Make 10? Make 11? Let's Call the Whole Thing Off

Can you decipher this first grader’s math assignment? BoingBoing’s Mark Frauenfelder wants a second opinion. [%comments]


seems sensible to me - assuming it was explained to them in class first. you have to make 10 and then see what else you've got. that's how you do addition isn't it?


Ahh, Investigations... how do I love thee - let me count the ways... let me try to remember a "counting strategy"...

Adam S

There were several good explanations over there. My wife is a 3rd grade teacher so I have some background knowledge. But the summary as I understand it is that they students are being taught number sense. The teachers want them to understand both the base ten system and place value as well as ways to rearrange numbers to make them easier to add together. There some pretty clear research that shows this works well when teachers teach it properly.



Just wait till they teach subtraction without borrowing and carrying over but adding and carrying to. And then subtraction with a list of eight numbers in a column.

Do you people really not get this? This is how i used to beat the boys in arithmetic races at the board in fifth grade.
(1970 or so). When they taught it to my kids in 1990 or so), i was a little confused at first because i didn't understand why they were teaching steps that you do in your head automatically.

think more numbers, think columns. think likely hood for error.

step by step.


Now we know why most Americans are economically illiterate, or innumerate.

The graduate degree in Education is the worst thing that ever happened to American education. (Hmmm....sounds like a topic begging for a chapter in HyperFreakonomics or MegaFreakonomics, whichever comes next.)

Paul Clapham

Usually if I'm given a set of instructions I manage to find an interpretation of those instructions which -- being the wrong interpretation -- causes some kind of chaos. But not with that example. It's pretty clear they are trying to teach children how to do addition with carrying.


I figured it out, but only because
(1) I took a college course in symbolic logic and I know that an arrow symbol translates roughly into English as "if...then...", and
(2) I assume the pedagogical purpose of the exercise is to condition children to think in base-10.

Thus they are expected to convert whatever two-digit sum they reach on the left (be it 11, 12, etc.) into an equation expressed as "10 + x." The drawings on the right should be completed to reflect the "10 + x" version of the equation.

Neil (SM)

I couldn't figure it out at first just by looking at it, but after reading one of the explanations over at the other site it made sense.

I would presume this was all explained in detail to the class during the day. Really all that's missing is a good set of written instructions or an example for when the kid gets home, forgets how to do it, and asks mom and dad for help.


No offense intended, but some teachers probably do not understand this well enough to teach it. All the research in the world can't help teachers teach. Since this is research driven, where is the research indicating which teachers actually teach this well and which do not teach this well?


That font looks suspiciously like my daughter's math book...we have had a couple long evenings of looking at worksheets coming home and thinking...WTF?


I realized its like teaching math using an abacus. Although not sure if its illustrated in an easy to understand manner


My son is in 2nd grade, and his math assignments look just like this. Last week, because of a very poorly written question, there were 2 right answers. I suggested that my kid mark down both answers and then explain to the teacher why they were both correct. He said that since one was clearly "more correct" than the other (also correct) answer, he would pick that one. ARGH!


This is what is confusing with most (math) teaching.

They tell you in younger grades when subtracting a larger number from a smaller number (make a deficit)

"you can't do that!"

Instead a person might say, "You probably aren't ready to do that kind of math. Wait until a few more grades. Let's just sticker with smaller numbers taken away from bigger numbers." Of course depending on the personality of the student, they might still say "NO!"

It depends on how you handle the inevitable time down the road a few years, when they say,

"You know when we said YOU CANNOT take big numbers away from small numbers? We lied. You can."

"You know when we said a coffee cup cannot be a doughnut? We lied. It can."

"You know when we said that insanity is doing the same thing over and over again? It is not. It is genius. We lied."


As a parent teaching my five-year-old son math, I get it.

He wouldn't.

There should have been instructions on it that says something along the lines of:

1) First, fill in the missing number/answer.

2) Then find out what number must be added to 10 to get the same answer.

3) Then fill in the squares to get the same answer.

Or some such.

My son gets homework like this on occasion. I'm tempted to send it back with all sorts of crazy comments, but then the adult takes over and I just use common sense.

Again, an adult can likely figure this out. A child would likely not.


"The graduate degree in Education is the worst thing that ever happened to American education."

And how, exactly, is this true?


I am familiar with this method of teaching. The idea behind it is to teach kids conceptual understanding of how numbers are constructed, rather than simply performing rote memorization of arithmetic facts. If this lesson and homework were used as intended, the child should already be familiar with the process and have performed this exact exercise in class, under the guidance and facilitation of the teacher. There are lots of mathematical concepts being presented here. Students who engage this type of work are likely to have a deeper conceptual understanding of number sense and addition and will be better able to problem-solve using arithmetic skills.

Is the sheet confusing to adults? Probably. Is it intended for adults? No. Again, if used properly, the sheet is intended to reinforce skills and concepts worked on in school, and should be something the child is familiar with.


Maybe I'm biased because I have to change number bases constantly in my day job (computers work in base-2 & base-16), but I saw the concept immediately and I'm confident I could explain to the little girl, using pennies, what she was supposed to do.

I actually like this assignment, because it's trying to teach through concept instead of solely by rote.


I worked through a book on how to use a Japanese style abacus, the soroban, and it teaches how to solve problems the way I would do math like this normally. You look at the given numbers and perform a memorized procedure to achieve the answer. Practice that enough and you can do addition and subtraction really fast.

No worrying about this weird grouping procedure, interesting though it may be, just accurate addition and subtraction done quickly. Which do you want?

Vincent Clement

@ AaronS

Don't be so sure that an adult can figure it out. I excelled in math in high school, but some of the math homework my kids bring home makes me want to pull my hair out.


If you follow the directions as stated, it makes sense, and is not too hard for a 6-year-old or an adult.

FIRST "Draw counters to solve." Finish the top group of ten, and then "spill over" into the bottom group of ten. In the first problem, you would draw in two counters to finish up the top group of ten, and then you draw one more counter in the bottom group of ten. That makes three counters you have drawn, and a solution of eleven (ten plus one).

THEN, "Write the missing numbers." Once you have drawn the counters, filling in the numbers is easy to understand. Just look at the picture and see that 8 + 3 = 11. And that is the same as 10 + 1 = 11.

The pictures and drawing act as manipulatives, or "concrete" objects to show what the numbers mean. It's a good thing for kids to learn how numbers can be broken apart (e.g. three is the same as two and one), and recombined (e.g. eight and two make ten). "Making ten" is a useful skill even for adults in everyday life - for quickly making change, for example.

Learning to think mathematically like this is important for kids to really understand math later, rather than just parroting back formulas without any understanding of how or why they work.