It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems.
But I am tired. Wiped out and exhausted.
So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?)
This plan worked beautifully for about five minutes.
She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook.
Me: What goes in the box?
Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is…
Hey! This doesn’t make any sense!
Me: Why not?
T: 8 plus 4 is something, then plus 5?
Me: What does the equal sign mean?
T: Is. Like 2 plus 2 is 4.
Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1?
T: No!
I let it go and we move on with our evening.
Later on, though, after putting on jammies but before toothbrushing, I follow up.
Me: Tabitha, I want to ask you a follow up math question.
T: OK.
Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1?
T: Yes! Of course! Easy!
Me: Can I let you in on a little secret?
T: A secret secret? Or not really a secret?
Me: Not really a secret. But something you might not know.
T: [rolls eyes] OK.
Me: The equal sign means “is the same as”.
T: Of course! I know that!
Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1.
T: Oh.
So what do we learn?
This is kind of a big deal.
We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!)
But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as.
Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right?
Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do.
The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right.
You can follow Tabitha’s and my adventures in equality in the coming weeks.
Starting the conversation
If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun!
Postscript
Coincidentally, a fourth-grade teacher wrote up his class’s explorations in equality today. If you’re interested in what this can look like in school (easily adaptable for homeschool), head on over.
I’ll try this with the little man, but equivalence statements are even hard (for a while) for my 8th graders. Once they get the idea however, they really roll with it.
Computer people call this “overloading”.
Here’s another one, for later : f(x) = x + sin(x)
This is not even a statement about equality, it is the definition of the function f
Have a look at my diatribe on numbers, at http://www.mathcomesalive.com\savingschoolmath.doc
It’s only 12 pages long!
Again!
You have to put all of http://www.mathcomesalive.com\savingschoolmath.doc
in the browser. WordPress just takes the site name
More:
Strictly speaking, in the statement 8 + 4 = box + 5 the = does not mean “the same as”, but “there is a number which when put in the box (in place of the box) makes the results of the calculations the same on each side”.
This might sound pedantic, but what does a 5 or 6 year old understand by “the same as” ?
Ok so I changed your problem because 8 plus 4 is one that takes my first grader a minute to think about. He can get the right answer but I wanted to use a problem that would roll off his tongue. So I picked
4 + 4 = __ + 7. He looked at it a minute and I asked him;
“What would you say goes in the box?”
“7 ,”he answered.
” Hmmm, why did you pick that? ”
” Because it’s a double just like 4+4.”
“Oh I can see why you might think that. Do you remember what we learned the equals sign means?”
“The same”
“Good. The same as. So if we want to make the sides the same as each other, what number do we need?”
“Eight.”
“That would give us fifteen over here and that does not equal 8 does it.”
“No.”
“So what do you think we need? to make both sides say 8”
A minute or two of thinking.
“Oh one!”
Not bad since we had just finished an hour math lesson and he was on break:)
Great post! I loved it!
Jennifer
One of the joys of having a reverse-Polish notation (RPN) calculator is that there is no equal sign. Typing “+” does an operation on the top two elements of the stack. HP has been making RPN calculators since 1972 (the first scientific pocket calculator, the HP-35 used RPN). I still have my HP-45 but both the battery pack and the charger died, so I had to get a new calculator (I now use an HP-32SII, which was sold from 1988-1991 for about $70).
I have your problem verbatim to my first grader and he wrote the seven without hesitation. I said “why is that the answer?” and he said “it’s like taking one away from the eight and giving it to the four.” Take that, “Jack’s dad, the engineer”! (Although I can’t really say what it means about his understanding of the equal sign…)
The working with two things independently and holding them in working memory (and knowing to do that) is a big deal for lots of my ‘remedial’ students.
Just tried the 8 + 4 = _ + 5 problem with my 8yo daughter (in UK Primary School Year 4).
“What goes in the box?”
“Huh?”
“What’s the missing number?”
“12.”
“No, not 12.”
“17.”
“What does the equals sign mean?”
“It means write the answer.”
We haven’t solved the problem yet, I’m leaving her to think – but you’ve certainly hit on an issue!
http://msan.wceruw.org/documents/researchBrief/SERP%20Misconceptions-Worked%20Examples%20Research%20Brief.pdf lots of middle schoolers believe the equals sign is “where you put the answer” too.
I absolutely loved this post! I recall being introduced to algebra in grade 9, where I suddenly went from straight A’s to failing every test. It just didn’t make sense to me! How could an answer have a plus sign in it?!! Finally, after half a semester, my dad saw me banging my head on the dining table and came over and said: “It’s solving for a formula, that you can then use for other things”. Woah! It was like a lightbulb went off for me, and I suddenly “got it”. I only wish it had’ve been explained to me earlier… and by my teacher.
Your explanation is an even better one, and it’s shocking to me that so many children go through years of schooling without understanding that “equal” (and the equal sign) means “the same as”.
I have a son in kindergarten who loves math, and I am so glad to have read this now.