution implies that they think they are close. If they fix one problem, the student will "crash" at another location. At some point, you just can't debug something correct. You have to go back and look at the code, line-by-line and rethink the whole process. Many of my old CS students could never fix their programs because there were too many errors and they refused to deal with fundamental design issues. They just kept debugging and looking for the one thing that will fix it all.

ChrisA said...

= sign is important. That's why you can do the same operation on both sides of the equation and it remains "equal".

Then there's the concept that to get the equation in a form that's easy to solve you might want to perform a particular action with a particular value! You might even have to "rinse and repeat" a few times.

Oh well.

FedUpMom said...

I wrote a blog post about the = sign here:

What is =?

SiouxGeonz said...

I got rather lost in the rambling post about the equals sign, but I think I've seen the same thing here at the community college.
Many of my students do see equations not as something to be read and interpreted, but as groups of images to be manipulated, and there are assorted rules for that manipulation based on looking like the examples. Then the nasty evil book or teacher changes how it looks (e.g., puts harder numbers in) and... they're lost.
The "equals sign" confusion confounded me at first, since it's so critical. has an interesting discussion about it from a numeracy email group I'm on.
Since I've been incorporating that "part to whole" idea into my tutoring, students are much less likely to add when confronted with x + 8 = 9.

Anonymous said...

This past year, I have started to tutor in math, chemistry, physics and biology for college students. The first discussion in all of my sessions is about the meaning of "=". For a few of my students, I am convinced that "=" means "the answer goes here".

Unlearning is the most difficult thing to do, especially when important concepts are learned incorrectly. I found this out teaching an advanced course in molecular genetics. Most of the students had genetics in the intro, microbiology and genetics courses. These students had a very difficult time since their misconceptions got in the way of acquiring any useful new knowledge.

At first, I was surprised to find that the chemistry and physics majors that took the class were the best performers. They had only been exposed to some of the very basics in the intro prerequisite course, and thus could learn things correctly from the beginning. I should not have been surprised at this, since I transitioned from a math major and chem minor to graduate biochemistry, biophysics and microbiology with only one undergraduate biology course (I almost flunked it).

My solution to teaching the course was to get away from the normal lecture of describing how things worked. I found that the students were very adept at memorizing the material, but their ingrained misconceptions prevented them from doing any simple reasoning. Luckily, I found a book that introduced the major concepts through the presentation of experiments that led to their development. I used PowerPoint to project the data, but the class was built upon the question, "What is this data telling you?"

The students were initially scared, but I just presented them with the face of confidence that they could do the work. I never said it was hard or easy, I just smiled. After about 6-8 weeks, I could see eyes pop open when I talked. "He is making sense." I had always made sense, but to learn something takes time. Since I approached everything from ground zero, their misconceptions never interfered with the learning process.

I have no idea how to approach this with math. I tried a tutoring a few high school students, but I had to stop since I found out that I had really no idea what they were trying to learn. It seemed to be all method -- there was no play at all in it. At least with college students, calculus is still calculus, chemistry is still chemistry and physics is still physics.

For my students, I emphasize the translation of symbols to what they represent, and conversely, taking real world examples and translating them back to symbols. Teaching them how to think for themselves is more like teaching someone how to dance. In the beginning, I think out loud while I am solving a problem. I am not so much demonstrating the algorithm to the solution but rather the approach to solving it. And just like dancing, the students start to get the hang of it.

I have to admit there was one conceptual stumbling block for me: electrons flow one way and current flows the other way. I would always screw up with a +/- somewhere and really screw thing up. Batteries in Pchem nearly killed me.


PS Great blog & good luck with your learning adventure.

allison said...

--My solution to teaching the course was to get away from the normal lecture of describing how things worked. I found that the students were very adept at memorizing the material, but their ingrained misconceptions prevented them from doing any simple reasoning.

Yes, this is the crux, the absolute crux! We cannot simply tell students "no, that's wrong; here's the right way to do X/the right equation of motion for the system/the right model." You must find out WHY they had the wrong idea in the first place, get to the bottom of it, and correct it there. Otherwise, the new idea will never stick, but be lost when the foundation of sand shifts underneath them.

And the only thing that can fix misconceptions is working out for themselves what's correct--so they have to be procedurally solid, but they have to be asked to interact with their data and results as you said--"what is the data telling us". Asking that paralyzes most students, from K all the way through undergrad. They have to get to where they aren't paralyzed to be asking themselves that question.

(sidenote: In physics, Prof. Hestenes has spent 2 decades trying to fix the misconceptions college students have in classical mechanics, and wa able to show that even at the high school level, students and their teachers have such grave misconceptions that all of the ability to compute an answer in the world isn't telling them anything about how the world actually works--they can't reason at all about classical mechanics. His work quantifying the misconceptions is detailed in the Force Concept Inventory (I wrote several blog posts on this subject a couple years back), and he's tried to create a high school teaching program to correct the misconceptions hs physics teachers have. )

Bill, the reason you can't figure out how to do this in high school math is that no one is asking students in K-12 to DO ANY REASONING at all. They've basically been given procedures and open ended problems, but they have been taught for at least all of k-8 that math defies reason, because it was taught without reasoning to them.

To fix k-12 math, we need to articulate and quantify the misconceptions held by the teachers---because we can't fix how the students don't understand until we fix what their teachers don't understand. The real question there is: are their misconceptions rigid (as young chemists misunderstand the bohr model, say), or are their misconceptions fluid and amorphous, because they never solidified any idea of what math was trying to tell them?